the eye carlo... · 2020. 1. 17. · quantum gravity quantum gravity is perhaps the most important...

Quantum Gravity

Quantum gravity is perhaps the most important open problem in fundamental physicsIt is the problem of merging quantum mechanics and general relativity the two greatconceptual revolutions in the physics of the twentieth century

This book discusses the many aspects of the problem and presents technical andconceptual advances towards a background-independent quantum theory of gravityobtained in the last two decades The first part of the book is an exploration on howto re-think basic physics from scratch in the light of the general-relativistic conceptualrevolution The second part is a detailed introduction to loop quantum gravity and thespinfoam formalism It provides an overview of the current state of the field includingresults on area and volume spectra dynamics extension of the theory to matter appli-cations to early cosmology and black-hole physics and the perspectives for computingscattering amplitudes The book is completed by a historical appendix which overviewsthe evolution of the research in quantum gravity from the 1930s to the present day

Carlo Rovel l i was born in Verona Italy in 1956 and obtained his PhD inPhysics in Padua in 1986 In 1996 he was awarded the Xanthopoulos InternationalPrize for the development of the loop approach to quantum gravity and for researchon the foundation of the physics of space and time Over the years he has taught andworked in the University of Pittsburgh Universite de la Mediterranee Marseille andUniversita La Sapienza Rome Professor Rovellirsquos main research interests lie in generalrelativity gravitational physics and the philosophy of space and time He has had over100 publications in international journals in physics and has written contributions formajor encyclopedias He is senior member of the Institut Universitaire de France

CAMBRIDGE MONOGRAPHS ONMATHEMATICAL PHYSICS

General editors P V Landshoff D R Nelson S Weinberg

S J Aarseth Gravitational N-Body Simulations

J Ambjorn B Durhuus and T Jonsson Quantum Geometry A Statistical Field Theory

Approach

A M Anile Relativistic Fluids and Magneto-Fluids

J A de Azcarrage and J M Izquierdo Lie Groups Lie Algebras Cohomology and Some

Applications in Physicsdagger

O Babelon D Bernard and M Talon Introduction to Classical Integrable Systems

V Belinkski and E Verdaguer Gravitational Solitons

J Bernstein Kinetic Theory in the Expanding Universe

G F Bertsch and R A Broglia Oscillations in Finite Quantum Systems

N D Birrell and PCW Davies Quantum Fields in Curved spacedagger

M Burgess Classical Covariant Fields

S Carlip Quantum Gravity in 2+1 Dimensions

J C Collins Renormalizationdagger

M Creutz Quarks Gluons and Latticesdagger

P D DrsquoEath Supersymmetric Quantum Cosmology

F de Felice and C J S Clarke Relativity on Curved Manifoldsdagger

B S DeWitt Supermanifolds 2nd editiondagger

P G O Freund Introduction to Supersymmetrydagger

J Fuchs Affine Lie Algebras and Quantum Groupsdagger

J Fuchs and C Schweigert Symmetries Lie Algebras and Representations A Graduate Course

for Physicistsdagger

Y Fujii and K Maeda The ScalarndashTensor Theory of Gravitation

A S Galperin E A Ivanov V I Orievetsky and E S Sokatchev Harmonic Superspace

R Gambini and J Pullin Loops Knots Gauge Theories and Quantum Gravitydagger

M Gockeler and T Schucker Differential Geometry Gauge Theories and Gravitydagger

C Gomez M Ruiz Altaba and G Sierra Quantum Groups in Two-dimensional Physics

M B Green J H Schwarz and E Witten Superstring Theory volume 1 Introductiondagger

M B Green J H Schwarz and E Witten Superstring Theory volume 2 Loop Amplitudes

Anomalies and Phenomenologydagger

V N Gribov The Theory of Complex Angular Momenta

S W Hawking and G F R Ellis The Large-Scale Structure of Space-Timedagger

F Iachello and A Arima The Interacting Boson Model

F Iachello and P van Isacker The Interacting BosonndashFermion Model

C Itzykson and J-M Drouffe Statistical Field Theory volume 1 From Brownian Motion to

Renormalization and Lattice Gauge Theorydagger

C Itzykson and J-M Drouffe Statistical Field Theory volume 2 Strong Coupling Monte

Carlo Methods Conformal Field Theory and Random Systemsdagger

C Johnson D-Branes

J I Kapusta Finite-Temperature Field Theorydagger

V E Korepin A G Izergin and N M Boguliubov The Quantum Inverse Scattering Method

and Correlation Functionsdagger

M Le Bellac Thermal Field Theorydagger

Y Makeenko Methods of Contemporary Gauge Theory

N Manton and P Sutcliffe Topological Solitons

N H March Liquid Metals Concepts and Theory

I M Montvay and G Munster Quantum Fields on a Latticedagger

L Orsquo Raifeartaigh Group Structure of Gauge Theoriesdagger

T Ortın Gravity and Strings

A Ozorio de Almeida Hamiltonian Systems Chaos and Quantizationdagger

R Penrose and W Rindler Spinors and Space-Time volume 1 Two-Spinor Calculus and

Relativistic Fieldsdagger

R Penrose and W Rindler Spinors and Space-Time volume 2 Spinor and Twistor Methods in

Space-Time Geometrydagger

S Pokorski Gauge Field Theories 2nd edition

J Polchinski String Theory volume 1 An Introduction to the Bosonic String

J Polchinski String Theory volume 2 Superstring Theory and Beyond

V N Popov Functional Integrals and Collective Excitationsdagger

R J Rivers Path Integral Methods in Quantum Field Theorydagger

R G Roberts The Structure of the Protondagger

C Rovelli Quantum Gravity

W C Saslaw Gravitational Physics of Stellar and Galactic Systemsdagger

H Stephani D Kramer M A H MacCallum C Hoenselaers and E Herlt Exact Solutions

of Einsteinrsquos Field Equations 2nd edition

J M Stewart Advanced General Relativitydagger

A Vilenkin and E P S Shellard Cosmic Strings and Other Topological Defectsdagger

R S Ward and R O Wells Jr Twistor Geometry and Field Theoriesdagger

J R Wilson and G J Mathews Relativistic Numerical Hydrodynamics

daggerIssued as a paperback

Quantum Gravity

CARLO ROVELLICentre de Physique Theorique de LuminyUniversite de la Mediterranee Marseille

cambridge university press Cambridge New York Melbourne Madrid Cape Town

Singapore Satildeo Paulo Delhi Tokyo Mexico City

Cambridge University Press The Edinburgh Building Cambridge CB2 8RU UK

Published in the United States of America by Cambridge University Press New York

wwwcambridgeorg Information on this title wwwcambridgeorg9780521715966

copy Cambridge University Press 2004

This publication is in copyright Subject to statutory exception and to the provisions of relevant collective licensing agreements no reproduction of any part may take place without the written

permission of Cambridge University Press

First published 2004Reprinted 2005

First paperback edition published with correction 2008 Reprinted 2010

A catalogue record for this publication is available from the British Library

Library of Congress Cataloguing in Publication data

isbn 978-0-521-83733-0 Hardback isbn 978-0-521-71596-6 Paperback

Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in

this publication and does not guarantee that any content on such websites is or will remain accurate or appropriate Information regarding prices travel

timetables and other factual information given in this work is correct at the time of first printing but Cambridge University Press does not guarantee

the accuracy of such information thereafter

Contents

Foreword by James Bjorken page xi

Preface xiii

Preface to the paperback edition xvii

Acknowledgements xix

Terminology and notation xxi

Part 1 Relativistic foundations 1

1 General ideas and heuristic picture 311 The problem of quantum gravity 3

111 Unfinished revolution 3112 How to search for quantum gravity 4113 The physical meaning of general relativity 9114 Background-independent quantum field theory 10

12 Loop quantum gravity 13121 Why loops 14122 Quantum space spin networks 17123 Dynamics in background-independent QFT 22124 Quantum spacetime spinfoam 26

13 Conceptual issues 28131 Physics without time 29

v

vi Contents

2 General Relativity 3321 Formalism 33

211 Gravitational field 33212 ldquoMatterrdquo 37213 Gauge invariance 40214 Physical geometry 42215 Holonomy and metric 44

22 The conceptual path to the theory 48221 Einsteinrsquos first problem a field theory for the

newtonian interaction 48222 Einsteinrsquos second problem relativity of motion 52223 The key idea 56224 Active and passive diffeomorphisms 62225 General covariance 65

23 Interpretation 71231 Observables predictions and coordinates 71232 The disappearance of spacetime 73

24 Complements 75241 Mach principles 75242 Relationalism versus substantivalism 76243 Has general covariance any physical content 78244 Meanings of time 82245 Nonrelativistic coordinates 87246 Physical coordinates and GPS observables 88

3 Mechanics 9831 Nonrelativistic mechanics 9832 Relativistic mechanics 105

321 Structure of relativistic systems partialobservablesrelativistic states 105

322 Hamiltonian mechanics 107323 Nonrelativistic systems as a special case 114324 Mechanics is about relations between observables 118325 Space of boundary data G and Hamilton function

S 120326 Evolution parameters 126327 Complex variables and reality conditions 128

33 Field theory 129331 Partial observables in field theory 129332 Relativistic hamiltonian mechanics 130333 The space of boundary data G and the Hamilton

function S 133

Contents vii

334 HamiltonndashJacobi 13734 Thermal time hypothesis 140

4 Hamiltonian general relativity 14541 EinsteinndashHamiltonndashJacobi 145

411 3d fields ldquoThe length of the electric field is theareardquo 147

412 Hamilton function of GR and its physical meaning 15142 Euclidean GR and real connection 153

421 Euclidean GR 153422 Lorentzian GR with a real connection 155423 Barbero connection and Immirzi parameter 156

43 Hamiltonian GR 157431 Version 1 real SO(3 1) connection 157432 Version 2 complex SO(3) connection 157433 Configuration space and hamiltonian 158434 Derivation of the HamiltonndashJacobi formalism 159435 Reality conditions 162

5 Quantum mechanics 16451 Nonrelativistic QM 164

511 Propagator and spacetime states 166512 Kinematical state space K and ldquoprojectorrdquo P 169513 Partial observables and probabilities 172514 Boundary state space K and covariant vacuum |0〉 174515 Evolving constants of motion 176

52 Relativistic QM 177521 General structure 177522 Quantization and classical limit 179523 Examples pendulum and timeless double

pendulum 18053 Quantum field theory 184

531 Functional representation 186532 Field propagator between parallel boundary

surfaces 190533 Arbitrary boundary surfaces 193534 What is a particle 195535 Boundary state space K and covariant vacuum |0〉 197536 Lattice scalar product intertwiners and spin

network states 19854 Quantum gravity 200

541 Transition amplitudes in quantum gravity 200542 Much ado about nothing the vacuum 202

viii Contents

55 Complements 204551 Thermal time hypothesis and Tomita flow 204552 The ldquochoicerdquo of the physical scalar product 206553 Reality conditions and scalar product 208

56 Relational interpretation of quantum theory 209561 The observer observed 210562 Facts are interactions 215563 Information 218564 Spacetime relationalism versus quantum

relationalism 220

Part II Loop quantum gravity 223

6 Quantum space 22561 Structure of quantum gravity 22562 The kinematical state space K 226

621 Structures in K 230622 Invariances of the scalar product 231623 Gauge-invariant and diffeomorphism-invariant

states 23363 Internal gauge invariance The space K0 234

631 Spin network states 234632 Details about spin networks 236

64 Diffeomorphism invariance The space Kdiff 238641 Knots and s-knot states 240642 The Hilbert space Kdiff is separable 241

65 Operators 242651 The connection A 242652 The conjugate momentum E 243

66 Operators on K0 246661 The operator A(S) 246662 Quanta of area 249663 n-hand operators and recoupling theory 250664 Degenerate sector 253665 Quanta of volume 259

67 Quantum geometry 262671 The texture of space weaves 268

7 Dynamics and matter 27671 Hamiltonian operator 277

711 Finiteness 280712 Matrix elements 282

Contents ix

713 Variants 28472 Matter kinematics 285

721 YangndashMills 286722 Fermions 287723 Scalars 288724 The quantum states of space and matter 289

73 Matter dynamics and finiteness 28974 Loop quantum gravity 291

741 Variants 293

8 Applications 29681 Loop quantum cosmology 296

811 Inflation 30182 Black-hole thermodynamics 301

821 The statistical ensemble 303822 Derivation of the BekensteinndashHawking entropy 308823 Ringing modes frequencies 311824 The BekensteinndashMukhanov effect 312

83 Observable effects 315

9 Quantum spacetime spinfoams 32091 From loops to spinfoams 32192 Spinfoam formalism 327

921 Boundaries 32893 Models 329

931 3d quantum gravity 331932 BF theory 340933 The spinfoamGFT duality 343934 BC models 348935 Group field theory 356936 Lorentzian models 359

94 Physics from spinfoams 361941 Particlesrsquo scattering and Minkowski vacuum 363

10 Conclusion 366101 The physical picture of loop gravity 366

1011 GR and QM 3661012 Observables and predictions 3671013 Space time and unitarity 3681014 Quantum gravity and other open problems 370

102 What has been achieved and what is missing 371

x Contents

Part III Appendices 375

Appendix A Groups and recoupling theory 377A1 SU(2) spinors intertwiners n-j symbols 377A2 Recoupling theory 383

A21 Penrose binor calculus 383A22 KL recoupling theory 385A23 Normalizations 388

A3 SO(n) and simple representations 390

Appendix B History 393B1 Three main directions 393B2 Five periods 395

B21 The Prehistory 1930ndash1957 398B22 The Classical Age 1958ndash1969 400B23 The Middle Ages 1970ndash1983 403B24 The Renaissance 1984ndash1994 407B25 Nowadays 1995ndash 410

B3 The divide 412

Appendix C On method and truth 415C1 The cumulative aspects of scientific knowledge 415C2 On realism 420C3 On truth 422

References 424

Index 452

Foreword

The problem of what happens to classical general relativity at the extremeshort-distance Planck scale of 10minus33 cm is clearly one of the most pressingin all of physics It seems abundantly clear that profound modifications ofexisting theoretical structures will be mandatory by the time one reachesthat distance scale There exist several serious responses to this chal-lenge These include effective field theory string theory loop quantumgravity thermogravity holography and emergent gravity Effective fieldtheory is to gravitation as chiral perturbation theory is to quantum chro-modynamics ndash appropriate at large distances and impotent at short Itsprimary contribution is the recognition that the EinsteinndashHilbert action isno doubt only the first term in an infinite series constructed out of higherpowers of the curvature tensor String theory emphasizes the possibleroles of supersymmetry extra dimensions and the standard-model inter-nal symmetries in shaping the form of the microscopic theory Loop grav-ity most directly attacks the fundamental quantum issues and featuresthe construction of candidate wave-functionals which are background in-dependent Thermogravity explores the apparent deep connection of semi-classical gravity to thermodynamic concepts such as temperature and en-tropy The closely related holographic ideas connect theories defined inbulk spacetimes to complementary descriptions residing on the bound-aries Finally emergent gravity suggests that the time-tested symbioticrelationship between condensed matter theory and elementary particletheory should be extended to the gravitational and cosmological contextsas well with more lessons yet to be learned

In each of the approaches difficult problems stand in the way of at-taining a fully satisfactory solution to the basic issues Each has its bandof enthusiasts the largest by far being the string community Most of theapproaches come with rather strong ideologies especially apparent whenthey are popularized The presence of these ideologies tends to isolate the

xi

xii Foreword

communities from each other In my opinion this is extremely unfortu-nate because it is probable that all these ideologies including my own(which is distinct from the above listing) are dead wrong The evidence ishistory from the Greeks to Kepler to Newton to Einstein there has beenno shortage of grand ideas regarding the Basic Questions In the presenceof new data available to us and not them only fragments of those grandvisions remain viable The clutter of thirty-odd standard model param-eters and the descriptive nature of modern cosmology suggests that wetoo have quite a way to go before ultimate simplicity is attained Thisdoes not mean abandoning ideologies ndash they are absolutely essential indriving us all to work hard on the problems But it does mean that anattitude of humility and of high sensitivity toward alternative approachesis essential

This book is about only one approach to the subject ndash loop quantumgravity It is a subject of considerable technical difficulty and the litera-ture devoted to it is a formidable one This feature alone has hindered thecross-fertilization which is as delineated above so essential for progressHowever within these pages one will find a much more accessible de-scription of the subject put forward by one of its leading architects anddeepest thinkers The existence of such a fine book will allow this im-portant subject quite likely to contribute significantly to the unknownultimate theory to be assimilated by a much larger community of the-orists If this does indeed come to pass its publication will become oneof the most important developments in this very active subfield since itsonset

James Bjorken

Preface

A dream I have long held was to write a ldquotreatiserdquo on quantum gravityonce the theory had been finally found and experimentally confirmed Weare not yet there There is neither experimental support nor sufficienttheoretical consensus Still a large amount of work has been developedover the last twenty years towards a quantum theory of spacetime Manyissues have been clarified and a definite approach has crystallized Theapproach variously denoted1 is mostly known as ldquoloop quantum gravityrdquo

The problem of quantum gravity has many aspects Ideas and resultsare scattered in the literature In this book I have attempted to collect themain results and to present an overall perspective on quantum gravity asdeveloped during this twenty-year period The point of view is personaland the choice of subjects is determined by my own interests I apologizeto friends and colleagues for what is missing the reason so much is missingis due to my own limitations for which I am the first to be sorry

It is difficult to over-estimate the vastitude of the problem of quantumgravity The physics of the early twentieth century has modified the veryfoundation our understanding of the physical world changing the meaningof the basic concepts we use to grasp it matter causality space and timeWe are not yet able to paint a consistent picture of the world in whichthese modifications taken together make sense The problem of quantumgravity is nothing less than the problem of finding this novel consistentpicture finally bringing the twentieth century scientific revolution to anend

Solving a problem of this sort is not just a matter of mathematical skillAs was the case with the birth of quantum mechanics relativity electro-magnetism and newtonian mechanics there are conceptual and founda-tional problems to be addressed We have to understand which (possibly

1See the notation section

xiv Preface

new) notions make sense and which old notions must be discarded inorder to describe spacetime in the quantum relativistic regime What weneed is not just a technique for computing say gravitonndashgraviton scatter-ing amplitudes (although we certainly want to be able to do so eventu-ally) We need to re-think the world in the light of what we have learnedabout it with quantum theory and general relativity

General relativity in particular has modified our understanding of thespatio-temporal structure of reality in a way whose consequences havenot yet been fully explored A significant part of the research in quantumgravity explores foundational issues and Part I of this book (ldquoRelativisticfoundationsrdquo) is devoted to these basic issues It is an exploration of howto rethink basic physics from scratch after the general-relativistic con-ceptual revolution Without this we risk asking any tentative quantumtheory of gravity the wrong kind of questions

Part II of the book (ldquoLoop quantum gravityrdquo) focuses on the loopapproach The loop theory described in Part II can be studied by itselfbut its reason and interpretation are only clear in the light of the generalframework studied in Part I Although several aspects of this theory arestill incomplete the subject is mature enough to justify a book A theorybegins to be credible only when its original predictions are reasonablyunique and are confirmed by new experiments Loop quantum gravity isnot yet credible in this sense Nor is any other current tentative theory ofquantum gravity The interest of the loop theory in my opinion is thatat present it is the only approach to quantum gravity leading to well-defined physical predictions (falsifiable at least in principle) and moreimportantly it is the most determined effort for a genuine merging ofquantum field theory with the world view that we have discovered withgeneral relativity The future will tell us more

There are several other introductions to loop quantum gravity Clas-sic reports on the subject [1ndash10 in chronological order] illustrate variousstages of the development of the theory For a rapid orientation and toappreciate different points of view see the review papers [11ndash15] Muchuseful material can be found in [16] Good introductions to spinfoam the-ory are to be found in [1117ndash19] This book is self-contained but I havetried to avoid excessive duplications referring to other books and reviewpapers for nonessential topics well developed elsewhere This book focuseson physical and conceptual aspects of loop quantum gravity ThomasThiemannrsquos book [20] which is going to be completed soon focuses onthe mathematical foundation of the same theory The two books are com-plementary this book can almost be read as Volume 1 (ldquoIntroduction andconceptual frameworkrdquo) and Thiemannrsquos book as Volume 2 (ldquoCompletemathematical frameworkrdquo) of a general presentation of loop quantumgravity

Preface xv

The book assumes that the reader has a basic knowledge of general rel-ativity quantum mechanics and quantum field theory In particular theaim of the chapters on general relativity (Chapter 2) classical mechan-ics (Chapter 3) hamiltonian general relativity (Chapter 4) and quantumtheory (Chapter 5) is to offer the fresh perspective on these topics whichis needed for quantum gravity to a reader already familiar with the con-ventional formulation of these theories

Sections with comments and examples are printed in smaller fonts (seeSection 131 for first such example) Sections that contain side or morecomplex topics and that can be skipped in a first reading without com-promising the understanding of what follows are marked with a star (lowast)in the title References in the text appear only when strictly needed forcomprehension Each chapter ends with a short bibliographical sectionpointing out essential references for the reader who wants to go into moredetail or to trace original works on a subject I have given up the immensetask of collecting a full bibliography on loop quantum gravity On manytopics I refer to specific review articles where ample bibliographic infor-mation can be found An extensive bibliography on loop quantum gravityis given in [9] and [20]

I have written this book thinking of a researcher interested in workingin quantum gravity but also of a good PhD student or an open-mindedscholar curious about this extraordinary open problem I have found thejourney towards general relativistic quantum physics towards quantumspacetime a fascinating adventure I hope the reader will see the beautyI see and that he or she will be capable of completing the journey Thelandscape is magic the trip is far from being over

Preface to the paperback edition

Three years have lapsed since the first edition of this book During thesethree years the research in loop gravity has been developing briskly and inseveral directions Remarkable new results are for instance the proof thatspinfoam and hamiltonian loop theory are equivalent in 3d the proof ofthe unicity of the loop representation (the ldquoLOSTrdquo theorem) the resolu-tion of the r = 0 black hole singularity major advances in loop cosmologythe result that in 3d loop quantum gravity plus matter yields an effectivenon-commutative quantum field theory the ldquomaster constraintrdquo programfor the definition of the quantum dynamics the idea of deriving parti-cles from linking the recalculation of the Immirzi parameter from blackhole thermodynamics and last but not least the first steps toward calcu-lating scattering amplitudes from the background independent quantumtheory I am certainly neglecting something that will soon turn out to beimportant

I have added notes and pointers to recent literature or recent reviewpapers where the interested reader can find updates on specific topicsIn spite of these rapid developments however it is too early for a full-fledged second edition of this book it seems to me that the book as itis still provides a comprehensive introduction to the field In fact severalof these developments reinforce the point of view of this book namelythat the lines of research considered form a coherent picture and definea common language in which a consistent quantum field theory without(background) space and time can be defined

When I feel pessimistic I see the divergence between research linesand the impressive number of problems that are still open When I feeloptimistic I see their remarkable coherence and I dream we might bewith respect to quantum gravity as Einstein was in 1914 with all themachinery ready trying a number of similar field equations Then itseems to me that a quantum theory of gravity (certainly not the final

xvii

xviii Preface to the paperback edition

theory of everything) is truly at hand maybe we have it maybe what weneed is just the right combination of techniques a few more details orone last missing key idea

Once again my wish is that among the readers of this paperback editionthere is she or he who will give us this last missing idea

Acknowledgements

I am indebted to the many people that have sent suggestions and cor-rections to the draft of this book posted online and to its first editionAmong them are M Carling Alexandru Mustatea Daniele Oriti JohnBaez Rafael Kaufmann Nedal Colin Hayhurst Jurgen Ehlers ChrisGauthier Gianluca Calcagni Tomas Liko Chang Chi-Ming YoungsubYoon Martin Bojowald and Gen Zhang Special thanks in particular toJustin Malecki Jacob Bourjaily and Leonard Cottrell

My great gratitude goes to the friends with whom I have had the priv-ilege of sharing this adventure

To Lee Smolin companion of adventures and friend His unique creativ-ity and intelligence intellectual freedom and total honesty are among thevery best things I have found in life

To Abhay Ashtekar whose tireless analytical rigor synthesis capacityand leadership have been a most precious guide Abhay has solidified ourideas and transformed our intuitions into theorems This book is a resultof Leersquos and Abhayrsquos ideas and work as much as my own

To Laura Scodellari and Chris Isham my first teachers who guided meinto mathematics and quantum gravity

To Ted Newman who with Sally parented the little boy just arrivedfrom the Empirersquos far provinces I have shared with Ted a decade ofintellectual joy His humanity generosity honesty passion and love forthinking are the example against which I judge myself

I would like to thank one by one all the friends working in this fieldwho have developed the ideas and results described in this book butthey are too many I can only mention my direct collaborators and afew friends outside this field Luisa Doplicher Simone Speziale ThomasSchucker Florian Conrady Daniele Colosi Etera Livine Daniele OritiFlorian Girelli Roberto DePietri Robert Oeckl Merced MontesinosKirill Krasnov Carlos Kozameh Michael Reisenberger Don Marolf

xx Acknowledgements

Berndt Brugmann Junichi Iwasaki Gianni Landi Mauro Carfora JormaLouko Marcus Gaul Hugo Morales-Tecotl Laurent Freidel Renate LollAlejandro Perez Giorgio Immirzi Philippe Roche Federico LaudisaJorge Pullin Thomas Thiemann Louis Crane Jerzy Lewandowski JohnBaez Ted Jacobson Marco Toller Jeremy Butterfield John Norton JohnBarrett Jonathan Halliwell Massimo Testa David Finkelstein GaryHorowitz John Earman Julian Barbour John Stachel Massimo PauriJim Hartle Roger Penrose John Wheeler and Alain Connes

With all these friends I have had the joy of talking about physics ina way far from problem-solving from outsmarting each other or frommaking weapons to make ldquousrdquo stronger than ldquothemrdquo I think that physicsis about escaping the prison of the received thoughts and searching fornovel ways of thinking the world about trying to clear a bit the mistylake of our insubstantial dreams which reflect reality like the lake reflectsthe mountainsForemost thanks to Bonnie ndash she knows why

Terminology and notation

bull In this book ldquorelativisticrdquo means ldquogeneral relativisticrdquo unless other-wise specified When referring to special relativity I say so explicitlySimilarly ldquononrelativisticrdquo and ldquoprerelativisticrdquo mean ldquonon-general -relativisticrdquo and ldquopre-general -relativisticrdquo The choice is a bit unusual(special relativity in this language is ldquononrelativisticrdquo) One reason forit is simply to make language smoother the book is about general rela-tivistic physics and repeating ldquogeneralrdquo every other line sounds too muchlike a Frenchman talking about de Gaulle But there is a more substantialreason the complete revolution in spacetime physics which truly deservesthe name of relativity is general relativity not special relativity This opin-ion is not always shared today but it was Einsteinrsquos opinion Einstein hasbeen criticized on this but in my opinion the criticisms miss the full reachof Einsteinrsquos discovery about spacetime One of the aims of this book isto defend in modern language Einsteinrsquos intuition that his gravitationaltheory is the full implementation of relativity in physics This point isdiscussed at length in Chapter 2

bull I often indulge in the physicistsrsquo bad habit of mixing up function fand function values f(x) Care is used when relevant Similarly I followstandard physicistsrsquo abuse of language in denoting a field such as theMaxwell potential as Aμ(x) A(x) or A where the three notations aretreated as equivalent manners of denoting the field Again care is usedwhere relevant

bull All fields are assumed to be smooth unless otherwise specified All state-ments about manifolds and functions are local unless otherwise specifiedthat is they hold within a single coordinate patch In general I do notspecify the domain of definition of functions clearly equations hold wherefunctions are defined

xxi

xxii Terminology and notation

bull Index notation follows the most common choice in the field Greek in-dices from the middle of the alphabet μ ν = 0 1 2 3 are 4d spacetimetangent indices Upper case Latin indices from the middle of the alphabetI J = 0 1 2 3 are 4d Lorentz tangent indices (In the special relativis-tic context the two are used without distinction) Lower case Latin indicesfrom the beginning of the alphabet a b = 1 2 3 are 3d tangent indicesLower case Latin indices from the middle of the alphabet i j = 1 2 3are 3d indices in R3 Coordinates of a 4d manifold are usually indicated asx y while 3d manifold coordinates are usually indicated as x y (alsoas τ) Thus the components of a spacetime coordinate x are

xμ = (t x) = (x0 xa)

while the components of a Lorentz vector e are

eI = (e0 ei)

bull ηIJ is the Minkowski metric with signature [minus+++] The indicesI J are raised and lowered with ηIJ δij is the Kronecker delta or theR3 metric The indices i j are raised and lowered with δij

bull For reasons explained at the beginning of Chapter 2 I call ldquogravita-tional fieldrdquo the tetrad field eIμ(x) instead of the metric tensor gμν(x) =ηIJ eIμ(x)eJν (x)

bull εIJKL or εμνρσ is the completely antisymmetric object with ε0123 = 1Similarly for εabc or εijk in 3d The Hodge star is defined by

F lowastIJ =

12εIJKL FKL

in flat space and by the same equation where FIJeIμe

Jν = Fμν and

F lowastIJe

Iμe

Jν = F lowast

μν in the presence of gravity Equivalently

F lowastμν =

radicminusdet g

12εμνρσ F ρσ = | det e| 1

2εμνρσ F ρσ

bull Symmetrization and antisymmetrization of indices is defined with a halfA(ab) = 1

2(Aab + Aba) and A[ab] = 12(Aab minusAba)

bull I call ldquocurverdquo on a manifold M a map

γ I rarr M

s rarr γa(s)

where I is an interval of the real line R (possibly the entire R) I callldquopathrdquo an oriented unparametrized curve namely an equivalence class of

Terminology and notation xxiii

curves under change of parametrization γa(s) rarr γprimea(s) = γa(sprime(s)) withdsprimeds gt 0

bull An orthonormal basis in the Lie algebras su(2) and so(3) is chosen onceand for all and these algebras are identified with R3 For so(3) the basisvectors (vi)jk can be taken proportional to εi

jk for su(2) the basis vec-

tors (vi)AB can be taken proportional to the Pauli matrices see AppendixA1 Thus an algebra element ω in su(2) sim so(3) has components ωi

bull For any antisymmetric quantity vij with two 3d indices i j I use alsothe one-index notation

vi = 12εijk vjk vij = εijk vk

the one-index and the two-indices notation are considered as definingthe same object For instance the SO(3) connections ωij and Aij areequivalently denoted ωi and Ai

Symbols Here is a list of symbols with their name and the equationchapter or section where they are introduced or defined

A area Section 214A YangndashMills connection Equation (230)AAi

μ(x) selfdual 4d gravitational connection Equation (219)AAi

a(x) selfdual or real 3d gravitational connection Sections 41142

C relativistic configuration space Section 321Dμ covariant derivative Equation (231)Diff lowast extended diffeomorphism group Section 622eIμ(x) gravitational field Equation (21)e determinant of eIμe edge (of spinfoam) Section 91EEa

i (x) gravitational electric field Section 411f face (of spinfoam) Section 91F curvature two-form Section 211g or U group elementG Newton constantG space of boundary data Sections 325ndash

333hγ U(A γ) Section 71H relativistic hamiltonian Section 32H0 nonrelativistic (conventional) hamiltonian Section 32H quantum state space Chapter 5H0 nonrelativistic quantum state space Chapter 5

xxiv Terminology and notation

in intertwiner on spin network node n Section 63ie intertwiner on spinfoam edge e Chapter 9j irreducible representation (for SU(2) spin)jl spin associated to spin network link l Section 621jf representation associated to spinfoam face f Chapter 9K kinematical quantum state space Section 52K0 SU(2) invariant quantum state space Section 623Kdiff diff-invariant quantum state space Section 623K boundary quantum space Sections 514

535l link (of spin network) Section 91lP Planck length

radicGcminus3

L length Section 214M spacetime manifoldn node (of spin network) Section 91pa relativistic momenta (including pt) Section 32pt momentum conjugate to t Section 32P the ldquoprojectorrdquo operator Section 52PG group G projector Equation

(9117)PH subgroup H projector Equation

(9119)P transition probability Chapter 5P path ordered Equation

(281)qa partial observables Section 32RI

J μν(x) curvature Equation(28)

R(j)αβ(g) matrix of group element g in representation j

R 3d region Section 214s s-knot abstract spin network Equation

(641)|s〉 s-knot state Equation

(641)SBH black-hole entropy Section 82S embedded spin network Section 63|S〉 spin network state Section 631S 2d surface Section 214S space of fast decrease functions Chapter 5S0 space of tempered distributions Chapter 5S[γ] action functional Section 32S(qa) HamiltonndashJacobi function Section 322S(qa qa0) Hamilton function Section 325

Terminology and notation xxv

tρ thermal time Sections 34551

T target space of a field theory Section 331U or g group elementU(A γ) holonomy Section 215v vertex (of spinfoam) Section 91V volume Section 214W (qa qprimea) propagator Chapter 5W transition amplitudes propagator Section 52x 4d spacetime coordinatesx 3d coordinatesZ partition function Chapter 9α loop closed pathβ inverse temperature Section 34γ pathγ motion (in C) Section 321γ Immirzi parameter Section 423γ motion in Ω Section 32Γ relativistic phase space Section 321Γ graph Section 62Γ two-complex Chapter 9θ PoincarendashCartan form on Σ Section 322θ Poincare form on Ω Equation (39)ηIJ ημν Minkowski metric = diag[minus1111]λ cosmological constant Equation (211)λ gauge parameter Section 213ρ statistical state Sections 34 551Σ constraint surface H = 0 Section 322σΣ 3d boundary surface Chapter 4σ spinfoam Chapter 9φ(x) scalar field Equation (232)ψ(x) fermion field Equation (235)ω presymplectic form on Σ Section 322ωIμJ(x) spin connection Equation (22)

ω symplectic form on Ω Section 322Ω space of observables and momenta Sections 32ndash3326j Wigner 6j symbol Equation (933)10j Wigner 10j symbol Equation (9103)15j Wigner 15j symbol Equation (956)|0〉 covariant vacuum in K Sections 514 535|0t〉 dynamical vacuum in Kt Sections 514 532|0M〉 Minkowski vacuum in H Sections 514 531

xxvi Terminology and notation

bull The name of the theory Finally a word about the name of the quantumtheory of gravity described in this book The theory is known as ldquoloopquantum gravityrdquo (LQG) or sometimes ldquoloop gravityrdquo for short Howeverthe theory is also designated in the literature using a variety of othernames I list here these other names and the variations of their use forthe benefit of the disoriented reader

ndash ldquoQuantum spin dynamicsrdquo (QSD) is used as a synonym of LQG WithinLQG it is sometimes used to designate in particular the dynamical as-pects of the hamiltonian theory

ndash ldquoQuantum geometryrdquo is also sometimes used as a synonym of LQGWithin LQG it is used to designate in particular the kinematical as-pects of the theory The expression ldquoquantum geometryrdquo is generic it isalso widely used in other approaches to quantum spacetime in particulardynamical triangulations [21] and noncommutative geometry

ndash ldquoNonperturbative quantum gravityrdquo ldquocanonical quantum gravityrdquo andldquoquantum general relativityrdquo (QGR) are often used to designate LQGalthough their proper meaning is wider

ndash The expression ldquoAshtekar approachrdquo is still used sometimes to desig-nate LQG it comes from the fact that a key ingredient of LQG is thereformulation of classical GR as a theory of connections developed byAbhay Ashtekar

ndash In the past LQG was also called ldquothe loop representation of quantumgeneral relativityrdquo Today ldquoloop representationrdquo and ldquoconnection repre-sentationrdquo are used within LQG to designate the representations of thestates of LQG as functionals of loops (or spin networks) and as functionalsof the connection respectively The two are related in the same manneras the energy (ψn = 〈n|ψ〉) and position (ψ(x) = 〈x|ψ〉) representationsof the harmonic oscillator states

Part I

Relativistic foundations

I know that I am mortal and the creature of aday but when I search out the massed wheeling circlesof the stars my feet no longer touch the earthside by side with Zeus himself I drink my fill ofambrosia food of the gods

Claudius Ptolemy Mathematical Syntaxis

1General ideas and heuristic picture

The aim of this chapter is to introduce the general ideas on which this book is based andto present the picture of quantum spacetime that emerges from loop quantum gravityin a heuristic and intuitive manner The style of the chapter is therefore conversationalwith little regard for precision and completeness In the course of the book the ideasand notions introduced here will be made precise and the claims will be justified andformally derived

11 The problem of quantum gravity

111 Unfinished revolution

Quantum mechanics (QM) and general relativity (GR) have greatlywidened our understanding of the physical world A large part of thephysics of the last century has been a triumphant march of exploration ofnew worlds opened up by these two theories QM led to atomic physics nu-clear physics particle physics condensed matter physics semiconductorslasers computers quantum optics GR led to relativistic astrophysicscosmology GPS technology and is today leading us hopefully towardsgravitational wave astronomy

But QM and GR have destroyed the coherent picture of the worldprovided by prerelativistic classical physics each was formulated in termsof assumptions contradicted by the other theory QM was formulatedusing an external time variable (the t of the Schrodinger equation) ora fixed nondynamical background spacetime (the spacetime on whichquantum field theory is defined) But this external time variable and thisfixed background spacetime are incompatible with GR In turn GR wasformulated in terms of riemannian geometry assuming that the metric isa smooth and deterministic dynamical field But QM requires that anydynamical field be quantized at small scales it manifests itself in discretequanta and is governed by probabilistic laws

3

4 General ideas and heuristic picture

We have learned from GR that spacetime is dynamical and we havelearned from QM that any dynamical entity is made up of quanta andcan be in probabilistic superposition states Therefore at small scales thereshould be quanta of space and quanta of time and quantum superpositionof spaces But what does this mean We live in a spacetime with quantumproperties a quantum spacetime What is quantum spacetime How canwe describe it

Classical prerelativistic physics provided a coherent picture of the phys-ical world This was based on clear notions such as time space matterparticle wave force measurement deterministic law This picture haspartially evolved (in particular with the advent of field theory and spe-cial relativity) but it has remained consistent and quite stable for threecenturies GR and QM have modified these basic notions in depth GRhas modified the notions of space and time QM the notions of causalitymatter and measurement The novel modified notions do not fit togethereasily The new coherent picture is not yet available With all their im-mense empirical success GR and QM have left us with an understandingof the physical world which is unclear and badly fragmented At the foun-dations of physics there is today confusion and incoherence

We want to combine what we have learnt about our world from the twotheories and to find a new synthesis This is a major challenge ndash perhapsthe major challenge ndash in todayrsquos fundamental physics GR and QM haveopened a revolution The revolution is not yet complete

With notable exceptions (Dirac Feynman Weinberg DeWitt WheelerPenrose Hawking rsquot Hooft among others) most of the physicists of thesecond half of the last century have ignored this challenge The urgencywas to apply the two theories to larger and larger domains The develop-ments were momentous and the dominant attitude was pragmatic Apply-ing the new theories was more important than understanding them Butan overly pragmatic attitude is not productive in the long run Towardsthe end of the twentieth century the attention of theoretical physics hasbeen increasingly focusing on the challenge of merging the conceptualnovelties of QM and GR

This book is the account of an effort to do so

112 How to search for quantum gravity

How to search for this new synthesis Conventional field quantizationmethods are based on the weak-field perturbation expansion Their appli-cation to GR fails because it yields a nonrenormalizable theory Perhapsthis is not surprising GR has changed the notions of space and time tooradically to docilely agree with flat space quantum field theory Somethingelse is needed

11 The problem of quantum gravity 5

In science there are no secure recipes for discovery and it is important toexplore different directions at the same time Currently a quantum theoryof gravity is sought along various paths The two most developed are loopquantum gravity described in this book and string theory Other researchdirections include dynamical triangulations noncommutative geometryHartlersquos quantum mechanics of spacetime (this is not really a specificquantum theory of gravity but rather a general theoretical frameworkfor general-relativistic quantum theory) Hawkingrsquos euclidean sum overgeometries quantum Regge calculus Penrosersquos twistor theory Sorkinrsquoscausal sets rsquot Hooftrsquos deterministic approach and Finkelsteinrsquos theoryThe reader can find ample references in the general introductions to quan-tum gravity mentioned in the note at the end of this chapter Here I sketchonly the general ideas that motivate the approach described in this bookplus a brief comment on string theory which is currently the most popularalternative to loop gravity

Our present knowledge of the basic structure of the physical universe issummarized by GR quantum theory and quantum field theory (QFT) to-gether with the particle-physics standard model This set of fundamentaltheories is inconsistent But it is characterized by an extraordinary em-pirical success nearly unique in the history of science Indeed currentlythere is no evidence of any observed phenomenon that clearly escapesquestions or contradicts this set of theories (or a minor modification ofthe same to account say for a neutrino mass or a cosmological constant)This set of theories becomes meaningless in certain physical regimes Inthese regimes we expect the predictions of quantum gravity to becomerelevant and to differ from the predictions of GR and the standard modelThese regimes are outside our experimental or observational reach at leastso far Therefore we have no direct empirical guidance for searching forquantum gravity ndash as say atomic spectra guided the discovery of quan-tum theory

Since quantum gravity is a theory expected to describe regimes that areso far inaccessible one might worry that anything could happen in theseregimes at scales far removed from our experience Maybe the search isimpossible because the range of the possible theories is too large Thisworry is unjustified If this was the problem we would have plenty ofcomplete predictive and coherent theories of quantum gravity Insteadthe situation is precisely the opposite we havenrsquot any The fact is that wedo have plenty of information about quantum gravity because we haveQM and we have GR Consistency with QM and GR is an extremely strictconstraint

A view is sometime expressed that some totally new radical and wildhypothesis is needed for quantum gravity I do not think that this isthe case Wild ideas pulled out of the blue sky have never made science


advance The radical hypotheses that physics has successfully adoptedhave always been reluctantly adopted because they were forced upon usby new empirical data ndash Keplerrsquos ellipses Bohrrsquos quantization ndash or bystringent theoretical deductions ndash Maxwellrsquos inductive current Einsteinrsquosrelativity (see Appendix C) Generally arbitrary novel hypotheses leadnowhere

In fact today we are precisely in one of the typical situations in whichtheoretical physics has worked at its best in the past Many of the moststriking advances in theoretical physics have derived from the effort offinding a common theoretical framework for two basic and apparently con-flicting discoveries For instance the aim of combining the keplerian or-bits with galilean physics led to newtonian mechanics combining Maxwelltheory with galilean relativity led to special relativity combining specialrelativity and nonrelativistic quantum theory led to the theoretical discov-ery of antiparticles combining special relativity with newtonian gravityled to general relativity and so on In all these cases major advances havebeen obtained by ldquotaking seriouslyrdquo1 apparently conflicting theories andexploring the implications of holding the key tenets of both theories fortrue Today we are precisely in one of these characteristic situations Wehave learned two new very general ldquofactsrdquo about Nature expressed byQM and GR we have ldquojustrdquo to figure out what they imply taken to-gether Therefore the question we have to ask is what have we reallylearned about the world from QM and from GR Can we combine theseinsights into a coherent picture What we need is a conceptual scheme inwhich the insights obtained with GR and QM fit together

This view is not the majority view in theoretical physics at presentThere is consensus that QM has been a conceptual revolution but manydo not view GR in the same way According to many the discovery of GRhas been just the writing of one more field theory This field theory isfurthermore likely to be only an approximation to a theory we do not yetknow According to this opinion GR should not be taken too seriously asa guidance for theoretical developments

I think that this opinion derives from a confusion the confusion betweenthe specific form of the EinsteinndashHilbert action and the modification of thenotions of space and time engendered by GR The EinsteinndashHilbert actionmight very well be a low-energy approximation of a high-energy theoryBut the modification of the notions of space and time does not depend onthe specific form of the EinsteinndashHilbert action It depends on its diffeo-morphism invariance and its background independence These properties

1In [22] Gell-Mann says that the main lesson to be learnt from Einstein is ldquoto lsquotakevery seriouslyrsquo ideas that work and see if they can be usefully carried much furtherthan the original proponent suggestedrdquo


(which are briefly illustrated in Section 113 below and discussed in de-tail in Chapter 2) are most likely to hold in the high-energy theory aswell One should not confuse the details of the dynamics of GR with themodifications of the notions of space and time that GR has determinedIf we make this confusion we underestimate the radical novelty of thephysical content of GR The challenge of quantum gravity is precisely tofully incorporate this radical novelty into QFT In other words the taskis to understand what is a general-relativistic QFT or a background-independent QFT

Today many physicists prefer disregarding or postponing these founda-tional issues and instead choose to develop and adjust current theoriesThe most popular strategy towards quantum gravity in particular isto pursue the line of research grown in the wake of the success of thestandard model of particle physics The failure of perturbative quantumGR is interpreted as a replay of the failure of Fermi theory2 Namely asan indication that we must modify GR at high energy With the inputof the grand-unified-theories (GUTs) supersymmetry and the KaluzandashKlein theory the search for a high-energy correction of GR free from badultraviolet divergences has led to higher derivative theories supergravityand finally to string theory

Sometimes the claim is made that the quantum theory of gravity hasalready been found and it is string theory Since this is a book about quan-tum gravity without strings I should say a few words about this claimString theory is based on a physical hypothesis elementary objects areextended rather than particle-like This hypothesis leads to a very richunified theory which contains much phenomenology including (with suit-able inputs) fermions YangndashMills fields and gravitons and is expected bymany to be free of ultraviolet divergences The price to pay for these theo-retical results is a gigantic baggage of additional physics supersymmetryextra dimensions an infinite number of fields with arbitrary masses andspins and so on

So far nothing of this new physics shows up in experiments Super-symmetry in particular has been claimed to be on the verge of beingdiscovered for years but hasnrsquot shown up Unfortunately so far the the-ory can accommodate any disappointing experimental result because it ishard to derive precise new quantitative physical predictions with whichthe theory could be falsified from the monumental mathematical appa-ratus of the theory Furthermore even recovering the real world is noteasy within the theory the search for a compactification leading to the

2Fermi theory was an empirically successful but nonrenormalizable theory of the weakinteractions just as GR is an empirically successful but nonrenormalizable theory ofthe gravitational interaction The solution has been the GlashowndashWeinbergndashSalamelectroweak theory which corrects Fermi theory at high energy


standard model with its families and masses and no instabilities has notyet succeeded as far as I know It is clear that string theory is a very inter-esting hypothesis but certainly not an established theory It is thereforeimportant to pursue alternative directions as well

String theory is a direct development of the standard model and isdeeply rooted in the techniques and the conceptual framework of flatspace QFT As I shall discuss in detail throughout this book manyof the tools used in this framework ndash energy unitary time evolutionvacuum state Poincare invariance S-matrix objects moving in a space-time Fourier transform ndash no longer make sense in the quantum grav-itational regime in which the gravitational field cannot be approxi-mated by a background spacetime ndash perhaps not even asymptotically3

Therefore string theory does not address directly the main challengeof quantum gravity understanding what is a background-independentQFT Facing this challenge directly before worrying about unificationleads instead to the direction of research investigated by loop quantumgravity4

The alternative to the line of research followed by string theory is givenby the possibility that the failure of perturbative quantum GR is not areplay of Fermi theory That is it is not due to a flaw of the GR actionbut instead it is due to the fact that the conventional weak-field quantumperturbation expansion cannot be applied to the gravitational field

This possibility is strongly supported a posteriori by the results of loopquantum gravity As we shall see loop quantum gravity leads to a pictureof the short-scale structure of spacetime extremely different from that ofa smooth background geometry (There are hints in this direction fromstring theory calculations as well [25]) Spacetime turns out to have anonperturbative quantized discrete structure at the Planck scale whichis explicitly described by the theory The ultraviolet divergences are curedby this structure The ultraviolet divergences that appear in the pertur-bation expansion of conventional QFT are a consequence of the fact that

3To be sure the development of string theory has incorporated many aspects of GRsuch as curved spacetimes horizons black holes and relations between different back-grounds But this is far from a background-independent framework such as the onerealized by GR in the classical context GR is not about physics on a curved space-time or about relations between different backgrounds it is about the dynamics ofspacetime A background-independent fundamental definition of string theory is beingactively searched for along several directions but so far the definition of the theoryand all calculations rely on background metric spaces

4It has been repeatedly suggested that loop gravity and string theory might mergebecause loop gravity has developed precisely the background-independent QFT meth-ods that string theory needs [23] Also excitations over a weave (see Section 671)have a natural string structure in loop gravity [24]


we erroneously replace this discrete Planck-scale structure with a smoothbackground geometry

If this is physically correct ultraviolet divergences do not require theheavy machinery of string theory to be cured On the other hand the con-ventional weak-field perturbative methods cannot be applied because wecannot work with a fixed smooth background geometry We must there-fore adapt QFT to the full conceptual novelty of GR and in particularto the change in the notion of space and time induced by GR What arethese changes I sketch an answer below leaving a complete discussion toChapter 2

113 The physical meaning of general relativity

GR is the discovery that spacetime and the gravitational field are thesame entity What we call ldquospacetimerdquo is itself a physical object in manyrespects similar to the electromagnetic field We can say that GR is thediscovery that there is no spacetime at all What Newton called ldquospacerdquoand Minkowski called ldquospacetimerdquo is unmasked it is nothing but a dy-namical object ndash the gravitational field ndash in a regime in which we neglectits dynamics

In newtonian and special-relativistic physics if we take away the dy-namical entities ndash particles and fields ndash what remains is space and time Ingeneral-relativistic physics if we take away the dynamical entities nothingremains The space and time of Newton and Minkowski are re-interpretedas a configuration of one of the fields the gravitational field This impliesthat physical entities ndash particles and fields ndash are not immersed in spaceand moving in time They do not live on spacetime They live so to sayon one another

It is as if we had observed in the ocean many animals living on anisland animals on the island Then we discover that the island itself is infact a great whale So the animals are no longer on the island just animalson animals Similarly the Universe is not made up of fields on spacetimeit is made up of fields on fields This book studies the far-reaching effectthat this conceptual shift has on QFT

One consequence is that the quanta of the field cannot live in spacetimethey must build ldquospacetimerdquo themselves This is precisely what the quantaof space do in loop quantum gravity

We may continue to use the expressions ldquospacerdquo and ldquotimerdquo to indicateaspects of the gravitational field and I do so in this book We are usedto this in classical GR But in the quantum theory where the field hasquantized ldquogranularrdquo properties and its dynamics is quantized and there-fore only probabilistic most of the ldquospatialrdquo and ldquotemporalrdquo features ofthe gravitational field are lost


Therefore to understand the quantum gravitational field we must aban-don some of the emphasis on geometry Geometry represents the classicalgravitational field but not quantum spacetime This is not a betrayal ofEinsteinrsquos legacy on the contrary it is a step in the direction of ldquorelativ-ityrdquo in the precise sense meant by Einstein Alain Connes has describedbeautifully the existence of two points of view on space the geometricone centered on space points and the algebraic or ldquospectralrdquo one cen-tered on the algebra of dual spectral quantities As emphasized by Alainquantum theory forces us to a complete shift to this second point of viewbecause of noncommutativity In the light of quantum theory continuousspacetime cannot be anything else than an approximation in which wedisregard quantum noncommutativity In loop gravity the physical fea-tures of space appear as spectral properties of quantum operators thatdescribe our (the observersrsquo) interactions with the gravitational field

The key conceptual difficulty of quantum gravity is therefore to accept theidea that we can do physics in the absence of the familiar stage of spaceand time We need to free ourselves from the prejudices associated withthe habit of thinking of the world as ldquoinhabiting spacerdquo and ldquoevolving intimerdquo Chapter 3 describes a language for describing mechanical systemsin this generalized conceptual framework

This absence of the familiar spacetime ldquostagerdquo is called the backgroundindependence of the classical theory Technically it is realized by the gaugeinvariance of the action under (active) diffeomorphisms A diffeomorphismis a transformation that smoothly drags all dynamical fields and particlesfrom one region of the four-dimensional manifold to another (the pre-cise definition of these transformations is given in Chapter 2) In turngauge invariance under diffeomorphism (or diffeomorphism invariance) isthe consequence of the combination of two properties of the action itsinvariance under arbitrary changes of coordinates and the fact that thereis no nondynamical ldquobackgroundrdquo field

114 Background-independent quantum field theory

Is quantum mechanics5 compatible with the general-relativistic notionsof space and time It is provided that we choose a sufficiently generalformulation For instance the Schrodinger picture is only viable for the-ories where there is a global observable time variable t this conflictswith GR where no such variable exists Therefore the Schrodinger pic-ture makes little sense in a background-independent context But there

5I use the expression ldquoquantum mechanicsrdquo to indicate the theory of all quantumsystems with a finite or infinite number of degrees of freedom In this sense QFT ispart of quantum mechanics


are formulations of quantum theory that are more general than theSchrodinger picture In Chapter 5 I describe a formulation of QM suf-ficiently general to deal with general-relativistic systems (For anotherrelativistic formulation of QM see [26]) Formulations of this kind aresometimes denoted ldquogeneralized quantum mechanicsrdquo I prefer to useldquoquantum mechanicsrdquo to denote any formulation of quantum theory ir-respective of its generality just as ldquoclassical mechanicsrdquo is used to des-ignate formalisms with different degrees of generality such as NewtonrsquosLagrangersquos Hamiltonrsquos or symplectic mechanics

On the other hand most of the conventional machinery of perturbativeQFT is profoundly incompatible with the general-relativistic frameworkThere are many reasons for this

bull The conventional formalism of QFT relies on Poincare invarianceIn particular it relies on the notion of energy and on the existence ofthe nonvanishing hamiltonian operator that generates unitary timeevolution The vacuum for instance is the state that minimizes theenergy Generally there is no global Poincare invariance no generalnotion of energy and no nonvanishing hamiltonian operator in ageneral-relativistic theory

bull At the root of conventional QFT is the physical notion of particleThe theoretical experience with QFT on curved spacetime [27] andon the relation between acceleration and temperature in QFT [28]indicates that in a generic gravitational situation the notion of par-ticle can be quite delicate (This point is discussed in Section 534)

bull Consider a conventional renormalized QFT The physical contentof the theory can be expressed in terms of its n-point functionsW (x1 xn) The n-point functions reflect the invariances of theclassical theory In a general-relativistic theory invariance under acoordinate transformation x rarr xprime = xprime(x) implies immediately thatthe n-point functions must satisfy

W (x1 xn) = W (xprime(x1) xprime(xn)) (11)

and therefore (if the points in the argument are distinct) it must bea constant That is

W (x1 xn) = constant (12)

Clearly we are immediately in a very different framework from con-ventional QFT

bull Similarly the behavior for small |xminus y| of the two-point function ofa conventional QFT

W (x y) =constant

|xminus y|d (13)


expresses the short-distance structure of the QFT More generallythe short-distance structure of the QFT is reflected in the operatorproduct expansion

O(x)Oprime(y) =sum

n

On(x)|xminus y|n (14)

Here |x minus y| is the distance measured in the spacetime metric Onflat space for instance |xminusy|2 = ημν(xμminusyμ)(xνminusyν) In a general-relativistic context these expressions make no sense since there isno background Minkowski (or other) metric ημν In its place there isthe gravitational field namely the quantum field operator itself Butthen if standard operator product expansion becomes meaninglessthe short-distance structure of a quantum gravitational theory mustbe profoundly different from that of conventional QFT As we shallsee in Chapter 7 this is precisely the case

There is a tentative escape strategy to circumvent these difficultieswrite the gravitational field e(x) as the sum of two terms

e(x) = ebackground(x) + h(x) (15)

where ebackground(x) is a background field configuration This may beMinkowski or any other Assume that ebackground(x) defines spacetimenamely it defines location and causal relations Then consider h(x) asthe gravitational field governed by a QFT on the spacetime backgrounddefined by ebackground For instance the field operator h(x) is assumed tocommute at spacelike separations where spacelike is defined in the geom-etry determined by ebackground(x) As a second step one may then considerconditions on ebackground(x) or relations between the formulations of thetheory defined by different choices of ebackground(x) This escape strategyleads to three orders of difficulties (i) Conventional perturbative QFTof GR based on (15) leads to a nonrenormalizable theory To get rid ofthe uncontrollable ultraviolet divergences one has to resort to the compli-cations of string theory (ii) As mentioned loop quantum gravity showsthat the structure of spacetime at the Planck scale is discrete Thereforephysical spacetime has no short-distance structure at all The unphysicalassumption of a smooth background ebackground(x) implicit in (15) maybe precisely the cause of the ultraviolet divergences (iii) The separationof the gravitational field from spacetime is in strident contradiction withthe very physical lesson of GR If GR is of any guide in searching for aquantum theory of gravity the relevant spacetime geometry is the onedetermined by the full gravitational field e(x) and the separation (15) ismisleading

12 Loop quantum gravity 13

A formulation of quantum gravity that does not take the escape strategy(15) is a background-independent or general covariant QFT The mainaim of this book is develop the formalism for background-independentQFT

12 Loop quantum gravity

I sketch here the physical picture of quantum spacetime that emergesfrom loop quantum gravity (LQG) The basic ideas and assumptions onwhich LQG is based are the following

(i) Quantum mechanics and general relativity QM suitably formulatedto be compatible with general covariance is assumed to be cor-rect The Einstein equations may be modified at high energy butthe general-relativistic notions of space and time are assumed to becorrect The motivation for these two assumptions is the extraordi-nary empirical success they have had so far and the absence of anycontrary empirical evidence

(ii) Background independence LQG is based on the idea that the quan-tization strategy based on the separation (15) is not appropriatefor describing the quantum properties of spacetime

To this we can add

(iii) No unification Nowadays a fashionable idea is that the problemof quantizing gravity has to be solved together with the problemof finding a unified description of all interactions LQG is a solu-tion of the first problem not the second6

(iv) Four spacetime dimensions and no supersymmetry LQG is com-patible with these possibilities but there is nothing in the theorythat requires higher dimensions or supersymmetry Higher space-time dimensions and supersymmetry are interesting theoreticalideas which as many other interesting theoretical ideas can bephysically wrong In spite of 15 years of search numerous pre-liminary announcements of discovery then turned out to be false

6A motivation for the idea that these two issues are connected is the expectation thatwe are ldquonear the end of physicsrdquo Unfortunately the expectation of being ldquonear theend of physicsrdquo has been present all along the three centuries of the history of modernphysics In the present situation of deep conceptual confusion on the fundamentalaspects of the world I see no sign indicating that we are close to the end of ourdiscoveries about the physical world When I was a student it was fashionable toclaim that the problem of finding a theory of the strong interactions had to be solvedtogether with the problem of getting rid of renormalization theory Nice idea Butwrong


and despite repeated proclamations that supersymmetry was go-ing to be discovered ldquonext yearrdquo so far empirical evidence has beensolidly and consistently against supersymmetry This might changebut as scientists we must take the indications of the experimentsseriously

On the basis of these assumptions LQG is a straightforward quantiza-tion of GR with its conventional matter couplings The program of LQGis therefore conservative and of small ambition The physical inputs ofthe theory are just QM and GR well-tested physical theories No majoradditional physical hypothesis or assumption is made (such as elementaryobjects are strings space is made by individual discrete points quantummechanics is wrong GR is wrong supersymmetry extra dimensions )No claim of being the final ldquoTheory Of Everythingrdquo is made

On the other hand LQG has a radical and ambitious side to merge theconceptual insight of GR into QM In order to achieve this we have togive up the familiar notions of space and time The space continuum ldquoonwhichrdquo things are located and the time ldquoalong whichrdquo evolution happensare semiclassical approximate notions in the theory In LQG this radicalstep is assumed in its entirety

LQG does not make use of most of the familiar tools of conventionalQFT because these become inadequate in a background-independent con-text It only makes use of the general tools of quantum theory a Hilbertspace of states operators related to the measurement of physical quanti-ties and transition amplitudes that determine the probability outcome ofmeasurements of these quantities Hilbert space of states and operatorsassociated to physical observables are obtained from classical GR follow-ing a rather standard quantization strategy A quantization strategy isa technique for searching for a solution to a well-posed inverse problemfinding a quantum theory with a given classical limit The inverse prob-lem could have many solutions As noticed presently the difficulty is notto discriminate among many complete and consistent quantum theoriesof gravity We would be content with one

121 Why loops

Among the technical choices to make in order to implement a quantiza-tion procedure is which algebra of field functions to promote to quan-tum operators In conventional QFT this is generally the canonical al-gebra formed by the positive and negative frequency components of thefield modes The quantization of this algebra leads to the creation and


annihilation operators a and adagger The characterization of the positive andnegative frequencies requires a background spacetime

In contrast to this what characterizes LQG is the choice of adifferent algebra of basic field functions a noncanonical algebra basedon the holonomies of the gravitational connection The holonomy (orldquoWilson looprdquo) is the matrix of the parallel transport along a closedcurve

The idea that holonomies are the natural variables in a gauge the-ory has a long history In a sense it can be traced back to the veryorigin of gauge theory in the physical intuition of Faraday Faraday un-derstood electromagnetic phenomena in terms of ldquolines of forcerdquo Twokey ideas underlie this intuition First that the relevant physical vari-ables fill up space this intuition by Faraday is the origin of field the-ory Second that the relevant variables do not refer to what happens ata point but rather refer to the relation between different points con-nected by a line The mathematical quantity that expresses this ideais the holonomy of the gauge potential along the line In the Maxwellcase for instance the holonomy U(Aα) along a loop α is simply theexponential of the line integral along α of the three-dimensional Maxwellpotential A

U(Aα) = e∮α A = exp

int 2π

0ds Aa(α(s))

dαa(s)ds

(16)

In LQG the holonomy becomes a quantum operator that creates ldquoloopstatesrdquo In the loop representation formulation of Maxwell theory forinstance a loop state |α〉 is a state in which the electric field vanisheseverywhere except along a single Faraday line α More precisely it is aneigenstate of the electric field with eigenvalue

Eα(x) =∮

dsdα(s)

dsδ3(x α(s)) (17)

where s rarr α(s) is the Faraday line in space This electric field vanisheseverywhere except on the loop α itself and at every point of α it is tangentto the loop see Figure 11 Notice that the vector distribution field E(x)defined in (17) is divergenceless that is it satisfies Coulomb law

div Eα(x) = 0 (18)


Fig 11 A loop α and the distributional electric field configuration Eα (repre-sented by the arrows)

in the sense of distributions In fact for any smooth function f we have

[div Eα](f) =int

d3x f(x) div Eα(x)

=int

d3x f(x)part

partxa

∮ds

dαa(s)ds

δ3(x α(s))

= minus∮

dsdαa(s)

dspart

partαaf(α(s))

= minus∮

αdf = minus

∮ds

dds

f(α(s)) = 0 (19)

Indeed intuitively Coulomb law requires precisely that an electric fieldat a point ldquocontinuesrdquo in the direction of the field itself namely that itdefines Faraday lines The state |α〉 is therefore a sort of minimal quantumexcitation satisfying (18) it is an elementary quantum excitation of asingle Faraday line

The idea that a YangndashMills theory is truly a theory of these loops hasbeen around for as long as such theories have been studied MandelstamPolyakov and Wilson among many others have long argued that loopexcitations should play a major role in quantum YangndashMills theories andthat we must get to understand quantum YangndashMills theories in terms ofthese excitations In fact much of the development of string theory hasbeen influenced by this idea

In lattice YangndashMills theory namely in the approximation to YangndashMills theory where spacetime is replaced by a fixed lattice loop stateshave finite norm In fact certain finite linear combinations of loop statescalled ldquospin networkrdquo states form a well-defined and well-understood or-thonormal basis in the Hilbert space of a lattice gauge theory

However in a QFT theory over a continuous background the idea offormulating the theory in terms of loop-like excitations has never provedfruitful The difficulty is essentially that loop states over a background areldquotoo singularrdquo and ldquotoo manyrdquo The quantum Maxwell state |α〉 describedabove for instance has infinite norm and an infinitesimal displacement of


a loop state over the background spacetime produces a distinct indepen-dent loop state yielding a continuum of loop states Over a continuousbackground the space spanned by the loop states is far ldquotoo bigrdquo forproviding a basis of the (separable) Hilbert space of a QFT

However loop states are not too singular nor too many in abackground-independent theory This is the key technical point on whichLQG relies The intuitive reason is as follows Spacetime itself is formedby loop-like states Therefore the position of a loop state is relevant onlywith respect to other loops and not with respect to the background Aninfinitesimal (coordinate) displacement of a loop state does not produce adistinct quantum state but only a gauge equivalent representation of thesame physical state Only a finite displacement carrying the loop stateacross another loop produces a physically different state Therefore thesize of the space of the loop states is dramatically reduced by diffeomor-phism invariance most of it is just gauge Equivalently we can think thata single loop has an intrinsic Planck-size ldquothicknessrdquo

Therefore in a general-relativistic context the loop basis becomesviable The state space of the theory called Kdiff is a separable Hilbertspace spanned by loop states More precisely as we shall see in Chapter 6Kdiff admits an orthonormal basis of spin network states which are formedby finite linear combinations of loop states and are defined precisely asthe spin network states of a lattice YangndashMills theory This Hilbert spaceand the field operators that act on it are described in Chapter 6 Theyform the basis of the mathematical structure of LQG

Therefore LQG is the result of the convergence of two lines of think-ing each characteristic of twentieth-century theoretical physics On theone hand the intuition of Faraday Yang and Mills Wilson MandelstamPolyakov and others that forces are described by lines On the otherhand the EinsteinndashWheelerndashDeWitt intuition of background indepen-dence and background-independent quantum states Truly remarkablyeach of these two lines of thinking is the solution of the blocking difficultyof the other On the one hand the traditional nonviability of the loop ba-sis in the continuum disappears because of background independence Onthe other hand the traditional difficulty of controlling diffeomorphism-invariant quantities comes under control thanks to the loop basis

Even more remarkably the spin network states generated by this happymarriage turn out to have a surprisingly compelling geometric interpre-tation which I sketch below

122 Quantum space spin networks

Physical systems reveal themselves by interacting with other systemsThese interactions may happen in ldquoquantardquo energy is exchanged with an


oscillator of frequency ν in discrete packets or quanta of size E = hνIf the oscillator is in the nth energy eigenstate we say that there are nquanta in it If the oscillator is a mode of a free field we say that thereare n ldquoparticlesrdquo in the field Therefore we can view the electromagneticfield as made up of its quanta the photons What are the quanta of thegravitational field Or since the gravitational field is the same entity asspacetime what are the quanta of space

The properties of the quanta of a system are determined by the spectralproperties of the operators representing the quantities involved in ourinteraction with the system The operator associated with the energy ofthe oscillator for instance has a discrete spectrum and the number ofquanta n labels its eigenvalues The set of its eigenstates form a basisin the state space of the quantum system this fact allows us to vieweach state of the system as a quantum superposition of states |n〉 formedby n quanta To understand the quantum properties of space we havetherefore to consider the spectral problem of the operators associatedwith the quantities involved in our interaction with space itself The mostdirect interaction we have with the gravitational field is via the geometricstructure of the physical space A measurement of length area or volumeis in fact according to GR a measurement of a local property of thegravitational field

For instance the volume V of a physical region R is

V =int

Rd3x | det e(x)| (110)

where e(x) is the (triad matrix representing the) gravitational field Inquantum gravity e(x) is a field operator and V is therefore an operatoras well

The volume V is a nonlinear function of the field e and the definitionof the volume operator implies products of local operator-valued distri-butions This can be achieved as a limit using an appropriate regulariza-tion procedure The development of regularization procedures that remainmeaningful in the absence of a background metric is a major technical toolon which LQG is based Using these techniques a well-defined self-adjointoperator V can be defined The computation of its spectral properties isthen one of the main results of LQG and will be derived in Section 665

The spectrum of V turns out to be discrete Therefore the spacetimevolume manifests itself in quanta of definite volume size given by theeigenvalues of the volume operator These quanta of space can be in-tuitively thought of as quantized ldquograinsrdquo of space or ldquoatoms of spacerdquoThe first intuitive picture of quantum space is therefore that of ldquograinsof spacerdquo These have quantized amounts of volume determined by thespectrum of the operator V


j

j

2= 12

s =i1 i2

1 = 1

j3

= 12

Fig 12 A simple spin network

The next element of the picture is the information on which grain isadjacent to which Adjacency (being contiguous being in touch beingnearby) is the basis of spatial relations If two spacetime regions are ad-jacent that is if they touch each other they are separated by a surfaceS Let A be the area of the surface S Area also is a function of the grav-itational field and is therefore represented by an operator like volumeThe spectral problem for this operator has been solved in LQG as wellIt is discussed in detail in Section 662 This spectrum turns out also tobe discrete Intuitively the grains of space are separated by ldquoquanta ofareardquo The principal series of the eigenvalues of the area for instance islabeled by multiplets of half-integers ji i = 1 n and turns out to begiven by

A = 8πγ Gsum

i

radicji(ji + 1) (111)

where γ the Immirzi parameter is a free dimensionless constant of thetheory

Consider a quantum state of space |s〉 formed by N ldquograinsrdquo of spacesome of which are adjacent to one another Represent this state as anabstract graph Γ with N nodes (By abstract graph I mean here anequivalence class under smooth deformations of graphs embedded in a3-manifold) The nodes of the graph represent the grains of space thelinks of the graph link adjacent grains and represent the surfaces separat-ing two adjacent grains The quantum state is then characterized by thegraph Γ and by labels on nodes and on links the label in on a node nis the quantum number of the volume and the label jl on a link l is thequantum number of the area

A graph with these labels is called an (abstract) ldquospin networkrdquos = (Γ in jl) see Figure 12 In Section 631 we will see that the quantumnumbers in and jl are determined by the representation theory of the localgauge group (SU(2)) More precisely jl labels unitary irreducible repre-sentations and in labels a basis in the space of the intertwiners betweenthe representations adjacent to the node n The area of a surface cutting


Fig 13 The graph of an abstract spinfoam and the ensemble of ldquochunks ofspacerdquo or quanta of volume it represents Chunks are adjacent when the corre-sponding nodes are linked Each link cuts one elementary surface separating twochunks

n links of the spin network with labels ji i = 1 n is then given by(111)

As shown in Section 631 the (kinematical) Hilbert space Kdiff admitsa basis labeled precisely by these spin networks This is a basis of statesin which certain area and volume operators are diagonal Its physicalinterpretation is the one sketched in Figure 13 a spin network state |s〉describes a quantized three-geometry

A loop state |α〉 is a spin network state in which the graph Γ hasno nodes namely is a single loop α and is labeled by the fundamentalrepresentation of the group In such a state the gravitational field hassupport just on the loop α itself as the electric field in (17)

In LQG physical space is a quantum superposition of spin networks inthe same sense that the electromagnetic field is a quantum superpositionof n-photon states The first and basic prediction of the (free) QFT ofthe electromagnetic field is the existence of the photons and the specificquantitative prediction of the energy and the momentum of the photonsof a given frequency Similarly the first prediction of LQG is the existenceof the quanta of area and volume and the quantitative prediction of theirspectrum

The theory predicts that any sufficiently accurate measurement of areaor volume would measure one of these spectral values So far verifyingthis prediction appears to be outside our technological capacities

Where is a spin network A spin network state does not have a positionIt is an abstract graph not a graph immersed in a spacetime manifold


Only abstract combinatorial relations defining the graph are significantnot its shape or its position in space

In fact a spin network state is not in space it is space It is not localizedwith respect to something else something else (matter particles otherfields) might be localized with respect to it To ask ldquowhere is a spinnetworkrdquo is like asking ldquowhere is a solution of the Einstein equationsrdquo Asolution of the Einstein equations is not ldquosomewhererdquo it is the ldquowhererdquowith respect to which anything else can be localized In the same waythe other dynamical objects such as YangndashMills and fermion fields liveon the spin network state

This is a consequence of diffeomorphism invariance Technically spinnetwork states are first defined as graphs embedded in a three-dimensionalmanifold then the implementation of the diffeomorphism gauge identifiestwo graphs that can be deformed into each other They are gauge equiv-alent This is like identifying two solutions of the Einstein equations thatare related by a change of coordinates Spin networks embedded in amanifold are denoted S and called ldquoembedded spin networksrdquo equiva-lence classes of these under diffeomorphisms are denoted s and are calledldquoabstract spin networksrdquo or s-knots A quantum state of space is deter-mined by an s-knot7

The fact that spin networks do not live in space but rather are spacehas far-reaching consequences Space itself turns out to have a discrete andcombinatorial character Notice that this is not imposed on the theory orassumed It is the result of a completely conventional quantum mechanicalcalculation of the spectrum of the physical quantities that describe thegeometry of space Since there is no spatial continuity at short scale thereis (literally) no room in the theory for ultraviolet divergencies The theoryeffectively cuts itself off at the Planck scale Space is effectively granularat the Planck scale and there is no infinite ultraviolet limit

Chapter 7 describes how YangndashMills and fermion fields can be coupledto the theory This can be obtained by enriching the structure of the spinnetworks s In the case of a YangndashMills theory with gauge group G forinstance links carry an additional quantum number labeling irreduciblerepresentations G The spin network itself behaves like the lattice of lat-tice YangndashMills theory In quantum gravity therefore the lattice itselfbecomes a dynamical variable But notice a crucial difference with re-spect to conventional lattice YangndashMills theory the lattice size is not tobe scaled down to zero it has physical Planck size

In summary spin networks provide a mathematically well-defined andphysically compelling description of the kinematics of the quantum grav-itational field They also provide a well-defined picture of the small-scale

7The expression ldquospin networkrdquo is used in the literature to designate both the embeddedand the abstract ones as well as to designate the quantum states they label


structure of space It is remarkable that this novel picture of space emergessimply from the combination of old YangndashMills theory ideas with general-relativistic background independence

123 Dynamics in background-independent QFT

The dynamics of the quantum gravitational field can be described givingamplitudes W (s) for spin network states Let me illustrate here in aheuristic manner the physical interpretation of these amplitudes and theway they are defined in the theory A major feature of this book is thatit is based on a general-relativistic way of thinking about observables +evolution This section sketches this view and may be somewhat harderto follow than the previous ones

Interpretation of the amplitude W (s) The quantum dynamics of a par-ticle is entirely described by the transition probability amplitudes

W (x t xprime tprime) = 〈x|eminus iH0(tminustprime)|xprime〉 = 〈x t|xprime tprime〉 (112)

where |x t〉 is the eigenstate of the Heisenberg position operator x(t)with eigenvalue x H0 is the hamiltonian operator and |x〉 = |x 0〉 Thepropagator W (x txprime tprime) depends on two events (x t) and (xprime tprime) thatbound a finite portion of a classical trajectory The space of the pairs ofevents (x t xprime tprime) is called G in this book

A physical experiment consists of a preparation at time tprime and a mea-surement at time t Say that in a particular experiment we have localizedthe particle in xprime at tprime and then found it in x at time t The set (x t xprime tprime)represents the complete set of data of a specific complete observationalset up including preparation and measurement The space G is the spaceof these data sets In the quantum theory we associate the complex am-plitude W (x t xprime tprime) which is a function on G with any such data set Asemphasized by Feynman this amplitude codes the full quantum dynamicsFollowing Feynman we can compute W (x t xprime tprime) with a sum-over-pathsthat take the values x and xprime at t and tprime respectively

If we measure a different observable than position we obtain statesdifferent from the states |x〉 Let |ψin〉 be the state prepared at time tprimeand let |ψout〉 be the state measured at time t The amplitude associatedto these measurements is

A = 〈ψout|eminusiH0(tminustprime)|ψin〉 (113)

The pair of states (ψin ψout) determines a state ψ = |ψin〉 otimes 〈ψout| inthe space Kttprime which is the tensor product of the Hilbert space of theinitial states and (the dual of) the Hilbert space of the final states The


propagator defines a (possibly generalized) state |0〉 in Kttprime by 〈0|(|xprime〉 otimes〈x|) = W (x t xprime tprime) The amplitude (113) can be written simply as

A = 〈0|ψ〉 (114)

Therefore we can express the dynamics from tprime to t in terms of a singlestate |0〉 in a Hilbert space Kttprime that represents outcomes of measurementson tprime and t The state |0〉 is called the covariant vacuum and should notbe confused with the state of minimal energy

Let us extend this idea to field theory In field theory the analog ofthe data set (x t xprime tprime) is the couple [Σ ϕ] where Σ is a 3d surfacebounding a finite spacetime region and ϕ is a field configuration on ΣThese data define a set of events (x isin Σ ϕ(x)) that bound a finite portionof a classical configuration of the field just as (x t xprime tprime) bound a finiteportion of the classical trajectory of the particle The data from a localexperiment (measurements preparation or just assumptions) must in factrefer to the state of the system on the entire boundary of a finite spacetimeregion The field theoretical space G is therefore the space of surfaces Σand field configurations ϕ on Σ Quantum dynamics can be expressed interms of an amplitude W [Σ ϕ] Following Feynmanrsquos intuition we canformally define W [Σ ϕ] in terms of a sum over bulk field configurationsthat take the value ϕ on the boundary Σ In fact in Section 53 I arguethat the functional W [Σ ϕ] captures the dynamics of a QFT

Notice that the dependence of W [Σ ϕ] on the geometry of Σ codes thespacetime position of the measuring apparatus In fact the relative posi-tion of the components of the apparatus is determined by their physicaldistance and the physical time lapsed between measurements and thesedata are contained in the metric of Σ

Consider now a background-independent theory Diffeomorphism in-variance implies immediately that W [Σ ϕ] is independent of Σ This isthe analog of the independence of W (x y) from x and y mentioned inSection 114 Therefore in gravity W depends only on the boundaryvalue of the fields However the fields include the gravitational field andthe gravitational field determines the spacetime geometry Therefore thedependence of W on the fields is still sufficient to code the relativedistance and time separation of the components of the measuring ap-paratus

What is happening is that in background-dependent QFT we have twokinds of measurements those that determine the distances of the partsof the apparatus and the time lapsed between measurements and the ac-tual measurements of the fieldsrsquo dynamical variables In quantum gravityinstead distances and time separations are on an equal footing with thedynamical fields This is the core of the general-relativistic revolutionand the key for background-independent QFT


We need one final step Notice from (112) that the argument of W isnot the classical quantity but rather the eigenstate of the correspond-ing operator The eigenstates of the gravitational field are spin networksTherefore in quantum gravity the argument of W must be a spin networkrepresenting the possible outcome of a measurement of the gravitationalfield (or the geometry) on a closed 3d surface Thus in quantum gravityphysical amplitudes must be expressed by amplitudes of the form W (s)These give the correlation probability amplitude associated with the out-come s in a measurement of a geometry just as W (x t xprime tprime) does for aparticle

A particularly interesting case is when we can separate the boundarysurface in two components then s = sout cup sin In this case W (sout sin)can be interpreted as the probability amplitude of measuring the quantumthree-geometry sout if sin was observed

Notice that a spin network sin is the analog of (x t) not just x aloneThe time variable is mixed up with the physical variables (Chapter 3) Thenotion of unitary quantum evolution in time is ill defined in this contextbut probability amplitudes remain well defined and physically meaningful(Chapter 5) The quantum dynamical information of the theory is entirelycontained in the spin network amplitudes W (s) Given a configuration ofspace and matter these amplitudes determine a correlation probabilityof observing it

Calculation of the amplitude W (s) In the relativistic formulation of clas-sical hamiltonian theory dynamics is governed by the relativistic hamilto-nian H8 This is discussed in detail in Chapter 3 The quantum dynamicsis governed by the corresponding quantum operator H In quantum grav-ity H is defined on the space of the spin networks There is no externaltime variable t in the theory and the quantum dynamical equation whichreplaces the Schrodinger equation is the equation HΨ = 0 called theWheelerndashDeWitt equation The space of the solutions of the WheelerndashDeWitt equation is denoted H There is an operator P Kdiff rarr H thatprojects Kdiff on the solutions of the WheelerndashDeWitt equation (for amathematically more precise statement see Section 52)

The transition amplitudes W (s sprime) are the matrix elements of the oper-ator P They define the physical scalar product namely the scalar productof the space H

W (s sprime) = 〈s|P |sprime〉Kdiff= 〈s|sprime〉H (115)

Thus the transition amplitude between two states is simply their physicalscalar product (Chapter 5) More generally there is a preferred state |empty〉

8H is sometimes called the ldquohamiltonian constraintrdquo or the ldquosuperhamiltonianrdquo


Fig 14 Scheme of the action of H on a node of a spin network

in Kdiff which is formed by no spin networks It represents a space withzero volume or more precisely no space at all The covariant vacuumstate which defines the dynamics of the theory is defined by |0〉=P |empty〉The amplitude of a spin network is defined by

W (s) = 〈0|s〉 = 〈empty|P |s〉 (116)

The construction of the operator H is a major task in LQG It is delicateand it requires a nontrivial regularization procedure in order to deal withoperator products Chapter 7 is devoted to this construction Remarkablythe limit in which the regularization is removed exists precisely thanksto diffeomorphism invariance (Section 71) This is a second major payoffof background independence At present more than one version of theoperator H has been constructed and it is not yet clear which variant(if any) is correct The remarks that follow refer to all of them

The most remarkable aspect of the hamiltonian operator H is that itacts only on the nodes A state labeled by a spin network without nodes ndashthat is in which the graph Γ is simply a collection of nonintersectingloops ndash is a solution of the WheelerndashDeWitt equation In fact the unex-pected fact that exact solutions of the WheelerndashDeWitt equation couldbe found at all was the first major surprise that raised interest in LQGin the first place in the late 1980s

Acting on a generic state |s〉 the action of the operator H turns out tobe discrete and combinatorial the topology of the graph is changed andthe labels are modified in the vicinity of a node A typical example of theaction of H on a node is illustrated in Figure 14 the action on a nodesplits the node into three nodes and multiplies the state by a number a(that depends on the labels of the spin network around the node) Labelsof links and nodes are not indicated in the figure

Notice the various manners in which the spin network basis is effectivein quantum gravity The states in the spin network basis

(i) diagonalize area and volume(ii) control diff-invariance diffeomorphism equivalence classes of states

are labeled by the s-knots(iii) simplify the action of H reducing it to a combinatorial action on

the nodes


The construction of the hamiltonian operator H completes the defini-tion of the general formalism of LQG in the case of pure gravity This isextended to matter couplings in Chapter 7 In Chapter 8 I describe someof the most interesting applications of the theory In particular I illustratethe application of LQG to cosmology (control of the classical initial sin-gularity inflation) and to black-hole physics (entropy emitted spectrum)I also mention some of its tentative applications in astrophysics

124 Quantum spacetime spinfoam

To be able to compute all the predictions of a theory it is not sufficient tohave the general definition of the theory A road towards the calculationof transition amplitudes in quantum gravity is provided by the spinfoamformalism

Following Feynmanrsquos ideas we can give W (s sprime) a representation as asum-over-paths This representation can be obtained in various mannersIn particular it can be intuitively derived from a perturbative expansionsumming over different histories of sequences of actions of H that send sprime

into sA path is then the ldquoworld-historyrdquo of a graph with interactions hap-

pening at the nodes This world-history is a two-complex as in Figure15 namely a collection of faces (the world-histories of the links) facesjoin at edges (the world-histories of the nodes) in turn edges join atvertices A vertex represents an individual action of H An example of avertex corresponding to the action of H of Figure 14 is illustrated inFigure 16 Notice that on moving from the bottom to the top a sectionof the two-complex goes precisely from the graph on the left-hand side ofFigure 14 to the one on the right-hand side Thus a two-complex is likea Feynman graph but with one additional structure A Feynman graph iscomposed by vertices and edges a spinfoam by vertices edges and faces

Faces are labeled by the area quantum numbers jl and edges by thevolume quantum numbers in A two-complex with faces and edges la-beled in this manner is called a ldquospinfoamrdquo and denoted σ Thus a spin-foam is a Feynman graph of spin networks or a world-history of spinnetworks A history going from sprime to s is a spinfoam σ bounded by sprime

and sIn the perturbative expansion of W (s sprime) there is a term associated

with each spinfoam σ bounded by s and sprime This term is the amplitude ofσ The amplitude of a spinfoam turns out to be given by (a measure termμ(σ) times) the product over the vertices v of a vertex amplitude Av(σ)The vertex amplitude is determined by the matrix element of H betweenthe incoming and the outgoing spin networks and is a function of the labels


v2

v1

5

56

7

8

8

1

3

7

63

3

si

sf

s1

Σi

Σf

Fig 15 A spinfoam representing the evolution of an initial spin network si toa final spin network sf via an intermediate spin network s1 Here v1 and v2 arethe interaction vertices

Fig 16 The vertex of a spinfoam

of the faces and the edges adjacent to the vertex This is analogous to theamplitude of a conventional Feynman vertex which is determined by thematrix element of the hamiltonian between the incoming and outgoingstates


The physical transition amplitudes W (s sprime) are then obtained by sum-ming over spinfoams bounded by the spin networks s and sprime

W (s sprime) simsum

σpartσ=scupsprime

μ(σ)prod

v

Av(σ) (117)

More generally for a spin network s representing a closed surface

W (s) simsum

σpartσ=s

μ(σ)prod

v

Av(σ) (118)

In general the Feynman path integral can be derived from Schrodingertheory by exponentiating the hamiltonian operator but it can also be di-rectly interpreted as a sum over classical trajectories of the particle Simi-larly the spinfoam sum (117) can be interpreted as a sum over spacetimesThat is the sum (117) can be seen as a concrete and mathematicallywell-defined realization of the (ill-defined) WheelerndashMisnerndashHawking rep-resentation of quantum gravity as a sum over four-geometries

W (3g 3gprime) simint

partg= 3gcup3gprime[Dg] e

iS[g] (119)

Because of their foamy structure at the Planck scale spinfoams canbe viewed as a mathematically precise realization of Wheelerrsquos intuitionof a spacetime ldquofoamrdquo In Chapter 9 I describe various concrete realiza-tions of (117) as well as the possibility of directly relating (117) with adiscretization of (119)

13 Conceptual issues

The search for a quantum theory of gravity raises questions such as Whatis space What is time What is the meaning of ldquobeing somewhererdquoWhat is the meaning of ldquomovingrdquo Is motion to be defined with respectto objects or with respect to space Can we formulate physics withoutreferring to time or to spacetime And also What is matter What iscausality What is the role of the observer in physics

Questions of this kind have played a central role in periods of majoradvances in physics For instance they played a central role for EinsteinHeisenberg Bohr and their colleagues but also for Descartes GalileoNewton and their contemporaries and for Faraday Maxwell and theircolleagues Today this manner of posing problems is often regarded asldquotoo philosophicalrdquo by many physicists

Indeed most physicists of the second half of the twentieth century haveviewed questions of this nature as irrelevant This view was appropriate

13 Conceptual issues 29

for the problems they were facing one does not need to worry aboutfirst principles in order to apply the Schrodinger equation to the heliumatom to understand how a neutron star holds together or to find out thesymmetry group governing the strong interactions During this periodphysicists lost interest in general issues As someone has said during thisperiod ldquodo not ask what the theory can do for you ask what you cando for the theoryrdquo That is do not ask foundational questions just keepdeveloping and adjusting the theory you happen to find in front of youWhen the basics are clear and the issue is problem-solving within a givenconceptual scheme there is no reason to worry about foundations theproblems are technical and the pragmatical approach is the most effectiveone

Today the kind of difficulties that we face have changed To understandquantum spacetime we have to return once more to those foundationalissues We have to find new answers to the old foundational questions Thenew answers have to take into account what we have learned with QM andGR This conceptual approach is not the one of Weinberg and Gell-Mannbut it is the one of Newton Maxwell Einstein Bohr Heisenberg FaradayBoltzmann and many others It is clear from the writings of the latterthat they discovered what they did discover by thinking about generalfoundational questions The problem of quantum gravity will not be solvedunless we reconsider these questions

Several of these questions are discussed in the text Here I only commenton one of these conceptual issues the role of the notion of time

131 Physics without time

The transition amplitudes W (s sprime) do not depend explicitly on time Thisis to be expected because the physical predictions of classical GR do notdepend explicitly on the time coordinate t either The theory predictscorrelations between physical variables not the way physical variablesevolve with respect to a preferred time variable But what is the meaningof a physical theory in which the time variable t does not appear

Let me tell a story It was Galileo Galilei who first realized that thephysical motion of objects on Earth could be described by mathematicallaws expressing the evolution of observable quantities ABC in timeThat is laws for the functions A(t) B(t) C(t) A crucial contributionby Galileo was to find an effective way to measure the time variable tand therefore provide an operational meaning to these functions In factGalileo gave a decisive contribution to the discovery of the modern clockby realizing as a young man that the small oscillations of a pendulumldquotake equal timerdquo The story goes that Galileo was staring at the slowoscillations of the big chandelier that can still be seen in the marvelous


Cathedral of Pisa9 He checked the period of the oscillations against hispulse and realized that the same number of pulses lapsed during anyoscillation of the chandelier This was the key insight the basis of themodern clock today virtually every clock contains an oscillator Laterin life Galileo used a clock to discover the first quantitative terrestrialphysical law in his historic experiments on descent down inclines

Now the puzzling part of the story is that while Galileo checked thependulum against his pulse not long afterwards doctors were checkingtheir patientrsquos pulse against a pendulum What is the actual meaning ofthe pendulum periods taking ldquoequal timerdquo An equal amount of t lapsesin any oscillation how do we know this if we can access t only via anotherpendulum

It was Newton who cleared up the issue conceptually Newton as-sumes that an unobservable quantity t exists which flows (ldquoabsolute andequal to itselfrdquo) We write equations of motion in terms of this t butwe cannot truly access t we can build clocks that give readings T1(t)T2(t) that according to our equations approximate t with the preci-sion we want What we actually measure is the evolution of other variablesagainst clocks namely A(T1) B(T1) Furthermore we can check clocksagainst one another by measuring the functions T1(T2) T2(T3) Thefact that all these observations agree with what we compute using evo-lution equations in t gives us confidence in the method In particularit gives us confidence that to assume the existence of the unobservablephysical quantity t is a useful and reasonable thing to do

Simply the usefulness of this assumption is lost in quantum gravity Thetheory allows us to calculate the relations between observable quantitiessuch as A(B) B(C) A(T1) T1(A) which is what we see But it doesnot give us the evolution of these observable quantities in terms of anunobservable t as Newtonrsquos theory and special relativity do In a sensethis simply means that there are no good clocks at the Planck scale

Of course in a specific problem we can choose one variable decide totreat it as the independent variable and call it ldquotherdquo time For instance acertain clock time a certain proper time along a certain particle historyetc The choice is largely arbitrary and generally it is only locally meaning-ful A generally covariant theory does not choose a preferred time variable

Here are two examples to illustrate this arbitrariness- Imagine we throw a precise clock upward and compare its lapsed reading tf when it

lands back with the lapsed reading te of a clock remaining on the Earth GR predictsthat the two clocks read differently and provides a quantitative relation between tf

9Nice story Too bad the chandelier was hung there a few decades after Galileorsquos dis-covery

Bibliographical notes 31

and te Is this about the observable tf evolving in the physical time te or about theobservable te evolving in the physical time tf

10

- The cosmological context is often indicated as one in which a natural choice oftime is available the cosmological time tc is the proper time from the Big Bang alongthe galaxiesrsquo worldlines But an event A happening on Andromeda at the same tc asours happens much later than an event B on Andromeda simultaneous to us in thesense of Einsteinrsquos definition of simultaneity11 So what is happening ldquoright nowrdquo onAndromeda A or B Furthermore the real world is not truly homogeneous when twogalaxies having two different ages relative to the Big Bang or two different massesme merge which of the two has the right time

So long as we remain within classical general relativity a given gravi-tational field has the structure of a pseudo-riemannian manifold There-fore the dynamics of the theory has no preferred time variable but wenevertheless have a notion of spacetime for each given solution But inquantum theory there are no classical field configurations just as thereare no trajectories of a particle Thus in quantum gravity the notion ofspacetime disappears in the same manner in which the notion of trajec-tory disappears in the quantum theory of a particle A single spinfoam canbe thought of as representing a spacetime but the history of the world isnot a single spinfoam it is a sum over spinfoams

The theory is conceptually well defined without making use of the no-tion of time It provides probabilistic predictions for correlations betweenthe physical quantities that we can observe In principle we can checkthese predictions against experiments12 Furthermore the theory providesa clear and intelligible picture of the quantum gravitational field namelyof a ldquoquantum geometryrdquo

Thus there is no background ldquospacetimerdquo forming the stage on whichthings move There is no ldquotimerdquo along which everything flows The worldin which we happen to live can be understood without using the notionof time

mdashmdash

Bibliographical notes

The fact that perturbative quantum general relativity is nonrenormaliz-able has been long believed but was proven only in 1986 by Goroff andSagnotti [29]

10If you are tempted to say that the lapsed reading te of the clock remaining on Earthgives the ldquotrue timerdquo recall that the pseudo-riemannian distance between the twoevents at which the clocks meet is tf not te it is the clock going up and down thatfollows a geodesic

11Thanks to Marc Lachieze-Rey for this observation12The special properties of a time variable may emerge only macroscopically This is

discussed in Sections 34 and 551


For an orientation on current research on quantum gravity see for in-stance the review papers [30ndash33] An interesting panoramic of points ofview on the problem is in the various contributions to the book [34] Ihave given a critical discussion on the present state of spacetime physicsin [35ndash37] A historical account of development of quantum gravity isgiven in Appendix B

As a general introduction to quantum gravity ndash a subject where nothingyet is certain ndash the student eager to learn is strongly advised to study alsothe classic reviews which are rich in ideas and present different points ofview such as John Wheeler 1967 [38] Steven Weinberg 1979 [39] StephenHawking 1979 and 1980 [4041] Karel Kuchar 1980 [42] and Chris Ishamrsquosmagistral syntheses [43ndash45] On string theory classic textbooks are GreenSchwarz and Witten and Polchinksi [46] For a discussion of the difficul-ties of string theory and a comparison of the results of strings and loopssee [47] written in the form of a dialog and [48] For a fascinating pre-sentation of Alain Connesrsquo vision see [49] Lee Smolinrsquos popular-sciencebook [50] provides a readable and enjoyable introduction to LQG

LQG has inspired novels and short stories Blue Mars by Kim StanleyRobinson [51] contains a description of the future evolution and merg-ing of loop gravity and strings I recommend the science fiction novelSchild Ladder by Greg Egan [52] which opens with one of the clearestpresentations of the picture of space given by loop gravity (Greg is a tal-ented writer and also a scientist who is contributing to the development ofLQG) and for those who can read Italian Anna prende il volo by EnricoPalandri [53] a charming novel with a gentle meditation on the meaning ofthe disappearance of time Literature has the capacity of delicately merg-ing the novel hard views that science develops into the common discourseof our civilization

2General Relativity

Lev Landau has called GR ldquothe most beautifulrdquo of the scientific theories The theoryis first of all a description of the gravitational force Nowadays it is very extensivelysupported by terrestrial and astronomical observations and so far it has never beenquestioned by an empirical observation

But GR is far more than that It is a complete modification of our understanding ofthe basic grammar of nature This modification does not apply solely to gravitationalinteraction it applies to all aspects of physics In fact the extent to which Einsteinrsquosdiscovery of this theory has modified our understanding of the physical world and thefull reach of its consequences have not yet been completely unraveled

This chapter is not an introduction to GR nor an exhaustive description of thetheory For this I refer the reader to the classic textbooks on the subject Here I givea short presentation of the formalism in a compact and modern form emphasizingthe reading of the theory which is most useful for quantum gravity I also discuss indetail the physical and conceptual basis of the theory and the way it has modified ourunderstanding of the physical world

21 Formalism

211 Gravitational field

Let M be the ldquospacetimerdquo four-dimensional manifold Coordinates onM are written as x x where x = (xμ) = (x0 x1 x2 x3) Indicesμ ν = 0 1 2 3 are spacetime tangent indices

bull The gravitational field e is a one-form

eI(x) = eIμ(x) dxμ (21)

with values in Minkowski space Indices I J = 0 1 2 3 label the com-ponents of a Minkowski vector They are raised and lowered with theMinkowski metric ηIJ

33

34 General Relativity

I call ldquogravitational fieldrdquo the tetrad field rather than Einsteinrsquos metric field gμν(x)There are three reasons for this (i) the standard model cannot be written in terms of gbecause fermions require the tetrad formalism (ii) the tetrad field e is nowadays moreutilized than g in quantum gravity and (iii) I think that e represents the gravitationalfield in a more conceptually clean way than g (see Section 223) The relation with themetric formalism is given in Section 215

bull The spin connection ω is a one-form with values in the Lie algebra ofthe Lorentz group so(3 1)

ωIJ(x) = ωI

μJ(x) dxμ (22)

where ωIJ = minusωJI It defines a covariant partial derivative Dμ on allfields that have Lorentz (I) indices

DμvI = partμv

I + ωIμJ vJ (23)

and a gauge-covariant exterior derivative D on forms For instance for aone-form uI with a Lorentz index

DuI = duI + ωIJ and uJ (24)

The torsion two-form is defined as

T I = DeI = deI + ωIJ and eJ (25)

A tetrad field e determines uniquely a torsion-free spin connection ω =ω[e] called compatible with e by

T I = deI + ω[e]IJ and eJ = 0 (26)

The explicit solution of this equation is given below in (291) or (292)

bull The curvature R of ω is the Lorentz algebra valued two-form1

RIJ = RI

J μν dxμ and dxν (27)

defined by2

RIJ = dωI

J + ωIK and ωK

J (28)

1Generally I write spacetime indices μν before internal Lorentz indices IJ But for thecurvature I prefer to stay closer to Riemannrsquos notation

2Sometimes the curvature of a connection ωIJ is written as RI

J = DωIJ If we naively

use the definition (24) for D we get an extra 2 in the quadratic term The point isthat the indices on the connection are not vector indices That is (24) defines theaction of D on sections of a vector bundle and a connection is not a section of a vectorbundle

21 Formalism 35

We have then immediately from (24)

D2uI = RIJ and uJ (29)

and from this equation and (26)

RIJ and eJ = 0 (210)

A region where the curvature is zero is called ldquoflatrdquo Equations (25) and(28) are called the Cartan structure equations

bull The Einstein equations ldquoin vacuumrdquo are

εIJKL (eI andRJK minus 23λ eI and eJ and eK) = 0 (211)

The equation (26) relating e and ω and the Einstein equations (211)are the field equations of GR in the absence of other fields They are theEulerndashLagrange equations of the action

S[e ω] =1

16πG

intεIJKL(

14eIandeJandR[ω]KLminus 1

12λ eIandeJandeKandeL) (212)

where G is the Newton constant3 and λ is the cosmological constantwhich I often set to zero below

bull Inverse tetrad Using the matrix eμI (x) defined to be the inverse of the matrixeIμ(x) we define the Ricci tensor

RIμ = RIJ

μν eνJ (213)

and the Ricci scalar

R = RIμ eμI (214)

and write the vacuum Einstein equations (211) as

RIμ minus 1

2ReIμ + λeIμ = 0 (215)

3The constant 16πG has no effect on the classical equations of motion (211) Howeverit governs the strength of the interaction with the matter fields described below andit also determines the quantum properties of the system In this it is similar to themass constant m in front of a free-particle action the classical equations of motion(x = 0) do not depend on m but the quantum dynamics of the particle does Forinstance the rate at which a wave packet spreads depends on m Similarly we willsee that the quanta of pure gravity are governed by this constant


bull Second-order formalism Replacing ω with ω[e] in (212) we get the equivalentaction

S[e] =1

16πG

intεIJKL (

1

4eI and eJ andR[ω[e]]KL minus 1

12λ eI and eJ and eK and eL) (216)

The formalisms in (212) where e and ω are independent is called the first-order for-malism The two formalism are not equivalent in the presence of fermions we do notknow which one is physically correct because the effect of gravity on single fermions ishard to measure

bull Selfdual formalism Consider the selfdual ldquoprojectorrdquo P iIJ given by

P ijk =

1

2εijk P i

0j = minusP ij0 =

i

2δij (217)

where i = 1 2 34 This verifies the two properties

1

2εIJ

KLP iKL = iP i

IJ P IJi P i

KL = P IJKL equiv 1

2δ I[K δLJ] +

i

4εIJKL (218)

where P IJKL is the projector on selfdual tensors Define the complex SO(3) connection

Aiμ = P i

IJ ωIJμ (219)

Equivalently

Ai = ωi + iω0i (220)

(We write ωi = 12εijkωjk See pg xxii) We can use the complex selfdual connection

Ai (three complex one-forms) instead of the real connection ωIJ (six real one-forms)

as the dynamical variable for GR (This is equivalent to describing a system with tworeal degrees of freedom x and y in terms of a single complex variable z = x + iy) Interms of Ai the vacuum Einstein equations read

PiIJ eI and (F i minus 2

3λ P i

KLeK and eL) = 0 (221)

where F i = dAi + εijkAjAk is the curvature of A5 These are the EulerndashLagrange

equations of the action

S[eA] =1

16πG

int(minusiPiIJ eI and eJ and F i minus 1

12λ εIJKL eI and eJ and eK and eL) (222)

which differs from the action (212) by an imaginary term that does not change theequations of motion The selfdual formalism is often used in canonical quantizationbecause it simplifies the form of the hamiltonian theory If we replace the imaginaryunit i in (217) with a real parameter γ (222) is called the Holst action [54] and givesrise to the Ashtekar-Barbero-Immirzi formalism γ is called the Immirzi parameter

Plebanski formalism The Plebanski selfdual two-form is defined as

Σi = P iIJ eI and eJ (223)

That isΣ1 = e2 and e3 + i e0 and e1 (224)

4The complex Lorentz algebra splits into two complex so(3) algebras called the selfdualand anti-selfdual components so(3 1C) = so(3C) oplus so(3C) The projector (217)reads out the selfdual component

5Because of the split mentioned in the previous footnote the curvature of the selfdualcomponent of the connection is the selfdual component of the curvature

21 Formalism 37

and so on cyclically A straightforward calculation shows that Σ satisfies

DΣi equiv dΣi + Aij and Σj = 0 (225)

where we write Aij = εijkA

k See pg xxii The algebraic equations for a triplet ofcomplex two-forms Σi

3 Σi and Σj = δij Σk and Σk = minusδij Σk and Σk Σi and Σ

j= 0 (226)

are solved by (223) where eI is an arbitrary real tetrad The GR action can thus bewritten as

S[Σ A] =minusi

16πG

int (Σi and F i +

1

3λ Σk and Σk)

(227)

where Σi satisfies the Plebanski constraints (226) The Plebanski formalism is often

used as a starting point for spinfoam models

212 ldquoMatterrdquo

In the general-relativistic parlance ldquomatterrdquo is anything which is not thegravitational field As far as we know the world is made up of the grav-itational field YangndashMills fields fermion fields and presumably scalarfields

bull Maxwell The electromagnetic field is described by the one-form fieldA the Maxwell potential

A(x) = Aμ(x) dxμ (228)

Its curvature is the two-form F = dA with components Fμν = partμAν minuspartνAμ Its dynamics is governed by the action

SM[eA] =14

intF lowast and F (229)

bull YangndashMills The above generalizes to a nonabelian connection A ina YangndashMills group G A defines a gauge covariant exterior derivative Dand curvature F The action is

SYM[eA] =14

inttr[F lowast and F ] (230)

where tr is a trace on the algebra

bull Scalar Let ϕ(x) be a scalar field possibly with values in a representa-tion of G The YangndashMills field A defines the covariant partial derivative

Dμϕ = partμϕ + AAμLAϕ (231)

where LA are the generators of the gauge algebra in the representationsto which ϕ belongs The action that governs the dynamics of the field is

Ssc[eA ϕ] =int

d4x e(ηIJ eμI Dμϕ eνJ Dνϕ + V (ϕ)

) (232)

where e is the determinant of eIμ and V (ϕ) is a self-interaction potential


bull Fermion A fermion field ψ is a field in a spinor representation ofthe Lorentz group possibly with values in a representation of G Thespin connection ω and the YangndashMills field A define the covariant partialderivative

Dμψ = partμψ + ωIμJL

JIψ + AA

μLAψ (233)

where LJI and LA are the generators of the Lorentz and gauge algebras

in the representations to which ψ belongs Define

Dψ = γIeμI Dμψ (234)

where γI are the standard Dirac matrices The action that governs thedynamics of the fermion field is

Sf [e ωA ϕ ψ] =int

d4x e(ψ Dψ + Y (ϕ ψ ψ)

)+ complex conjugate

(235)where the second term is a polynomial interaction potential with a scalarfield

bull The ldquolagrangian of the worldrdquo the standard model As far as we knowthe world can be described in terms of a set of fields e ωA ψ ϕ whereG = SU(3) times SU(2) times U(1) and ψ and ϕ are in suitable multiplets andis governed by the action

S[e ωA ψ ϕ] = SGR[e ω] + SYM[eA] + Sf [e ωA ψ] + Ssc[eA ϕ]= SGR[e ω] + Smatter[e ωA ϕ ψ] (236)

with suitable polynomials V and Y The equations of motion that followfrom this action by varying e are the Einstein equations (211) with asource term namely

εIJKL (eI andRJK minus 23λ eI and eJ and eK) = 2πG TL (237)

where the energy-momentum three-form

TI =det e3

TμI εμνρσdxν and dxρ and dxσ (238)

is defined by

TI(x) =δSmatter

δeI(x) (239)

Equivalently the Einstein equations (237) can be written as

RIμ minus 1

2ReIμ + λeIμ = 8πG T I

μ (240)

21 Formalism 39

T Iμ(x) is called the energy-momentum tensor It is the sum of the individ-

ual energy-momentum tensors of the various matter terms6

bull Particles The trajectory xμ(s) of a point particle is an approximate notion Macro-scopic objects have finite size and elementary particles are quantum entities and there-fore have no trajectories At macroscopic scales the notion of a point-particle trajectoryis nevertheless very useful

In the absence of nongravitational forces the equations of motion for the worldlineγ s rarr xμ(s) of a particle are determined by the action

S[e γ] = m

intds

radicminusηIJvI(s)vJ(s) (241)

wherevI(s) = eIμ(x(s))vμ(s) (242)

and vμ is the particle velocity

vμ(s) = xμ(s) equiv dxμ(s)

ds (243)

This action is independent of the way the trajectory is parametrized and thereforedetermines the path not its parametrization With the parametrization choice vIv

I =minus1 the equations of motion are

xμ = minusΓμνρ xν xρ (244)

whereΓσμν = eρJe

Jσ(eρIpart(μeIν) + eνIpart[μe

Iρ] + eμIpart[νe

Iρ]) (245)

is called the LevindashCivita connection In an arbitrary parametrization the equations ofmotion are

xμ + Γμνρ xν xρ = I(s) xμ (246)

where I(s) is an arbitrary function of s

Minkowski solution Consider a regime in which we can assume that theNewton constant G is small that is a regime in which we can neglectthe effect of matter on the gravitational field Assume also that withinour approximation the cosmological constant λ is negligible The Ein-stein equations (211) then admit (among many others) the particularlyinteresting solution

eIμ(x) = δIμ ωIμJ(x) = 0 (247)

which is called the Minkowski solution This solution is everywhere flatAssume that the gravitational field is in this configuration What are

the equations of motion of the matter interacting with this particular

6The energy-momentum tensor defined as the variation of the action with respect tothe gravitational field may differ by a total derivative from the one conventional inMinkowski space defined as the Noether current of translations


gravitational field These are easily obtained by inserting the Minkowskisolution (247) into the matter action (236)

S[Aϕ ψ] = Smatter[e = δ ω = 0 A ϕ ψ] (248)

The action S[Aϕ ψ] is the action of the standard model used in high-energy physics This action is usually written in terms of the spacetimeMinkowski metric ημν This metric is obtained from the Minkowski value(247) of the tetrad field For instance in the action of a scalar field (232)the combination ηIJeμI (x)eνJ(x) becomes

ηIJeμI (x)eνJ(x) = ηIJδμI δνJ = ημν (249)

on this solutionThe Minkowski metric ημν of special-relativistic physics is nothing but

a particular value of the gravitational field It is one of the solutions ofthe Einstein equation within a certain approximation

213 Gauge invariance

The general definition of a system with a gauge invariance and the onewhich is most useful for understanding the physics of gauge systems is thefollowing which is due to Dirac Consider a system of evolution equationsin an evolution parameter t The system is said to be ldquogaugerdquo invariantif evolution is under-determined that is if there are two distinct solu-tions that are equal for t less than a certain t see Figure 21 These twosolutions are said to be ldquogauge equivalentrdquo Any two solutions are saidto be gauge equivalent if they are gauge equivalent (as above) to a thirdsolution The gauge group G is a group that acts on the physical fields andmaps gauge-equivalent solutions into one another Since classical physics isdeterministic under-determined evolution equations are physically consis-tent only under the stipulation that only quantities invariant under gaugetransformations are physical predictions of the theory These quantitiesare called the gauge-invariant observables

The equations of motion derived by the action (236) are invariant underthree groups of gauge transformations (i) local YangndashMills gauge trans-formations (ii) local Lorentz transformations and (iii) diffeomorphismtransformations They are described below Gauge-invariant observablesmust be invariant under these three groups of transformations

(i) Local G transformations G is the YangndashMills group A local G transformationis labeled by a map λ M rarr G It acts on ϕψ and the connection A in the

21 Formalism 41

t

j

t

~j (t)j(t)

Fig 21 Dirac definition of gauge two different solutions of the equations ofmotion must be considered gauge equivalent if they are equal for t lt t

well-known form while e and ω are invariant

λ ϕ(x) rarr Rϕ(λ(x)) ϕ(x) (250)

ψ(x) rarr Rψ(λ(x)) ψ(x) (251)

Aμ(x) rarr R(λ(x)) Aμ(x) + λ(x)partμλminus1(x) (252)

eIμ(x) rarr eIμ(x) (253)

ωIμJ(x) rarr ωI

μJ(x) (254)

Here Rϕ and Rψ are the representations of G to which ϕ and ψ belong and Ris the adjoint representation

(ii) Local Lorentz transformations A local Lorentz transformation is labeled bya map λ M rarr SO(3 1) It acts on ϕψ and the connection ω precisely as aYangndashMills local transformation with YangndashMills group G=SO(3 1) Scalars ϕbelong to the trivial representation fermions ψ belong to the spinor representa-tions S The gravitational field e transforms in the fundamental representationExplicitly writing an element of SO(3 1) as λI

J we have

λ ϕ(x) rarr ϕ(x) (255)

ψ(x) rarr S(λ(x)) ψ(x) (256)

Aμ(x) rarr Aμ(x) (257)

eIμ(x) rarr λIJ(x) eJμ(x) (258)

ωIμJ(x) rarr λI

K(x)ωKμL(x)λL

J(x) + λ IK (x)partμλ

KJ(x) (259)

(iii) Diffeomorphisms Third and most important is the invariance under diffeo-morphisms A diffeomorphism gauge transformation is labeled by a smoothinvertible map φ M rarr M (that is by a ldquodiffeomorphismrdquo of M)7 It actsnonlocally on all the fields by pulling them back according to their form char-

7There is an unfortunate terminological imprecision A map φ M rarr M is called adiffeomorphism The associated transformations (260)ndash(264) on the fields are alsooften loosely called a diffeomorphism (also in this book) instead of diffeomorphismgauge transformations This tends to generate confusion


acter ϕ and ψ are zero forms e ω and A are one-forms8

φ ϕ(x) rarr ϕ(φ(x)) (260)

ψ(x) rarr ψ(φ(x)) (261)

Aμ(x) rarr partφν(x)

partxμAν(φ(x)) (262)

eIμ(x) rarr partφν(x)

partxμeIν(φ(x)) (263)

ωIμJ(x) rarr partφν(x)

partxμωIνJ(φ(x)) (264)

These three groups of transformations send solutions of the equationsof motion into other solutions of the equations of motion They are gaugetransformations because we can take these transformations to be the iden-tity before a given coordinate time t and different from the identity af-terwards Therefore they are responsible for the under-determination ofthe evolution equations Following Diracrsquos argument given above physicalpredictions of the theory must be given by quantities invariant under allthree of these transformations

In particular let a local quantity in spacetime be a quantity dependenton a fixed given point x Notice that such a quantity cannot be invariantunder a diffeomorphism Therefore no local quantity in spacetime (in thissense) is a gauge-invariant observable in GR The meaning of this fact andthe far-reaching consequences of diffeomorphism invariance are discussedbelow in Section 232

214 Physical geometry

At each point x of the spacetime manifold M the gravitational field eIμ(x)defines a map from the tangent space TxM to Minkowski space The mapsends a vector vμ in TxM into the Minkowski vector uI = eIμ(x)vμ TheMinkowski length |u| =

radicminusu middot u =radic

minusηIJuIuJ defines a norm |v| of thetangent vector vμ

|v| equiv |u| =radicminusηIJ(eIμ(x)vμ) (eJν (x)vν) (265)

8Under this definition internal Lorentz spinor and gauge indices do not transformunder a diffeomorphism Alternatively one should consider fiber-preserving diffeo-morphisms of the Lorentz and gauge bundle This alternative can be viewed as math-ematically more clean and physically more attractive because it makes more explicitthe fact that local inertial frames or local gauge choices at different spacetime pointscannot be identified (see later) However the mathematical description of a diffeo-morphism becomes more complicated while the two choices are ultimately physicallyequivalent due to the gauge invariance under local Lorentz and gauge transforma-tions The proper mathematical transformation of a spinor under diffeomorphisms isdiscussed in [55] and [56]

21 Formalism 43

|v| is called the ldquophysical lengthrdquo of the tangent vector v The tangentvector v is called timelike (spacelike or lightlike) if u is timelike (spacelikeor lightlike)

This fact allows us to assign a size to any d-dimensional surface in M At any point x on the surface the gravitational field maps the tangentspace of the surface into a surface in Minkowski space This surface carriesa volume form which can be pulled back to the tangent space of x andthen to the surface itself and integrated In particular

The length L of a curve γ s rarr xμ(s) is the line integral of the norm ofits tangent

L[e γ] =int

|dγ| =int

ds |u(s)| =int

dsradic

minusηIJ uI(s)uJ(s) (266)

whereuI(s) = eIμ(γ(s))

dxμ(s)ds

(267)

This can be written as the line integral of the norm of the one-formeI(x) = eIμ(x)dxμ along γ

L[e γ] =int

γ|e| (268)

The length is independent of the parametrization and the orien-tation of γ A curve is called timelike if its tangent is everywheretimelike Notice that the action of a particle (241) is nothing butthe length of its path in spacetime

S[e γ] = m L[e γ] (269)

The area A of a two-dimensional surface S σ= (σi) rarr xμ(σi) i= 1 2immersed in M is

A[eS] =int ∣∣d2S

∣∣ =int

Sd2σ

radicdet (ui middot uj) (270)

whereuIi (σ) = eIμ(γ(σ))

partxμ(σ)partσi

(271)

and the determinant is over the i j indices That is

A[eS] =int

d2σradic

(u1 middot u1)(u2 middot u2) minus (u1 middot u2)2 (272)

A surface is called spacelike if its tangents are all spacelike


The volume V of a three-dimensional region R σ = (σi) rarr xμ(σi) i =1 2 3 immersed in M is

V[eR] =int ∣

∣d3R∣∣ =

int

Rd3σ

radicn middot n (273)

wherenI = εIJKL uJ1u

K2 uL3 (274)

is normal to the surface A region is called spacelike if n is every-where timelike

The quantities L A and V are particular functions of the grav-itational field e The reason they have these geometric names isdiscussed below in Section 223

215 Holonomy and metric

In GR quantities close to observations such as lengths and areas arenonlocal in the sense that they depend on finite but extended regions inspacetime such as lines and surfaces Another natural nonlocal quantitywhich plays a central role in the quantum theory is the holonomy U ofthe gravitational connection (ω or its selfdual part A) along a curve γ

Definition of the holonomy Given a connection A in a group G overa manifold M the holonomy is defined as follows Let a curve γ be acontinuous piecewise smooth map from the interval [0 1] into M

γ [0 1] minusrarr M (275)s minusrarr xμ(s) (276)

The holonomy or parallel propagator U [A γ] of the connection A alongthe curve γ is the element of G defined by

U [A γ](0) = 11 (277)dds

U [A γ](s) minus γμ(s)Aμ

(γ(s)

)U [A γ](s) = 0 (278)

U [A γ] = U [A γ](1) (279)

where γμ(s) equiv dxμ(s)ds is the tangent to the curve (In the mathematicalliterature the term ldquoholonomyrdquo is generally used for closed curves only Inthe quantum gravity literature it is commonly employed for open curvesas well) The formal solution of this equation is

U [A γ] = P expint 1

0ds γμ(s) Ai

μ

(γ(s)

)τi equiv P exp

int

γA (280)

21 Formalism 45

where τi is a basis in the Lie algebra of the group G and the path orderedP is defined by the power series expansion

P expint 1

0dsA

(γ(s)

)

=infinsum

n=0

int 1

0ds1

int s1

0ds2 middot middot middot

int snminus1

0dsnA

(γ(sn)

)middot middot middotA

(γ(s1)

) (281)

The connection A is a rule that defines the meaning of parallel-transporting a vector in a representation R of G from a point of M toa nearby point the vector v at x is defined to be parallel to the vectorv +R(Aadxμ)v at x+ dx A vector is parallel-transported along γ to thevector R(U(A γ))v

An important property of the holonomy is that it transforms homoge-neously under the gauge transformation (252) of A That is U [Aλ γ] =λ(xγf )U [A γ]λminus1(xγi ) where xγif are the initial and final points of γ

A technical remark that we shall need later on the holonomy of anycurve γ is well defined even if there are (a finite number of) points whereγ is nondifferentiable and A is ill defined The reason is that we canbreak γ into components where everything is differentiable and define theholonomy of γ as the product of the holonomies of the components whichare well defined by continuity

Physical interpretation of the holonomy Consider two left-handed neutri-nos that meet at the spacetime point A separate and then meet again atthe spacetime point B Assume their spins are parallel at A and evolve un-der the sole influence of the gravitational field What is their relative spinat B A left-handed neutrino lives in the selfdual representation of theLorentz group and therefore its spin is parallel-transported by the selfdualconnection A Let γ1 and γ2 be the worldlines of the two neutrinos fromA to B and let γ = γminus1

2 γ1 be the loop formed by the two worldlines Ifthe first neutrino has spin ψ at B the second has spin ψprime = U(A γ)ψ Byhaving the two neutrinos interact we can in principle measure a quantitysuch as α = 2Re〈ψ|ψprime〉 which (assuming |ψ| = 1) gives the trace of theholonomy α = tr U [A γ]

Metric notation Einstein wrote GR in terms of the metric field HereI give the translation to metric variables Notice however that this isnecessarily incomplete since the fermion equations of motion cannot bewritten in terms of the metric field


The metric field g is a symmetric tensor field defined by

gμν(x) = eIμ(x) eJν (x) ηIJ (282)

At each point x of M g defines a scalar product in the tangent space TxM

(u v) = gμν(x)uμvν u v isin TxM (283)

and therefore maps TxM into T lowastxM In other words gμν and its inverse gμν can be used

to raise and lower tangent indices The fact that eμI (x) equiv ηIJgμνeJν (x) is the inverse

matrix of eIμ(x) is then a result not a definition

The metric-preserving linear connection Γ is the field Γρμν(x) defined by

Γρμν = eρI(partμe

Iν + ωI

μJ eJν ) (284)

It defines a covariant partial derivative Dμ on all fields that have tangent (μ) indices

Dμvν = partμv

ν + Γνμρv

ρ (285)

Together with ω it defines a covariant partial derivative Dμ on all objects thathave Lorentz as well as tangent indices In particular notice that (284) yieldsimmediately

DμeIν = partμe

Iν + ωI

μJ eJν minus Γρμν eIρ = 0 (286)

The antisymmetric part T ρμν = Γρ

μν minus Γρνμ of the linear connection gives the torsion

T I = eIρTρμνdxμdxν defined in (25)

The LevindashCivita connection is the (metric-preserving) linear connection determinedby e and ω[e] That is it is defined by

partμeIν + ω[e]IμJ eJν minus Γρ

μν eIρ = 0 (287)

whose solution is (245) It is torsion-free Notice that the antisymmetric part of thisequation is the first Cartan structure equation with vanishing torsion namely (26)which is sufficient to determine ω[e] as a function of e

The LevindashCivita connection is uniquely determined by g it is the unique torsion-freelinear connection that is metric preserving namely that satisfies

Dμgνρ = 0 (288)

or equivalently

partμgνρ minus Γσμνgσρ minus Γσ

μρgνσ = 0 (289)

This equation is solved by (245) or

Γρμν =

1

2gρσ(partμgσν + partνgμσ minus partσgμν) (290)

Notice that equations (287) and (290) allow us to write the explicit solution of theGR equation of motion (26)

ω[e]IμJ = eνJ(partμeIν minus Γρ

μνeIρ) (291)

21 Formalism 47

where Γ is given by (290) and g by (282) Explicitly this gives with a bit of algebra

ω[e]IJμ = 2 eν[Ipart[μeν]J] + eμKeνIeσJpart[σeν]

K (292)

The Riemann tensor can be defined via

Rμνρσ eIμ = RI

J ρσ eJν (293)

The Ricci tensor is

Rμν = RIμ eIν (294)

where RIμ is defined in (213) The energy-momentum tensor (see footnote 6 after (240))

Tμν = T Iμ eIν (295)

In terms of these quantities the Einstein equations (240) read

Rμν minus 1

2Rgμν + λgμν = 8πG Tμν (296)

The Minkowski solution isgμν(x) = ημν (297)

where we see clearly that the spacetime Minkowski metric is nothing but a particularvalue of the gravitational field With a straightforward calculation the action (212)reads

S[g] =1

16πG

int(R + λ)

radicminus det g d4x (298)

The matter action cannot be written in metric variables

Riemann geometry The tensor g equips the spacetime manifold M with a metric struc-ture it defines a distance between any two points and this distance is a smooth functionon M (More precisely it defines a pseudo-metric structure as distance can be imag-inary) Riemann studied the structure defined by (M g) called today a riemannianmanifold and defined the Riemann curvature tensor as a generalization of Gauss the-ory of curved surfaces to an arbitrary number of dimensions Riemann presented thismathematical theory as a general theory of ldquogeometryrdquo that generalizes Euclidean ge-ometry Einstein utilized this mathematical theory for describing the physical dynamicsof the gravitational field In retrospect the reason this was possible is because as un-derstood by Einstein the euclidean structure of the physical space in which we liveis determined by the local gravitational field Therefore elementary physical geometryis simply a description of the local properties of the gravitational field as revealed bymatter (rigid bodies) interacting with it This point is discussed in more detail belowin Section 223

mdashmdashmdashndash

The basic equations of GR presented in this section do not look too dif-ferent from the equations of a prerelativistic9 field theory such as QED orthe standard model But the similarity can be very misleading The phys-ical interpretation of a general-relativistic theory is very different fromthe interpretation of a prerelativistic one In particular the meaning of

9Recall that in this book ldquorelativisticrdquo means general relativistic


the coordinates xμ is different than in prerelativistic physics and thegauge-invariant observables are not related to the fields as they are inprerelativistic physics

The process of understanding the physical meaning of the GR formal-ism has taken many decades and perhaps it is not entirely concluded yetFor several decades after Einsteinrsquos discovery of the theory for instanceit was not clear whether or not the theory predicted gravitational wavesThe prevailing opinion was that wave solutions were only a coordinate ar-tifact and did not represent physical waves capable of carrying energy andmomentum or as Bondi put it capable of ldquoboiling a glass of waterrdquo Thisopinion was wrong of course Einstein himself badly misinterpreted themeaning of the Schwarzschild singularity Wrong high-precision measure-ments of the EarthndashMoon distance have been in the literature for a whilebecause of a mistake due to a conceptual confusion between physical andcoordinate distance

I do not want to give the impression that GR is ldquofoggyrdquo Quite thereverse the fact that in all these and similar instances consensus haseventually emerged indicates that the conceptual structure of GR is se-cure But to understand this conceptual structure to understand how touse the equations of GR correctly and how to relate the quantities ap-pearing in these equations to the numbers measured in the laboratoryor observed by the astronomers is definitely a nontrivial problem Moregenerally the problem is to understand what precisely GR says about theworld Clarity in this respect is essential if we want to understand thequantum physics of the theory

In order to shed light on this problem it is illuminating to retrace theconceptual path and the problems that led to the discovery of the theoryThis is done in the following Section 22 The impatient reader may skipSection 22 and jump to Section 23 where the interpretation of GR iscompactly presented (but impatience slows understanding)

22 The conceptual path to the theory

The roots of GR are in two distinct problems Einsteinrsquos genius was tounderstand that the two problems solve each other

221 Einsteinrsquos first problem a field theory for the newtonianinteraction

It was Newton who discovered dynamics But to a large extent it wasDescartes who a generation earlier fixed the general rules of the modernscience of nature or the Scientia Nova as it was called at the time One of

22 The conceptual path to the theory 49

Descartesrsquo prescriptions was the elimination of all the ldquoinfluences from farawayrdquo that plagued mediaeval science According to Descartes physicalinteractions happen only between contiguous entities ndash as in collisionspushes and pulls Newton violated this prescription describing gravity asthe instantaneous ldquoaction-at-a-distancerdquo of the force

F = Gm1m2

d2 (299)

Newton did not introduce action-at-a-distance with a light heart he callsit ldquorepugnantrdquo His violation of the cartesian prescriptions was one of thereasons for the strong initial opposition to newtonianism For many hislaw of gravitation sounded too much like the discredited ldquoinfluences fromthe starsrdquo of the ineffective science of the Middle Ages But the empiricalsuccess of Newtonrsquos dynamics and gravitational theory was so immensethat most worries about action-at-a-distance dissipated

Two centuries later it is another Briton who finds the way to ad-dress the problem afresh in an effort to understand electric and magneticforces Faraday introduces a new notion10 which is going to revolution-ize modern physics the notion of field For Faraday the field is a setof lines filling space The Faraday lines begin and end on charges inthe absence of charges each line closes forming a loop In his wonderfulbook which is one of the pillars of modern physics and has virtually noequations Faraday discusses whether the field is a real physical entity11

Maxwell formalizes Faradayrsquos powerful physical intuition into a beautiful

10Many ideas of modern science have been resuscitated from hellenistic science [57]Is the FaradayndashMaxwell notion of field a direct descendant of the notion of πνευμα(pneuma) that appears for instance in Hipparchus as the carrier of the attractionof the Moon on the oceans causing the tides and which also appears in contextsrelated to magnetism [58] Did Faraday know this notion

11ldquoWith regards to the great point under consideration it is simply whether the linesof force have a physical existence or not I think that the physical nature of thelines must be grantedrdquo [59] Strictly speaking we can translate the problem in modernterms as to whether the field has degrees of freedom independent from the chargesor not But this doesnrsquot diminish the ontological significance of Faradayrsquos questionwhich seems to me transparent in these lines Faradayrsquos continuation is lovely ldquoAndthough I should not have raised the argument unless I had thought it both importantand likely to be answered ultimately in the affirmative I still hold the opinion withsome hesitation with as much indeed as accompanies any conclusion I endeavor todraw respecting points in the very depths of sciencerdquo I think that Faradayrsquos greatnessshines in this ldquohesitationrdquo which betrays his full awareness of the importance of thestep he is taking (virtually all of modern fundamental physics comes out of theselines) as well as the full awareness of the risk of taking any major novel step


mathematical theory ndash a field theory At each spacetime point Maxwellelectric and magnetic fields represent the tangent to the Faraday lineThere is no action-at-a-distance in the theory the Coulomb descriptionof the electric force between two charges namely the instantaneous action-at-a-distance law

F = kq1 q2d2

(2100)

is understood to be correct only in the static limit A charge q1 at distanced from another charge q2 does not produce an instantaneous force on q2because if we move q1 rapidly away it takes a time t = dc before q2begins to feel any change This is the time the interaction takes to moveacross space at a finite speed in a manner remarkably consistent withDescartesrsquo prescription

When Einstein studies physics Maxwell theory is only three decadesold In his writings Einstein rhapsodizes on the beauty of Maxwell the-ory and the profound impression it made upon him Given the formalsimilarity of the Newton and Coulomb forces (299) and (2100) it iscompletely natural to suspect that (299) also is only correct in the staticlimit Namely that the gravitational force is not instantaneous either ifa neutron star rushing at great speed from the deep sky smashed awaythe Sun it would take a finite time before any effect be felt on EarthThat is it is natural to suspect that there is a field theory behind New-ton theory as well Einstein set out to find this field theory GR is what hefound

Special relativity In fact the need for a field theory behind Newton law(299) is not just suggested by the CoulombndashMaxwell analogy it is indi-rectly required by Maxwell theory The reason is that Maxwell theory notonly eliminated the apparent action-at-a-distance of Coulomb law (2100)but it also led to a reorganization of the notions of space and time whichin turn renders any action-at-a-distance inconsistent This reorganizationof the notions of space and time is special relativity a key step towardsGR

In spite of its huge empirical success Maxwell theory had an appar-ent flaw if taken as a fundamental theory12 it is not galilean invariantGalilean invariance is a consequence of the equivalence of inertial frames ndashat least it had always been understood as such Inertial frame equivalenceor the fact that velocity is a relative notion is one of the pillars of dy-namics The story goes that in the silent halls of Warsawrsquos University an

12Rather than as a phenomenological theory of the disturbances of a mechanical etherwhose dynamics is still to be found


old and grave professor stormed out of his office like a madman shoutingldquoEureka Eureka The new Archimedes is bornrdquo when he saw Einsteinrsquos1905 paper offering the solution of this apparent contradiction The wayEinstein solves the problem is an example of theoretical thinking at itsbest I think it should be kept in mind as an exemplar when we considerthe apparent contradictions between GR and QM

Einstein maintains his confidence in the galilean discovery that physicsis the same in all moving inertial frames and also maintains his confidencethat Maxwell equations are correct in spite of the apparent contradictionHe realizes that there is contradiction only because we implicitly hold athird assumption By dropping this third assumption the contradictiondisappears The third assumption regards the notion of time It is the ideathat it is always meaningful to say which of two distant events A andB happens first Namely that simultaneity is well defined in a mannerindependent of the observer Einstein observes that this is a prejudice wehave on the structure of reality We can drop this prejudice and accept thefact that the temporal ordering of distant events may have no meaningIf we do so the picture returns to consistency

The success of special relativity was rapid and the theory is todaywidely empirically supported and universally accepted Still I do notthink that special relativity has really been fully absorbed even nowthe large majority of cultivated people as well as a surprisingly highnumber of theoretical physicists still believe deep in their heart thatthere is something happening ldquoright nowrdquo on Andromeda that there isa single universal time ticking away the life of the Universe Do you myreader

An immediate consequence of special relativity is that action-at-a-distance is not just ldquorepugnantrdquo as Newton felt it is a nonsense Thereis no (reasonable) sense in which we can say that the force due to themass m1 acts on the mass m2 ldquoinstantaneouslyrdquo If special relativity iscorrect (299) is not just likely to be the static limit of a field theoryit has to be the static limit of a field theory When the neutron starhits the Sun there is no ldquonowrdquo at which the Earth could feel the effectThe information that the Sun is no longer there must travel from Sun toEarth across space carried by an entity This entity is the gravitationalfield

Maxwell rarr Einstein Therefore shortly after having worked out the keyconsequences of special relativity Einstein attacks what is obviously thenext problem searching the field theory that gives (299) in the staticlimit His aim is to do for (299) what Faraday and Maxwell had done for(2100) The result in brief is the following expressed in modern language


Maxwellrsquos solution to the problem is tointroduce the one-form field Aμ(x)

The force on the particles is

xμ = eFμν xν (2101)

where F is constructed with the firstderivatives of AA satisfies the (Maxwell) field equations

partμFνμ = Jν (2102)

a system of second-order partial differ-ential equations for A with the chargecurrent Jν as sourceMore generally the field equations canbe obtained as EulerndashLagrange equa-tions of the action

S[Amatt] =1

4

intF lowast and F

+Smatt[Amatt] (2103)

where F is the curvature of A

Smatt is obtained from the matter actionby replacing derivatives with covariantderivatives

It follows that the source of the fieldequations is

Jμ =δ

δAμSmatt[Amatt] (2104)

Einsteinrsquos solution is to introduce thefield eIμ(x) a one-form with value inMinkowski spaceThe force on the particles is (eq (244))


where Γ is constructed with the firstderivatives of e (equation (245))e satisfies the (Einstein) field equations(eq (237) here with λ = 0)

RIμ minus 1

2eIμR = 8πG T I

μ (2106)

a system of second order partial differ-ential equations for e with the energymomentum tensor T I

μ as sourceMore generally the field equations canbe obtained as EulerndashLagrange equa-tions of the action ((236) in second or-der form)

S[ematt] =1

16πG

inteIandeJandRKLεIJKL

+ Smatt[ematt] (2107)

where R is the curvature of the connec-tion ω compatible with eSmatt is obtained from the matter ac-tion by replacing derivatives with co-variant derivatives and the Minkowskimetric with the gravitational metricIt follows that the source of the fieldequations is (237)

T Iμ =

δ

δeIμSmatt[ematt] (2108)

The structural similarity between the theories of Maxwell and Einsteintheories is evident However this is only half of the story

222 Einsteinrsquos second problem relativity of motion

To understand Einsteinrsquos second problem we have to return again to theorigin of modern physics In the western culture there are two traditionalways of understanding what is ldquospacerdquo as an entity or as a relation

ldquoSpace is an entityrdquo means that space still exists when there is nothingelse besides space It exists by itself and objects move in it Thisis the way Newton describes space and is called absolute spaceIt is also the way spacetime (rather than space) is understood in


special relativity Although considered since ancient times (in thedemocritean tradition) this way of understanding space was not thetraditional dominant view in western culture The dominant viewfrom Aristotle to Descartes was to understand space as a relation

ldquoSpace is a relationrdquo means that the world is made up of physical objectsor physical entities These objects have the property that they canbe in touch with one another or not Space is this ldquotouchrdquo orldquocontiguityrdquo or ldquoadjacencyrdquo relation between objects Aristotle forinstance defines the spatial location of an object as the (internal)boundary of the set of the objects that surround it This is relationalspace

Strictly connected to these two ways of understanding space there aretwo ways of understanding motion

ldquoAbsolute motionrdquo If space is an entity motion can be defined as goingfrom one part of space to another part of space This is how Newtondefines motion

ldquoRelative motionrdquo If space is a relation motion can only be defined asgoing from the contiguity of one object to the contiguity of anotherobject This is how Descartes13 and Aristotle14 define motion

For a physicist the issue is which of these two ways of thinking aboutspace and motion allows a more effective description of the world

For Newton space is absolute and motion is absolute15 This is a sec-ond violation of cartesianism Once more Newton does not take this step

13ldquoWe can say that movement is the transference of one part of matter or of one bodyfrom the vicinity of those bodies immediately contiguous to it and considered at restinto the vicinity of some othersrdquo (Descartes Principia Philosophiae Section II-25p 51) [60]

14Aristotle insists that motion is relative He illustrates the point with the example of aman walking on a boat The man moves with respect to the boat which moves withrespect to the water of the river which moves with respect to the ground Aristotlersquosrelationalism is tempered by the fact that there are preferred objects that can be usedas a preferred reference the Earth at the center of the Universe and the celestialspheres in particular one of the fixed stars Thus we can say that something ismoving ldquoin absolute termsrdquo if it moves with respect to the Earth However thereare two preferred frames in ancient cosmology the Earth and the fixed stars andthe two rotate with respect to each other The thinkers of the Middle Ages did notmiss this point and discussed at length whether the stars rotate around the Earthor the Earth rotates under the stars Remarkably in the fourteenth century Buridanconcluded that neither view is more true than the other on grounds of reason andOresme studied the rotation of the Earth more than a century before Copernicus

15ldquoSo it is necessary that the definition of places and hence local motion be referredto some motionless thing such as extension alone or space in so far as space is seentruly distinct from moving bodiesrdquo [61] This is in open contrast with Descartesdefinition given in footnote 13


with a light heart he devotes a long initial section of the Principia toexplain the reasons of his choice The strongest argument in Newtonrsquosfavor is entirely a posteriori his theoretical construction works extraor-dinarily well Cartesian physics was never as effective But this is notNewtonrsquos argument Newton resorts to empirical evidence discussing afamous experiment with a bucket

Newtonrsquos bucket Consider a ldquobucket full of water hung by a long cordso often turned about that the cord is strongly twistedrdquo Whirl thebucket so that it starts rotating and the cord untwisting At first

(i) the bucket rotates (with respect to us) and the water remainsstill The surface of the water is flatThen the motion of the bucket is transmitted to the water byfriction and thus the water starts rotating together with thebucket At some time

(ii) the water and the bucket rotate together The surface of thewater is no longer flat it is concave

We know from experience that the concavity of the water is caused byrotation Rotation with respect to what Newtonrsquos bucket experimentshows something subtle about this question If motion is change of placewith respect to the surrounding objects as Descartes demands then wemust say that in (i) water rotates (with respect to the bucket whichsurrounds it) while in (ii) water is still (with respect to the bucket) Butobserves Newton the concavity of the surface appears in (ii) not in (i)It appears when the water is still with respect to the bucket not whenthe water moves with respect to the bucket Therefore the rotation thatproduces the physical effect is not the rotation with respect to the bucketIt is the rotation with respect to what

It is rotation with respect to space itself answers Newton The concav-ity of the water surface is an effect of the absolute motion of the waterthe motion with respect to absolute space not to the surrounding bodiesThis claims Newton proves the existence of absolute space

Newtonrsquos argument is subtle and for three centuries nobody had been able to defeatit To understand it correctly we should lay to rest a common misunderstanding Rela-tionalism namely the idea that motion can be defined only in relation to other objectsshould not be confused with galilean relativity Galilean relativity is the statementthat ldquorectilinear uniform motionrdquo is a priori indistinguishable from stasis Namely thatvelocity (just velocity) is relative to other bodies Relationalism on the other handholds that any motion (however zig-zagging) is a priori indistinguishable from stasisThe very formulation of galilean relativity assumes a nonrelational definition of motionldquorectilinear and uniformrdquo with respect to what

Now when Newton claimed that motion with respect to absolute space is real andphysical he in a sense overdid it insisting that even rectilinear uniform motion is


absolute This caused a painful debate because there are no physical effects of inertialmotion and therefore the bucket argument fails for this particular class of motions16

Therefore inertial motion and velocity are to be considered relative in newtonian me-chanics

What Newton needed for the foundation of dynamics ndash and what we are discussinghere ndash is not the relativity of inertial motion it is whether accelerated motion exem-plified by the rotation of the water in the bucket is relative or absolute The questionhere is not whether or not there is an absolute space with respect to which velocity canbe defined The question is whether or not there is an absolute space with respect towhich acceleration can be defined Newtonrsquos answer supported by the bucket argumentwas positive Without this answer Newtonrsquos main law

F = ma (2109)

wouldnrsquot even make sense

Opposition to Newtonrsquos absolute space was even stronger than oppo-sition to his action-at-a-distance Leibniz and his school argued fierilyagainst Newton absolute motion and Newtonrsquos use of absolute accelera-tion17 Doubts never really disappeared down through subsequent cen-turies and a lingering feeling remained that something was wrong inNewtonrsquos argument At the end of the nineteenth century Ernst Machreturned to the issue suggesting that Newtonrsquos bucket argument could bewrong because the water does not rotate with respect to absolute space itrotates with respect to the full matter content of the Universe I will com-ment on this idea and its influence on Einstein in Section 241 But asfor action-at-a-distance the immense empirical triumph of newtonianismcould not be overcome

Or could it After all in the early twentieth century 43 seconds of arcin Mercuryrsquos orbit were observed which Newtonrsquos theory didnrsquot seem tobe able to account for

Generalize relativity Einstein was impressed by galilean relativity Thevelocity of a single object has no meaning only the velocity of objectswith respect to one another is meaningful Notice that in a sense thisis a failure of Newtonrsquos program of revealing the ldquotrue motionsrdquo It is aminor but significant failure For Einstein this was a hint that there issomething wrong in the newtonian (and special-relativistic) conceptualscheme

16Newton is well aware of this point which is clearly stated in the Corollary V ofthe Principia but he chooses to ignore it in the introduction to Principia I thinkhe did this just to simplify his argument which was already hard enough for hiscontemporaries

17Leibniz had other reasons of complaint with Newton The two were fighting overthe priority for the invention of calculus ndash scientistsrsquo frailties remain the same in allcenturies


In spite of its immense empirical success Newtonrsquos idea of an abso-lute space has something deeply disturbing in it As Leibniz Mach andmany others emphasized space is a sort of extrasensorial entity that actson objects but cannot be acted upon Einstein was convinced that theidea of such an absolute space was wrong There can be no absolutespace no ldquotrue motionrdquo Only relative motion and therefore relative ac-celeration must be physically meaningful Absolute acceleration shouldnot enter physical equations With special relativity Einstein had suc-ceeded in vindicating galilean relativity of velocities from the challenge ofMaxwell theory He was then convinced that he could vindicate the entirearistotelianndashcartesian relativity of motion In Einsteinrsquos terms ldquothe lawsof motion should be the same in all reference frames not just in the iner-tial framesrdquo Things move with respect to one another not with respect toan absolute space there cannot be any physical effect of absolute motion

According to many contemporary physicists this is excessive weightgiven to ldquophilosophicalrdquo thinking which should not play a role in physicsBut Einsteinrsquos achievements in physics are far more effective than the onesobtained by these physicists

223 The key idea

The question addressed in Newtonrsquos bucket experiment is the followingThe rotation of the water has a physical effect ndash the concavity of thewater surface with respect to what does the water ldquorotaterdquo Newtonargues that the relevant rotation is not the rotation with respect to thesurrounding objects (the bucket) therefore it is rotation with respect toabsolute space Einsteinrsquos new answer is simple and fulgurating

The water rotates with respect to a local physical entity the gravitational field

It is the gravitational field not Newtonrsquos inert absolute space that tellsobjects if they are accelerating or not if they are rotating or not Thereis no inert background entity such as newtonian space there are onlydynamical physical entities Among these are the fields Among the fieldsis the gravitational field

The flatness or concavity of the water surface in Newtonrsquos bucket is notdetermined by the motion of the water with respect to absolute spaceIt is determined by the physical interaction between the water and thegravitational field

The two lines of Einsteinrsquos thinking about gravity (finding a field the-ory for the newtonian interaction and getting rid of absolute acceleration)meet here Einsteinrsquos key idea is that Newton has mistaken the gravita-tional field for an absolute space


What leads Einstein to this idea Why should newtonian accelerationbe defined with respect to the gravitational field The answer is givenby the special properties of the gravitational interaction18 These canbe revealed by a thought experiment called Einsteinrsquos elevator I presentbelow a modern and more realistic version of Einsteinrsquos elevator argument

An ldquoelevatorrdquo argument newtonian cosmology Here is a simple physical situation thatillustrates that inertia and gravity are the same thing The model is simple but com-pletely realistic It leads directly to the physical intuition underlying GR

In the context of newtonian physics consider a universe formed by a very largespherical cloud of galaxies Assume that the galaxies are ndash and remain ndash uniformlydistributed in space with a time-dependent density ρ(t) and that they attract eachother gravitationally Let C be the center of the cloud Consider a galaxy A (say ours)at a distance r(t) from the center C As is well known the gravitational force on A dueto the galaxies outside a sphere of radius r around C cancels out and the gravitationalforce due to the galaxies inside this sphere is the same as the force due to the samemass concentrated in C Therefore the gravitational force on A is

F = minusGmA

43πr3(t) ρ(t)

r2(t) (2110)

ord2r

dt2= minusG

4

3π r(t)ρ(t) (2111)

If the density remains spatially constant it scales uniformly as rminus3 That is ρ(t) =ρ0r

minus3(t) where ρ0 is a constant equal to the density at r(t) = 1 Therefore

d2r

dt2= minus4

3πGρ0

1

r2(t)= minus c

r2(t) (2112)

where

c =4πGρO

3(2113)

is a constant Equation (2112) is the Friedmann cosmological equation which governsthe expansion of the universe (It is the same equation that one obtains from full GRin the spatially flat case)

In the newtonian model we are considering the galaxy C is in the center of theuniverse and defines an inertial frame while the galaxy A is not in the center and isnot inertial Assume that the cloud is so large that its boundary cannot be observedfrom C or A If you are in one of these two galaxies how can you tell in which youare That is how can you tell whether you are in the inertial reference frame C or inthe accelerated frame A

The answer is very remarkably that you cannot Since the entire cloud expandsor contracts uniformly the picture of the local sky looks uniformly expanding or con-tracting precisely in the same manner from all galaxies But you cannot detect if youare in the inertial galaxy C or in the accelerated galaxy A by local experiments eitherIndeed to detect if you are in an accelerated frame you have to observe inertial forces

18Gravity is ldquospecialrdquo in the sense that newtonian absolute space is a configuration ofthe gravitational field Once we get rid of the notion of absolute space the gravita-tional interaction is no longer particularly special It is one of the fields forming theworld But it is a very different world from that of Newton and Maxwell


such as the ones that make the water surface of Newtonrsquos bucket concave The A frameacceleration is

a =c

r2(t)u (2114)

where u is a unit vector pointing towards C Therefore there is an inertial force

Finertial = minus c

r2(t)u (2115)

on all moving masses This is the force that should allow us to detect that the frame isnot inertial However all masses feel besides the local forces Flocal also the cosmologicalgravitational pull towards C

Fcosmological =c

r2(t)u (2116)

so that their motion in the accelerated A frame is governed by

ma = Flocal + Finertial + Fcosmological (2117)

= Flocal (2118)

because (2115) and (2116) cancel out exactly Therefore the local dynamics in Alooks precisely as if it were inertial The parabola of a falling stone in A seen from theaccelerated A frame looks as a straight line There is no way of telling if you are thecenter and no way of telling if you are inertial or not

How do we interpret this impossibility of detecting the inertial frameAccording to newtonian physics the dynamics in C or A should be com-pletely different But this difference is not physically observable In thenewtonian conceptual scheme A is noninertial there are gravitationalforces and inertial forces but there is a sort of conspiracy that hides bothof them In fact the situation is completely general in a sufficiently smallregion inertial and gravitational forces cancel to any accuracy in a free-falling reference system19 It is clear that there should be a better wayof understanding this physical situation without resorting to all theseunobservable forces

The better way is to drop the newtonian preferred global frame andto realize each galaxy has its own local inertial reference frame We candefine local inertial frame by the absence of observable inertial effects asin newtonian physics Each galaxy then has its local inertial frame These

19This is the equivalence principle By the way Newton the genius knew it ldquoIf bodiesmoved among themselves are urged in the direction of parallel lines by equal acceler-ative forces they will all continue to move among themselves after the same manneras if they had not been urged by those forcesrdquo (Newton Principia Corollary VI tothe ldquoLaws of Motionrdquo) [62] Newton uses this corollary for computing the complicatedmotion of the Moon in the Solar System In the frame of the Earth inertial forcesand the solar gravity cancel out with good approximation and the Moon follows akeplerian orbit


frames are determined by the gravitational force That is it is gravitythat determines at each point what is inertial Inertial motion is suchwith respect to the local gravitational field not with respect to absolutespace

Gravity determines then the way the frames of different galaxies fallwith respect to one another The gravitational field expresses the rela-tion between the various inertial frames It is the gravitational field thatdetermines inertial motion Newtonrsquos true motion is not motion with re-spect to absolute space it is motion with respect to a frame determinedby the gravitational field It is motion relative to the gravitational fieldEquation (2109) governs the motion of objects with respect to the grav-itational field

The form of the gravitational field Recall that Einsteinrsquos problem wasto describe the gravitational field The discussion above indicates thatthe gravitational field can be viewed as the field that determines at eachpoint of spacetime the preferred frames in which motion is inertial Letus write the mathematics that expresses this intuition

Return to the cloud of galaxies Since we have dropped the idea ofa global inertial reference system let us coordinatize events in the cloudwith arbitrary coordinates x= (xμ) The precise physical meaning of thesecoordinates is discussed in detail in the next section Let xμA be coordinatesof a particular event A say in our galaxy Since these coordinates arearbitrarily chosen motion described in the coordinates xμ is in generalnot inertial in our galaxy For instance particles free from local forcesdo not follow straight lines But we can find a locally inertial referenceframe around A Let us denote the coordinates it defines as XI and takethe event A as the origin so that XI(A) = 0 The coordinates XI can beexpressed as functions

XI = XI(x) (2119)

of the arbitrary coordinates x In the x coordinates the noninertiality ofthe motion in A is gravity Gravity in A is the information of the changeof coordinates that takes us to inertial coordinates This information iscontained in the functions (2119) But only the value of these functions ina small neighborhood around A is relevant because if we move away thelocal inertial frame will change Therefore we can Taylor-expand (2119)and keep only the first nonvanishing term As XI(A) = 0 to first non-vanishing order we have

XI(x) = eIμ(xA) xμ (2120)


where we have defined

eIμ(xA) =partXI(x)partxμ

∣∣∣∣x=x(A)

(2121)

The quantity eIμ(xA) contains all the information we need to know thelocal inertial frame in A The construction can be repeated at each pointx The quantity

eIμ(x) =partXI(x)partxμ

∣∣∣∣x

(2122)

where XI are now inertial coordinates at x is the gravitational field atx This is the form of the field introduced in Section 211

The gravitational field eIμ(x) is therefore the jacobian matrix of thechange of coordinates from the x coordinates to the coordinates XI thatare locally inertial at x The field eIμ(x) is also called the ldquotetradrdquo fieldfrom the Greek word for ldquofourrdquo or the ldquosoldering formrdquo because it ldquosol-dersrdquo a Minkowski vector bundle to the tangent bundle or followingCartan the ldquomoving framerdquo although there is nothing moving about it

Transformation properties If the coordinate system XI defines a localinertial system at a given point so does any other local coordinate sys-tem Y J = ΛJ

IXI where Λ is a Lorentz transformation Therefore the

index I of eIμ(x) transforms as a Lorentz index under a local Lorentztransformation and the two fields eIμ(x) and

eprimeJμ(x) = ΛJI(x)eIμ(x) (2123)

represent the same physical gravitational field Thus this description ofgravity has a local Lorentz gauge invariance

What happens if instead of using the physical coordinates x we hadchosen coordinates y = y(x) The chain rule determines the field eprimeIν(y)that we would have found had we used coordinates y

eprimeIν(y) =partxμ(y)partyν

eIμ(x(y)) (2124)

The transformation properties (2123) and (2124) are precisely the trans-formation properties (258) and (263) under which the GR action is in-variant

These transformation laws are also the ones of a one-form field valued in a vectorbundle P over the spacetime manifold M whose fiber is Minkowski space M associatedwith a principal SO(3 1) Lorentz bundle This is a natural geometric setting for thegravitational field The connection ω defined in Section 211 is a connection of this


bundle This setting realizes the physical picture of a patchwork of Minkowski spacessuggested by the cloud of galaxies carrying Lorentz frames at each galaxy More pre-cisely the gravitational field can be viewed as map e TM rarr P that sends tangentvectors to Lorentz vectors

Matter Finally consider a particle moving in spacetime along a worldlinexμ(τ) If a particle has velocity vμ = dxμdτ at a point x its velocity inlocal Minkowski coordinates XI at x is

uI =partXI(x)partxμ

∣∣∣∣x

vμ = eIμ(x)vμ (2125)

In this local Minkowski frame the infinitesimal action along the trajectoryis

dS = mradicminusηIJuIuJ dτ (2126)

Therefore the action along the trajectory is the one given in (241) Thesame argument applies to all matter fields the action is a sum over space-time of local terms which can be inferred from their Minkowski spaceequivalent

Metric geometry In Section 214 we saw that the gravitational field e defines a metricstructure over spacetime One is often tempted to give excessive significance to thisstructure as if distance was an essential property of reality But there is no a priorikantian notion of distance needed to understand the world We could have developedphysics without ever thinking about distances and still have retained the completepredictive and descriptive power of our theories

What is the physical meaning of the spacetime metric structure What do we meanwhen we say that two points are 3 centimeters apart or two events are 3 seconds apart

The answer is in the dynamics of matter interacting with the gravitational field Letus first consider Minkowski space Consider two objects A and B that are 3 centimetersapart This means that if we put a ruler between the two points the part of the rulerthat fits between the two is marked 3 cm The shape of the ruler is determined bythe Maxwell and Schrodinger equations at the atomic level These equations containthe Minkowski tensor ηIJ They have stable solutions in which the molecules maintainpositions (better vibrate around equilibrium positions) at a fixed ldquodistancerdquo L fromone another L is determined by the constants in these equations This means that themolecules maintain positions at points with coordinate distances ΔxI such that

ηIJΔxIΔxJ = L2 (2127)

We exploit this peculiar behavior of condensed matter for coordinatizing spacetimelocations That is ldquodistancerdquo is nothing but a convenient manner for labeling locationsdetermined by material objects (the ruler) whose dynamics is governed by certainequations We could avoid mentioning distance by saying a number N =3 [cm]L of


molecules obeying the Maxwell and Schrodinger equations with given initial values fitbetween A and B

Consider now the same situation in a gravitational field e Again the fact that twopoints A and B are 3 centimeters apart means that we can fit the N molecules of theruler between A and B But now the dynamics of the molecules is determined by theirinteraction with the gravitational field The Maxwell and Schrodinger equations havestable solutions in which the molecules keep themselves at coordinate distances Δxμ

such thatηIJe

Iμ(x)eJν (x)ΔxμΔxν = L2 (2128)

Thus a measure of distance is a measurement of the local gravitational field performedexploiting the peculiar way matter interacts with gravity

The same is true for temporal intervals Consider two events A and B that happenin time The meaning that 3 seconds have elapsed between A and B is that a second-ticking clock has ticked three times in this time interval The physical system that weuse as a clock interacts with the gravitational field The pace of the clock is determinedby the local value of e Thus a clock is nothing but a device measuring an extensivefunction of the gravitational field along a worldline going from A to B

Imagine that a particle falls along a timelike geodesic from A to B We know fromspecial relativity that the increase of the action of the particle in the particle frame is

dS = mdt (2129)

where m is the particle mass Therefore a clock comoving with the particle will measurethe quantity

T =1

mS =

int B

A

dτradic

minusηIJeIμeJν xμxν (2130)

Thus a clock is a device for measuring a function T of the gravitational field Ingeneral any metric measurement is nothing but a measurement of a nonlocal functionof the gravitational field

This is true in an arbitrary gravitational field e as well as in flat space In flatspace we can use these measurements for determining positions with respect to thegravitational field Since the flat-space gravitational field is Newton absolute spacethese measurements locate points in spacetime

224 Active and passive diffeomorphisms

Before getting to the last and main step in Einsteinrsquos discovery of GRwe need the notion of active diffeomorphism I introduce this notion withan example

Consider the surface of the Earth and call it M At each point P isin M on Earthsay the city of Paris there is a certain temperature T (P ) The temperature is a scalarfunction T M rarr R on the Earthrsquos surface Imagine a simplified model of weatherevolution in which the only factor determining temperature change was the displace-ment of air due to wind By this I mean the following Fix a time interval say we callT the temperature on May 1st and T the temperature on May 2nd During this timeinterval the winds move the air which is over a point Q = φ(P ) to the point P Ifsay Q is the French village of Quintin this means that the winds have blown the airof Quintin to Paris Assume the temperature T (P ) of Paris on May 2nd is equal to thetemperature T (Q) of Quintin the day before The ldquowindrdquo map φ is a map from the


Earthrsquos surface to itself which associates with each point P the point Q from whichthe air has been blown by the wind From May 1st to May 2nd the temperature fieldchanges then as follows

T (P ) rarr T (P ) = T (φ(P )) (2131)

Assuming it is smooth and invertible the map φ M rarr M is an active diffeomorphismThe scalar field T on M is transformed by this active diffeomorphism as in (2131)it is ldquodraggedrdquo along the surface of the Earth by the diffeomorphism φ Notice thatcoordinates play no role in all this

Now imagine that we choose certain geographical coordinates x to coordinatize thesurface of the Earth For instance latitude and longitude namely the polar coordinatesx = (θ ϕ) with ϕ = 0 being Greenwich Using these coordinates the temperature isrepresented by a function of the coordinates T (x) The May 1st temperature T (x) andthe May 2nd temperature T (x) are related by

T (x) = T (φ(x)) (2132)

For instance if the wind has blown uniformly westward by 220prime degrees (Quintin is220prime west of Paris) then

T (θ ϕ) = T (θ ϕ + 220prime) (2133)

Of course there is nothing sacred about this choice of coordinates For instance theFrench might resent that the origin of the coordinates is Greenwich and have it passthrough Paris instead Thus the French would describe the same temperature fieldthat the British describe as T (θ ϕ) by means of different polar coordinates defined byϕ = 0 being Paris Since Paris is 220prime degrees East of Greenwich for the French thetemperature field on May 1st is

T prime(θ ϕ) = T (θ ϕ + 220prime) (2134)

This is a change of coordinates or a passive diffeomorphismNow the two equations (2133) and (2134) look precisely the same But it would

be silly to confuse them In (2133) T (θ ϕ) is the temperature on May 2nd while in(2134) T prime(θ ϕ) is the temperature on May 1st but written in French coordinatesIn summary the first equation represents a change in the temperature field due tothe wind the second equation represents a change in convention The first equationdescribes an ldquoactive diffeomorphismrdquo the second a change of coordinates also calleda ldquopassive diffeomorphismrdquo

Given a manifold M an active diffeomorphism φ is a smooth invertiblemap from M to M A scalar field T on M is a map T M rarr R Givenan active diffeomorphism φ we define the new scalar field T transformedby φ as

T (P ) = T (φ(P )) (2135)

Coordinates play no role in thisA coordinate system x on a d-dimensional manifold M is an invertible

differentiable map from (an open set of) M to Rd Given a field T on M this map determines the function t Rd rarr R defined by t(x) = T (P (x))called ldquothe field T in coordinates xrdquo20 A passive diffeomorphism is an

20In the physics literature the two maps T M rarr R and t = T xminus1 Rd rarr R


invertible differentiable map φ Rd rarr Rd that defines a new coordi-nate system xprime on M by x(P ) = φ(xprime(P )) The value of the field T incoordinates xprime is given by

tprime(xprime) = t(φ(xprime)) (2136)

Beware the formal similarity between (2135) and (2136)The above extends immediately to all structures on M For instance

an active diffeomorphism φ carries a one-form field e on M to the newone-form field e = φlowaste the pull-back of e under φ and so on

In particular a metric d M timesM rarr R+ is an assignment of a distanced(AB) between any two points A and B of M An active diffeomorphismdefines the new metric d given by d(AB) equiv d(φminus1(A) φminus1(B)) Thetwo metrics d and d are isometric but distinct21 An equivalence class ofmetrics under active diffeomorphisms is sometimes called a ldquogeometryrdquoGiven a coordinate system we can represent a (Riemannian) metric dby means of a tensor field on Rd Riemannrsquos metric tensor gμν(x) orequivalently the tetrad field eIμ(x) Under a change of coordinate systemthe same metric is represented by a different gμν(x) or eIμ(x)

The example of the Earthrsquos temperature given above illustrates a pecu-liar relation between active and passive diffeomorphisms given two tem-perature fields T and T related by an active diffeomorphism we can al-ways find a coordinate transformation such that in the new coordinatesT is represented by the same function as T in the old coordinates Thissimple mathematical observation is at the root of Einsteinrsquos argumentsthat I will describe below (The argument will be essentially that a the-ory that does not distinguish coordinate systems cannot distinguish fieldsrelated by active diffeomorphisms either)

More precisely the relation between active and passive diffeomorphismsis as follows The group of the active diffeomorphisms acts on the space

are always indicated with the same symbol generating confusion between active andpassive diffeomorphisms In this paragraph I use distinct notations In the rest of thetext however I shall adhere to the standard notation and indicate the field and itscoordinate representation with the same symbol

21Here is an example of isometric but distinct metrics The 2001 Shell road-map saysthat the distances between New York (NY) Chicago (C) and Kansas City (KC) ared(NY C) = 100 miles d(C KC) = 50 miles d(KC NY) = 100 miles while the 2002Lonely Planet tourist guide claims that these distances are d(NY C) = 100 milesd(C KC) = 100 miles d(KC NY) = 50 miles Obviously these are not the samedistances But they are isometric the two are transformed into each other by theactive diffeomorphism φ(NY) = C φ(C) = KC φ(KC) = NY


Space of metrics dSpace of functions g (x)mν

Orbit of the passive

diffeomorphism groupOrbit of the active

diffeomorphism group

Coordinate system S

Coordinate system S prime

Fig 22 Active and passive diffeomorphisms

of metrics d The group of passive diffeomorphisms acts on the spaceof functions gμν(x) The orbits of the first group are in natural one-to-one correspondence with the orbits of the second However the relationbetween the individual metrics d and the individual functions gμν(x) de-pends on the coordinate system chosen The situation is illustrated inFigure 22

225 General covariance

Around 1912 using the idea that any motion is relative Einstein hadfound the form of the gravitational field as well as the equations of motionsof matter in a given gravitational field This was already a remarkableachievement but the field equations for the gravitational field were stillmissing In fact the best part of the story had yet to come

Two problems remained open the field equations and understandingthe physical meaning of the coordinates xμ introduced above Einsteinstruggled with these two problems during the years 1912ndash1915 trying sev-eral solutions and changing his mind repeatedly Einstein has called this


search his ldquostruggle with the meaning of the coordinatesrdquo The strugglewas epic The result turned out to be amazing In Einsteinrsquos words it wasldquobeyond my wildest expectationsrdquo

To increase Einsteinrsquos stress Hilbert probably the greatest mathemati-cian at the time was working on the same problem trying to be first tofind the gravitational field equations The fact that Hilbert with his farsuperior mathematical skills could not find these equations first testifiesto the profound differences between fundamental physical problems andmathematical problems

In his search for the field equations Einstein was guided by severalpieces of information First the static limit of the field equations mustyield the Newton law as the static limit of Maxwell theory yields theCoulomb law Second the source of Coulomb law is charge and the chargedensity is the temporal component of four-current Jμ(x) which is thesource of Maxwell equations The source of the Newtonian interaction ismass Einstein had understood with special relativity that mass is in facta form of energy and that the energy density is the temporal compo-nent of the energy-momentum tensor Tμν(x) Therefore Tμν(x) had tobe the likely source of the field equations Third the introduction of thegravitational field was based on the use of arbitrary coordinates there-fore there should be some form of covariance under arbitrary changes ofcoordinates in the field equations Einstein searched for covariant second-order equations as relations between tensorial quantities since these areunaffected by coordinate change He learned from Riemannian geometrythat the only combination of second derivatives of the gravitational fieldthat transforms tensorially is the Riemann tensor Rμ

νρσ(x) This was infact Riemannrsquos major result Einstein knew all this in 1912 To deriveEinsteinrsquos field equations (297) from these ideas is a simple calculationpresented in all GR textbooks and which a good graduate student cantoday repeat easily Still Hilbert couldnrsquot do it and Einstein got stuckfor several years What was the problem

The problem was ldquothe meaning of the coordinatesrdquo Here is the story

1 Einstein for general covariance At first Einstein demands the fieldequations for the gravitational field eIμ(x) to be generally covariant onM This means that if eIμ(x) is a solution then eprimeIν(y) defined in (2124)should also be a solution For Einstein this requirement (unheard of atthe time) was the formalization of the idea that the laws of nature must bethe same in all reference frames and therefore in all coordinate systems

2 Einstein against general covariance In 1914 however Einstein con-vinces himself that the field equations should not be generally covariant


t = 0

M

AB

M

AB

f

e e~

(a) (b)

Fig 23 The active diffeomorphism φ drags the nonflat (wavy) gravitationalfield from the point B to the point A

[63] Why Because Einstein rapidly understands the physical conse-quences of general covariance and he initially panics in front of themThe story is very instructive because it reveals the true magic hiddeninside GR Einsteinrsquos argument against general covariance is the follow-ing22

Consider a region of spacetime containing two spacetime points A andB Let e be a gravitational field in this region Say that around the pointA the field is flat while at the point B it is not (see Figure 23(a)) Nextconsider a map φ from M to M that maps the point A to the point BConsider the new field e = φlowaste which is pulled back by this map Thevalue of the field e at A is determined by the value of e at B and thereforethe field e will not be flat around A (see Figure 23(b))

Now if e is a solution of the equations of motion and if the equationsof motion are generally covariant then e is also a solution of the equationsof motion This is because of the relation between active diffeomorphismsand changes of coordinates we can always find two different coordinatesystems on M say x and y such that the function eIμ(x) that represents ein the coordinate system x is the same function as the function eIμ(y) thatrepresents e in the coordinate systems y Since the equations of motion

22At first Einstein got discouraged about generally covariant field equations becauseof a mistake he was making while deriving the static limit the calculation yieldedthe wrong limit But this is of little importance here given the powerful use thatEinstein has been routinely capable of making of general conceptual arguments


are the same in the two coordinate systems the fact that this functionsatisfies the Einstein equations implies that e as well as e are physicalsolutions

Let me repeat the argument in a different form We have found in theprevious section that if eIμ(x) is a solution of the Einstein equations thenso is eprimeIν(y) defined in (2124) But the function eprimeIν can be interpreted intwo distinct manners First as the same field as e expressed in a differentcoordinate system Second as a different field e expressed in the samecoordinate system That is we can define the new field as

eIμ(x) = eprimeIμ(x) (2137)

This new field e is genuinely different from e In general it will not beflat around A In particular the scalar curvature R of e at A is

R|A = R(xA) = R(φ(xA)) = R|B (2138)

In other words if the equations of motion are generally covariant they arealso invariant under active diffeomorphisms

Given this Einstein makes the following famous observation

The ldquoholerdquo argument Assume the gravitational field-equations aregenerally covariant Consider a solution of these equations in whichthe gravitational field is e and there is a region H of the uni-verse without matter (the ldquoholerdquo represented as the white regionin Figure 23) Assume that inside H there is a point A where eis flat and a point B where it is not flat Consider a smooth mapφ M rarr M which reduces to the identity outside H and such thatφ(A) = B and let e = φlowaste be the pull-back of e under φ The twofields e and e have the same past are both solutions of the fieldequations but have different properties at the point A Thereforethe field equations do not determine the physics at the spacetimepoint A Therefore they are not deterministic But we know that(classical) gravitational physics is deterministic Therefore either

(i) the field equations must not be generally covariant or(ii) there is no meaning in talking about the physical spacetime

point A

On the basis of this argument Einstein searched for nongenerally covari-ant field equations for three years in a frantic race against Hilbert

3 Einsteinrsquos return to general covariance Then rather suddenly in 1915Einstein published generally covariant field equations What had hap-pened Why had Einstein changed his mind Is there a mistake in the


t = 0

M

AB

M

AB

f

e x x~

e xa xb~~

a b

Fig 24 The diffeomorphism moves the nonflat region as well as the intersectionpoint of the two particles a and b from the point B to the point A

hole argument No the hole argument is correct The correct physicalconclusion however is (ii) not (i) This point hit Einstein like a flashof lightning the precise conceptual discovery to which all his previousthoughts had led

Einsteinrsquos way out from the difficulty raised by the hole argument isto realize that there is no meaning in referring to ldquothe point Ardquo or ldquotheevent Ardquo without further specifications

Let us follow Einsteinrsquos explanation in detail

Spacetime coincidences Consider again the solution e of the field equa-tions but assume that in the universe there are also the two particles aand b Say that the worldlines (xa(τa) xb(τb)) of the two particles intersectat the spacetime point B see Figure 24

Now for given initial conditions the worldlines of the particles are de-termined by the gravitational field They are geodesics of e or if otherforces are involved they satisfy the geodesic equation with an addi-tional force term Consider the field e = φlowaste The particlesrsquo worldlines(xa(τa) xb(τb)) are no longer solutions of the particlesrsquo equations of mo-tion in this gravitational field If the gravitational field is e instead of ethe particlesrsquo motions over M will be different But it is easy to find themotion of the particles determined by e precisely because the completeset of equations of motion is generally covariant Therefore an active dif-feomorphism on the gravitational field and the particles sends solutionsinto solutions Thus the motion of the particles in the field e is given by


the worldlines

xa(τa) = φminus1(xa(τa)) xb(τb) = φminus1(xb(τb)) (2139)

Then the particles a and b no longer intersect in B They intersect inA = φminus1(B)

Now instead of asking whether or not the field is flat at A let us askwhether or not the field is flat at the point where the particles meetClearly the result is the same for the two cases (e xa xb) and (e xa xb)Formally assuming the intersection point is at τa = τb = 0

R|inters = R(xa(0)) = R(φ(xa(0)))= R(φ(φminus1(xa(0)))) = R(xa(0)) = R|inters (2140)

This prediction is deterministic There are not two contradictory predic-tions therefore there is determinism so long as we restrict ourselves tothis kind of prediction Einstein calls ldquospacetime coincidencesrdquo this wayof determining points

Einstein observes that this conclusion is general the theory does notpredict what happens at spacetime points (like newtonian and special-relativistic theories do) Rather it predicts what happens at locations de-termined by the dynamical elements of the theory themselves In Einsteinrsquoswords

All our space-time verifications invariably amount to a determinationof space-time coincidences If for example events consisted merely inthe motion of material points then ultimately nothing would be ob-servable but the meeting of two or more of these points Moreover theresults of our measuring are nothing but verifications of such meetingsof the material points of our measuring instruments with other mate-rial points coincidences between the hands of a clock and points onthe clock dial and observed point-events happening at the same placeat the same time The introduction of a system of reference serves noother purpose than to facilitate the description of the totality of suchcoincidences [64]

The two solutions (e xa xb) and (e xa xb) are only distinguished by theirlocalization on the manifold They are different in the sense that they as-cribe different properties to manifold points However if we demand thatlocalization is defined only with respect to the fields and particles them-selves then there is nothing that distinguishes the two solutions physi-cally In fact concludes Einstein the two solutions represent the samephysical situation The theory is gauge invariant in the sense of Diracunder active diffeomorphisms there is a redundancy in the mathematicalformalism the same physical world can be described by different solutionsof the equations of motion

23 Interpretation 71

It follows that localization on the manifold has no physical meaningThe physical picture is completely different from the example of the tem-perature field on the Earthrsquos surface illustrated in the previous section Inthat example the cities of Paris and Quintin were real distinguishable en-tities independent from the temperature field In GR general covarianceis compatible with determinism only assuming that individual spacetimepoints have no physical meaning by themselves It is like having only thetemperature field without the underlying Earth

What disappears in this step is precisely the background spacetime thatNewton believed to have been able to detect with great effort beyond theapparent relative motions

Einsteinrsquos step toward a profoundly novel understanding of nature isachieved Background space and spacetime are effaced from this new un-derstanding of the world Motion is entirely relative Active diffeomor-phism invariance is the key to implement this complete relativizationReality is not made up of particles and fields on a spacetime it is madeup of particles and fields (including the gravitational field) that can onlybe localized with respect to one another No more fields on spacetimejust fields on fields Relativity has become general

23 Interpretation

General covariance makes the relation between formalism and experimentfar more indirect than in conventional field theories

Take Maxwell theory as an example We assume that there is a back-ground spacetime We have special objects at our disposal (the walls ofthe lab the Earth) that define an inertial frame to a desired approxima-tion These objects allow us to designate locations relative to backgroundspacetime We have two kinds of measuring devices (a) meters and clocksthat measure distance and time intervals from these reference objects and(b) devices that measure the electric and magnetic fields The reading ofthe devices (a) gives us xμ The reading of the devices (b) gives us Fμν We measure the two and say that the field has the value Fμν at the pointxμ The theory can predict the value Fμν at the point xμ

We cannot do the same in GR The theory does not predict the value ofthe field at the point xμ So how do we compare theory and observations

231 Observables predictions and coordinates

As discussed at the end of the previous section a physical state does notcorrespond to a solution e(x) of Einsteinrsquos equations but to an equivalenceclass of solutions under active diffeomorphisms Therefore the quantitiesthat the theory predicts are all and only the quantities that are well


defined on these equivalence classes That is only the quantities that areinvariant under diffeomorphisms These quantities are independent fromthe coordinates xμ

In concrete applications of the theory these quantities are generallyobtained by solving away the coordinates x from solutions to the equationsof motion Here are a few examples

Solar System Consider the dynamics of the Solar System The vari-ables are the gravitational field e(x) and the worldlines of the plan-ets xn(τn) Fix a solution (e(x) xn(τn)) to the equations of motionWe want to derive physical predictions from this solution and com-pare them with observations Choose for simplicity τn = x0 so thatthe solution is expressed by (e(x) xn(x0)) Consider the worldlineof the Earth Compute the distance dn(x0) between the Earth andthe planet n defined as the proper time elapsed along the Earthrsquosworldline while a null geodesic (a light pulse) leaving the Earth atx0 travels from Earth to the planet and back

The functions (dn(x0)) can be computed from the given solutionsto the equations of motion Consider a space C with coordinates(dn) The functions (dn(x0)) define a curve γ on this space

We can associate a measuring device with each dn a laser ap-paratus that measures the distance to planet n These quantitiescan be measured together We obtain the event (dn) which can berepresented by a point in C The theory predicts that this point willfall on the curve γ A sequence of these events can be comparedwith the curve γ and in this way we can test the given solutionsto the equations of motion against experience (In the terminologyof Chapter 3 the quantities dn are partial observables) Notice thatthis can be done with arbitrary precision and that distant starsinertial systems preferred coordinates or choice of time variableplay no role

Clocks Consider the gravitational field around the Earth Consider twoworldlines Let the first be the worldline of an object fixed on theEarthrsquos surface Let the second be the worldline of an object in freefall on a keplerian orbit around the Earth that is a satellite Fixan arbitrary initial point P on the worldline of the orbiting objectand let T1 be the proper time from P along this worldline Send alight signal from P to the object on Earth let Q be the point onthe Earthrsquos worldline when the signal is received and let T2 be theproper time from Q along this worldline Then let T2(T1) be thereception proper time on Earth of a signal sent at T1 proper timein orbit GR allows us to compute the function T2(T1) for any T1


It is easy to associate measuring devices to T1 and T2 these area clock on Earth and a clock in orbit If the orbiting object sendsa signal at fixed proper times T1 the reception times T2 can becompared with the predictions of the theory Here T1 and T2 arethe partial observables I let you decide which one of the two is theldquotrue time variablerdquo

Solar System with a clock We can add a clock to the Solar Systemmeasurements described above Fixing arbitrarily an initial eventon Earth (a particular eclipse the birth of Jesus or the death ofJohn Lennon) we can compute the proper time T (x0) lapsed fromthis event along the Earthrsquos worldline The partial observable Tcan be added to the partial observables dn giving the set (dn T ) ofpartial observables If we do so it may be convenient to express thecorrelations (dn T ) as functions dn(T ) A complete gauge-invariantobservable fully predicted by the theory is the value dn(T ) of aplanet distance at a certain given Earth proper time T from theinitial event Notice that T is not a coordinate It is a complicatednonlocal function of the gravitational field to which a measuringdevice (measuring a partial observable) has been attached The useof a clock on Earth to determine a local temporal localization isjust a matter of convenience

Binary pulsar Consider a binary-star system in which one of the twostars is a pulsar Because of a Doppler effect the frequency of thepulsing signal oscillates with the orbital period of the system Thisfact allows us to count the number of pulses in each orbit Let Nn

be the number of pulses we receive in the nth orbit A theoreticalmodel of the pulsar allows us to compute the expected decrease inorbital period due to gravitational wave emission and therefore theexpected sequence Nn which can be compared with the observedone Doing this with sufficient care won JH Taylor and RA Hulsethe 1993 Nobel Prize

Notice that in all these examples the coordinates xμ have disappearedfrom the observable quantities This is true in general A theoretical modelof a physical system is made using coordinates xμ but then observablequantities are independent of the coordinates xμ23

232 The disappearance of spacetime

In the mathematical formalism of GR we utilize the ldquospacetimerdquo man-ifold M coordinatized by x However a state of the universe does not

23Unless we gauge-fix them to given partial observables see Section 246


correspond to a configuration of fields on M It corresponds to an equiva-lence class of field configurations under active diffeomorphisms An activediffeomorphism changes the localization of the field on M by dragging itaround Therefore localization on M is just gauge it is physically irrele-vant

In fact M itself has no physical interpretation it is just a mathematicaldevice a gauge artifact Pre-general-relativistic coordinates xμ designatepoints of the physical spacetime manifold ldquowhererdquo things happen (see adetailed discussion below in Section 245) in GR there is nothing of thesort The manifold M cannot be interpreted as a set of physical ldquoeventsrdquoor physical spacetime points ldquowhererdquo the fields take value It is meaning-less to ask whether or not the gravitational field is flat around the point Aof M because there is no physical entity ldquospacetime point Ardquo Contraryto Newton and to Minkowski there are no spacetime points where parti-cles and fields live There are no spacetime points at all The Newtoniannotions of space and time have disappeared

In Einsteinrsquos words

the requirement of general covariance takes away from space andtime the last remnant of physical objectivity [64]

Einstein justifies this conclusion in the immediate continuation of thistext which is the paragraph I quoted at the end of the previous sectionwith the observation that all observations are spacetime coincidences

In newtonian physics if we take away the dynamical entities whatremains is space and time In general-relativistic physics if we take awaythe dynamical entities nothing remains The space and time of Newtonand Minkowski are reinterpreted as a configuration of one of the fieldsthe gravitational field

Concretely this radically novel understanding of spatial and temporalrelations is implemented in the theory by the invariance of the field equa-tions under diffeomorphisms Because of background independence ndash thatis since there are no nondynamical objects that break this invariance inthe theory ndash diffeomorphism invariance is formally equivalent to generalcovariance namely the invariance of the field equations under arbitrarychanges of the spacetime coordinates x and t

Diffeomorphism invariance implies that the spacetime coordinates xand t used in GR have a different physical meaning to the coordinatesx and t used in prerelativistic physics In prerelativistic physics x andt denote localization with respect to appropriately chosen reference ob-jects These reference objects are chosen in such a way that they make thephysical influence of background spacetime manifest In particular theirmotion can be chosen to be inertial In GR on the other hand the space-time coordinates x and t have no physical meaning physical predictionsof GR are independent of the coordinates x and t

24 Complements 75

A physical theory should not describe the location in space and theevolution in time of dynamical objects It describes relative location andrelative evolution of dynamical objects Newton introduced the notion ofbackground spacetime because he needed the acceleration of a particle tobe well defined (so that F = ma could make sense) In the newtoniantheory and in special relativity a particle accelerates when it does sowith respect to a fixed spacetime in which the particle moves In generalrelativity a particle (a dynamical object) accelerates when it does so withrespect to the local values of the gravitational field (another dynamicalobject) There is no meaning for the location of the gravitational field orthe location of the particle only the relative location of the particle withrespect to the gravitational field has physical meaning

What remains of the prerelativistic notion of spacetime is a relationbetween dynamical entities we can say that two particlesrsquo worldlines ldquoin-tersectrdquo that a field has a certain value ldquowhererdquo another field has a certainvalue or that we measure two partial observables ldquotogetherrdquo This is pre-cisely the modern realization of Descartesrsquo notion of contiguity and it isthe basis of spatial and temporal notions in GR

As Whitehead put it we cannot have spacetime without dynamicalentities anymore than saying that we can have the catrsquos grin withoutthe cat The world is made up of fields Physically these do not liveon spacetime They live so to say on one another No more fields onspacetime just fields on fields It is as outlined in the metaphor in Section113 where we no longer had animals on the island just animals on thewhale animals on animal Our feet are no longer in space we have to ridethe whale

24 Complements

I close this chapter by discussing a certain number of issues related to the interpretationof GR

241 Mach principles

The ideas of Ernst Mach had a strong influence on Einsteinrsquos discovery of GR Machpresented a number of acute criticisms to Newtonrsquos motivations for introducing absolutespace and absolute time In particular he pointed out that in Newtonrsquos bucket argumentthere is a missing element he observed that the inertial reference frame (the referenceframe with respect to which rotation has detectable physical effects) is also the referenceframe in which the fixed stars do not rotate Mach then suggested that the inertialreference frame is not determined by absolute space but rather it is determined by theentire matter content of the Universe including distant stars He suggested that if wecould repeat the experiment with a very massive bucket the mass of the bucket wouldaffect the inertial frame and the inertial frame would rotate with the bucket

In the light of GR the observation is certainly pertinent and it is clear that the ar-gument may have played a role in Einsteinrsquos dismissal of Newtonrsquos argument However


for some reason the precise relation between Machrsquos suggestion and GR has generateda vast debate Machrsquos suggestion that inertia is determined by surrounding matter hasbeen called ldquothe Mach principlerdquo and much ink has been employed to discuss whetheror not GR implements this principle whether or not ldquoGR is machianrdquo Remarkably inthe literature one finds arguments and proofs in favor as well as against the conclusionthat GR is machian Why this confusion

Because there is no well-defined ldquoMach principlerdquo Mach provided a very importantbut vague suggestion that Einstein developed into a theory not a precise statement thatcan be true or false Every author that has discussed ldquothe Mach principlerdquo has actuallyconsidered a different principle Some of these ldquoMach principlesrdquo are implemented inGR others are not

In spite of the confusion or perhaps thanks to it the discussion on how machianGR is sheds some light on the physical content of GR Here I list several versions ofthe Mach principle that have been considered in the literature and for each of theseI comment on whether this particular Mach principle is True or False in GR In thefollowing ldquomatterrdquo means any dynamical entity except the gravitational field

bull Mach principle 1 Distant stars can affect the local inertial frameTrue Because matter affects the gravitational field

bull Mach principle 2 The local inertial frame is completely determined by thematter content of the UniverseFalse The gravitational field has independent degrees of freedom

bull Mach principle 3 The rotation of the inertial reference frame inside the bucketis in fact dragged by the bucket and this effect increases with the mass of thebucketTrue In fact this is the LensendashThirring effect a rotating mass drags the inertialframes in its vicinity

bull Mach principle 4 In the limit in which the mass of the bucket is large theinternal inertial reference frame rotates with the bucketDepends It depends on the details of the way the limit is taken

bull Mach principle 5 There can be no global rotation of the UniverseFalse Einstein believed this to be true in GR but Godelrsquos solution is a counter-example

bull Mach principle 6 In the absence of matter there would be no inertiaFalse There are vacuum solutions of the Einstein field equations

bull Mach principle 7 There is no absolute motion only motion relative to some-thing else therefore the water in the bucket does not rotate in absolute terms itrotates with respect to some dynamical physical entityTrue This is the basic physical idea of GR

bull Mach principle 8 The local inertial frame is completely determined by thedynamical fields in the UniverseTrue In fact this is precisely Einsteinrsquos key idea

242 Relationalism versus substantivalism

In contemporary philosophy of science there is an interesting debate on the inter-pretation of GR The two traditional theses about space ndash absolute and relational ndashsuitably edited to take into account scientific progress continue under the names

24 Complements 77

of substantivalism and relationalism Here I present a few considerations on theissue

GR changes the notion of spacetime in physics in the sense of relationalism In pre-relativistic physics spacetime is a fixed nondynamical entity in which physics happensIt is a sort of structured container which is the home of the world In relativistic physicsthere is nothing of the sort There are only interacting fields and particles The onlynotion of localization which is present in the theory is relative dynamical objects canbe localized only with respect to one another This is the notion of space defendedby Aristotle and Descartes against which Newton wrote the initial part of PrincipiaNewton had two points the physical reality of inertial effects such as the concavityof the water in the bucket and the immense empirical success of his theory based onabsolute space Einstein provided an alternative interpretation for the cause of the con-cavity ndash interaction with the local gravitational field ndash and a theory based on relationalspace that has better empirical success than Newton theory After three centuriesthe European culture has returned to a fully relational understanding of space andtime

At the basis of cartesian relationalism is the notion of ldquocontiguityrdquo Two objects arecontiguous if they are close to one another Space is the order of things with respectto the contiguity relation At the basis of the spacetime structure of GR is essentiallythe same notion Einsteinrsquos ldquospacetime coincidencesrdquo are analogous to Descartes ldquocon-tiguityrdquo

A substantivalist position can nevertheless still be defended to some extent Ein-steinrsquos discovery is that newtonian spacetime and the gravitational field are the sameentity This can be expressed in two equivalent ways One states that there is no space-time there is only the gravitational field This is the choice I have made in this bookThe second states that there is no gravitational field it is spacetime that has dynamicalproperties This choice is common in the literature I prefer the first because I find thatthe differences between the gravitational field and other fields are more accidental thanessential But the choice between the two points of view is only a matter of choice ofwords and thus ultimately personal taste If one prefers to keep the name ldquospacetimerdquofor the gravitational field then one can still hold a substantivalist position and claimthat according to GR spacetime is an entity not a relation Furthermore localizationcan be defined with respect to the gravitational field and therefore the substantivalistcan say that spacetime is an entity that defines localization For an articulation of thisthesis see for instance [65]

However this is a very weakened substantivalist position One is free to call ldquospace-timerdquo anything with respect to which we define position But to what extent is space-time different from any arbitrary continuum of objects used to define position New-tonrsquos acute formulation of his substantivalism already mentioned in footnote 15 abovecontains a precise characterization of ldquospacerdquo

so it is necessary that the definition of places and hence of localmotion be referred to some motionless thing such as extension aloneor ldquospacerdquo in so far as space is seen to be truly distinct from movingbodies24

The characterizing feature of space is that of being truly distinct from moving bodiesthat is in modern terms and after the FaradayndashMaxwell conceptual revolution that of

24I Newton De Gravitatione et aequipondio fluidorum [61]


being truly distinct from dynamical entities such as particles or fields This is clearlynot the case for the spacetime of GR If the modern substantivalist is happy to give upNewtonrsquos strong substantivalism and identify the thesis that ldquospacetime is an entityrdquowith the thesis that ldquospacetime is the gravitational field which is a dynamical entityrdquothen the distinction between substantivalism and relationalism is completely reducedto one of semantics

When two opposite positions in a long-standing debate have come so close that theirdistinction is reduced to semantics one can probably say that the issue is solved I thinkone can say that in this sense GR has solved the long-standing issue of the relationalversus substantivalist interpretations of space

243 Has general covariance any physical content Kretschmannrsquosobjection

Virtually any field theory can be reformulated in a generally covariant form An exampleof a generally covariant reformulation of a scalar field theory on Minkowski spacetime ispresented below This fact has led some people to wonder whether general covariancehas any physical significance at all The argument is as follows if any theory canbe formulated in a general covariant language then general covariance is not a principlethat selects a particular class of theories therefore it has no physical content Thisargument was presented by Kretschmann shortly after Einsteinrsquos publication of GRIt is heard among some philosophers of science and sometimes used also by somephysicists that dismiss the conceptual novelty of GR

I think that the argument is wrong The non sequitur is the idea that a formal prop-erty that does not restrict the class of admissible theories has no physical significanceWhy should that be Formalism is flexible and we can artificially give a theory a cer-tain formal property especially if we accept byzantine formulations But it does notfollow from this that the use of one formalism or another is irrelevant Physics is thesearch for the more effective formalism to read Nature The relevant question is notwhether general covariance restricts the class of admissible theories but whether GRcould have been conceived or understood at all without general covariance Let meillustrate this point with the example of rotational invariance

Kretschmannrsquos objection applied to rotational symmetry Ancient physics assumed thatspace has a preferred direction The ldquouprdquo and the ldquodownrdquo were considered absolutelydefined This changes with newtonian physics where space has rotational symmetryall spatial directions are a priori equivalent and only contingent circumstances ndash suchas the presence of a nearby mass like the Earth ndash can make one direction particularPhysicists often say that rotational invariance limits the admissible forces But strictlyspeaking this is not true Kretschmannrsquos objection applies equally well to rotationalinvariance given a theory which is not rotationally invariant we can reformulate it asa rotationally invariant theory just by adding some variable For instance consider aphysical theory T in which all bodies are subject to a force in the z-direction F = minusgwhere g is a constant (such as gravity) This is a nonrotationally-invariant theory Nowconsider another theory T prime in which there is a dynamical vector quantity v of lengthunity and a force F = gv The theory T prime is rotationally invariant but in each solutionthe vector v will take a particular value in a particular direction Calling z this directionwe have precisely the same phenomenology as theory T

24 Complements 79

The example shows that we can express a nonrotationally invariant theory T ina rotationally invariant formalism T prime Therefore rotational invariance does not trulyrestrict the class of admissible theories Shall we conclude with Kretschmann thatrotational invariance has no physical significance

Obviously not Modern physics has made real progress with respect to ancientphysics in understanding that space is rotationally invariant Where is the progress Itis in the fact that the discovery of the rotational invariance of space puts us in a farmore effective position for understanding Nature We can say that we have discoveredthat in general there is no preferred ldquouprdquo and ldquodownrdquo in the Universe Equivalentlywe can say that a rotationally invariant physical formalism is far more effective forunderstanding Nature than a nonrotationally invariant one

There are two key issues here First it would have been difficult to find newtoniantheory within a conceptual framework in which the ldquouprdquo and the ldquodownrdquo are consideredabsolute Second reformulating the theory T in the rotationally invariant form T prime

modifies our understanding of it we have to introduce the dynamical vector v Fromthe point of view of the two theories T and T prime the vector v is a byzantine constructionwithout much sense But notice that from the point of view of understanding Naturethe introduction of v points to the physically correct direction we are led to investigatethe nature and the dynamics of this vector v is indeed the local gravitational fieldand this is precisely the right track towards a more effective understanding of NatureThis is the strength of having understood rotational invariance

In fact if there is rotational invariance in the Universe there should be a rota-tionally invariant manner of understanding ancient physics which in its limited ex-tent was effective Theory T prime above represents precisely this better understanding ofancient physics More than that the reinterpretation itself indicates a new effectiveway of understanding the world In conclusion the fact that the effective but non-rotationally invariant theory T admits the byzantine rotationally invariant formula-tion T prime is not an argument for the physical irrelevance of rotational invariance Farfrom that it is something that is required for us to have confidence in rotationalinvariance

On the one hand rotational invariance is interesting because it enlarges not be-cause it restricts the kind of physics we can naturally describe On the other handrotational invariance does drastically reduce the kind of theories that we are willingto consider Not because it forbids us to write certain theories ndash such as theory T prime

ndash but because if we want to describe a theory such as T we have to pay a priceHere the introduction of the vector v It is up to the theoretician to judge whetherthis price is worth paying that is whether v is in fact a physical entity worthwhileconsidering

The value of a novel idea or a novel language in theoretical physics is not in the factthat old physics cannot be expressed in the new language It is simply in the fact thatit is more effective for describing reality A physical theoretical framework is a map ofreality If the symbols of the map are better chosen the map is more effective A newlanguage by itself rarely truly restricts the kind of theories that can be expressed Butit renders certain theories far simpler and others awkward It orients our investigationon Nature This and nothing else is scientific knowledge

Let me come back to general covariance Like rotational invariance general covari-ance is a novel language which expresses a general physical idea about the worldIt is possible to express Newtonian physics in a generally covariant language It isalso possible to express GR physics in a nongenerally covariant language (by gauge-fixing the coordinates) But newtonian physics expressed in a covariant language or GR


expressed in a noncovariant language are both monsters formulated in a form far moreintricate than what is possible Nobody would have found them

What Einstein discovered is that two classes of entities previously considered dis-tinct are in fact entities of the same kind Newton taught us that (an effective wayto understand the world is to think that) the world is made up of two clearly distinctclasses of entities of very different nature The first class is formed by space and timeThe second class includes all dynamical entities moving in space and in time In new-tonian physics these two classes of entities are different in many respects and enterthe formalism of physical models in very different manners Einstein has understoodthat (a more effective way to understand the world is to think that) the world is notmade up of two distinct kinds of entities There is only one type of entity dynamicalfields General covariance is the language for describing a world without distinctionbetween the spacetime entities and the dynamical entities It is the language that doesnot assume this distinction

We can reinterpret prerelativistic physics in a generally covariant language It sufficesto rewrite the newtonian absolute space and absolute time as a dynamical field andthen write generally covariant equations that fix them to their flat-space values But ifwe do so we are not denying the physical content of Einsteinrsquos idea On the contrary weare simply reinterpreting the world in Einsteinrsquos terms In other words we are showingthe strength not the weakness of general covariance Furthermore in so doing weintroduce a new physical field and we find ourselves in the funny situation of havingto write equations of motion for this field that constrain it to a single value Thus wehave a theory where one of the dynamical fields is strangely constrained to a singlevalue This immediately suggests that perhaps we can relax these equations and allowa full dynamics for this field If we do so we are directly on the track of GR Again farfrom showing the physical irrelevance of general covariance this indicates its enormouscognitive strength

I think that the mistake behind Kretschmannrsquos argument is an excessively legalisticreading of the scientific enterprise It is the mistake of taking certain common physi-cistsrsquo statements too literally Physicists often write that a certain symmetry or a certainprinciple ldquouniquely determinesrdquo a certain theory At a close reading these statementsare almost always much exaggerated The uniqueness only holds under a vast number ofother assumptions that are left implicit and which are facts or ideas the physicist con-siders natural and does not bother detailing The typical physicist carelessly dismissescounter-examples by saying that they would be unphysical implausible or completelyartificial The connection between general physical ideas general principles intuitionssymmetries is a burning melt of powerful ideas not the icy demonstration of a math-ematical theorem What is at stake is finding the most effective language for thinkingthe world not writing axioms It is language in formation not bureaucracy25

25Historically the entire issue might be the result of a misunderstanding Kretschmannattacked Einstein in a virulent form In particular he attacked Einsteinrsquos coincidencessolution of the hole argument Now Einstein probably learned the idea that coinci-dences are the only observables precisely from Kretschmann but didnrsquot give muchcredit to Kretschmann for this I suppose this should have made Kretschmann quitebitter I think that Kretschmannrsquos subtext in saying that general covariance is emptywas not that general covariance was no progress with respect to old physics it wasthat general covariance was no progress with respect to what he himself had alreadyrealized before Einstein

24 Complements 81

Generally covariant flat-space field theory Consider the field theory of a free masslessscalar field φ(x) on Minkowski space The theory is defined by the action

S[φ] =

intd4x ηαβpartαφ partβφ (2141)

The equation of motion is the flat-space KleinndashGordon equation

ηαβpartαpartβφ = 0 (2142)

and the theory is obviously not generally covariantA trivial way to reformulate this theory in generally covariant language is to intro-

duce the tetrad field eαμ(x) and write the equations

partμ(e ηαβeμαeνβpartνφ) = 0 (2143)

Rαβμν = 0 (2144)

The solution of (2144) is that e is flat Since the system is covariant we can choose agauge in which eαμ(x) = δαμ In this gauge (2143) becomes (2141)

A more interesting way is as follows Consider a field theory for five scalar fieldsΦA(x) where A = 1 5 Use the notation

VA = εABCDE partμΦBpartνΦCpartρΦDpartσΦEεμνρσ (2145)

where εμνρσ and εABCDE are the 4-dimensional and 5-dimensional completely antisym-metric pseudo-tensors Consider the theory defined by the action

S[ΦA] =

intd4x V minus1

5 (V4V4 minus V3V3 minus V2V2 minus V1V1) (2146)

where V5 is assumed never to vanish The theory is invariant under diffeomorphismsIndeed VA transforms as a scalar density (because εμνρσ is a scalar density) hence theintegrand is a scalar density and the integral is invariant For α = 1 2 3 4 define thematrix

Eαμ (x) = partμΦα(x) (2147)

its inverse Eμα and its determinant E Varying Φ5 we obtain the equation of motion

partμ(E ηαβEμαE

νβpartνΦ5) = 0 (2148)

This is the massless KleinndashGordon equation (2143) interacting with a gravitationalfield Eα

μ Varying Φα we do not obtain independent equations We obtain the energy-momentum conservation law implied by (2148) The fact that there is only one inde-pendent equation is a consequence of the fact that there is a four-fold gauge invarianceWe can choose a gauge in which

Φa(x) = xa (2149)

We then have immediately Eaμ = δaμ and (2148) becomes (2142) The other four

equations are

parta(partaΦ5partbΦ

5 minus 1

2δab partcΦ

5partcΦ5) = 0 (2150)

Even better we may not fix the gauge and consider the gauge-invariant function offour variables φ(Xa) defined by

φ(Φa(x)) = Φ5(x) (2151)

This function satisfies the Minkowski-space KleinndashGordon equation (2142)How to interpret such a theory The theory (2141) is not generally covariant there-

fore its coordinates x are (partial) observables The theory is defined by five partial


observables four xμ and φ To interpret the theory we must have measuring proceduresassociated with these five quantities The relation between these observables is governedby (2141) On the other hand the theory (2146) is generally covariant therefore thecoordinates x are not observable The theory is defined by five partial observablesthe five φA We must have measuring procedures associated with these five quantitiesThe relation between these observables is governed again by (2141) Therefore in thetwo cases we have the same partial observables identified by ΦA harr (xa φ) relatedby the same equation

There is only one subtle but important difference between theory (2146) and theory(2141) Theory (2141) separates the five partial observables (x φ) into two sets theindependent ones (x) and the dependent one (φ) Theory (2146) treats the five partialobservables ΦA on an equal footing Thus in a strict sense theory (2141) containsone extra item of information a distinction between dependent and independent par-tial observables Because of this difference the two theories reflect two quite differentinterpretations of the world The first describes a worldrsquos ontology split into spacetimeand matter The second describes a world where the spacetime structure is interpretedas relational

244 Meanings of time

The concept of time used in natural language carries many properties Within a giventheoretical framework (say newtonian mechanics) time maintains some of these prop-erties and loses others In different theoretical frameworks time has different proper-ties The best-known example is probably the directionality of time absent in me-chanics present in thermodynamics But many other features of time lack in onetheory and are present in others For instance a property of time in newtonian me-chanics is uniqueness there is a unique time interval between any two events Con-versely in special relativity there are as many time variables as there are Lorentzobservers (x0 xprime0 ) Another attribute of time in newtonian mechanics is globalityevery solution of the equations of motion ldquopassesrdquo through every value of newtoniantime t once and only once In some cosmological models on the other hand thereis no choice of time variable with such a property there is ldquono timerdquo if we demandthat being global is an essential property of time In other words we use the wordldquotimerdquo to denote quite different concepts that may or may not include this or thatproperty

Here I describe a simple classification of possible attributes of time Below I identifyand list nine properties of time Then I describe and tabulate ten separate levels ofincreasing complexity of the notion of time corresponding to an increasing number ofproperties Theories typically fall in one of these levels according to the set of attributesthat the theory ascribes to the notion of time it uses The ten-fold arrangement isconventional the main point I intend to emphasize is that a single clear and purenotion of ldquotimerdquo does not exist

Properties of time Consider an infinite set S without any structure Add to S a topol-ogy and a differential structure dx Thus S becomes a manifold assume that thismanifold is one-dimensional and denote the set S together with its differentiable struc-ture as the line L = (Sdx) Next assume we add a metric structure d to L denotethe resulting metric line as M = (Sdx d) Next fix an ordering lt (a direction) inM Denote the resulting oriented line as the affine line A = (Sdx dlt) Next fix a

24 Complements 83

preferred point of A as the origin 0 the resulting space is isomorphic to the real lineR = (Sdx dlt 0)

The real line R is the traditional metaphor for the idea of time Time is frequentlyrepresented by a variable t in R The structure of R corresponds to an ensemble ofproperties that we naturally associate to the notion of time as follows (a) The existenceof a topology on the set of the time instants namely the existence of a notion of two timeinstants being close to each other and the fact that time is ldquoone-dimensionalrdquo (b) Theexistence of a metric Namely the possibility of stating that two distinct time intervalsare equal in magnitude time is ldquometricrdquo (c) The existence of an ordering relationbetween time instants Namely the possibility of distinguishing the past direction fromthe future direction (d) The existence of a preferred time instant the present theldquonowrdquo To capture these properties in mathematical language we describe time as areal line R An affine line A describes time up to the notion of present a metric lineM describes time up to the notions of present and pastfuture distinction a line Ldescribes time up to the notion of metricity

In newtonian mechanics we begin by representing time as a variable in R but thenthe equations are invariant both under t rarr minust and under t rarr t + a Thus the theoryis actually defined in terms of a variable t in a metric line M Newtonian mechanicsin fact incorporates both the notions of topology of the set of time instants and (ina very essential way) the fact that time is metric but it does not make any use ofthe notion of present nor the direction of time This is well known Note that Newtontheory is not inconsistent with the introduction of the notions of a present and of time-directionality it simply does not make any use of these notions These notions are notpresent in Newton theory

The properties listed above do not exhaust the different ways in which the notionof time enters physical theories the development of theoretical physics has modifiedsubstantially the natural notion of time A first modification was introduced by specialrelativity Einsteinrsquos definition of the time coordinate of distant events yields a notion oftime which is observer dependent An invariant structure can be maintained at the priceof relaxing the 1d character of time and the 3d character of space in favor of a notion of4d spacetime Alternatively we may say that the notion of a single time is replaced bya three-parameter family of times tv one for each Lorentz observer Therefore the timewe use in special relativity is not unique as is the time in newtonian mechanics Ratherthan a single line we have a three-parameter family of lines (the straight lines throughthe origin that fill the light cone of Minkowski space) Denote this three-parameter setof lines as M3

Times in GR There are several distinct possibilities of identifying ldquotimerdquo in GR Eachsingles out a different notion of time Each of these notions reduces to the standardnonrelativistic or special-relativistic time in appropriate limits but each lacks at leastsome of the properties of nonrelativistic time The most common ways of identifyingtime within GR are the following

Coordinate time x0 Coordinate time can be arbitrarily rescaled and does not pro-vide a way of identifying two time intervals as equal in duration Therefore it isnot metric in the sense defined above In addition the possibility of changingthe time coordinate freely from point to point implies that there is an infinite-dimensional choice of equally good coordinate times Finally unlike prerelativis-tic time x0 is not an observable quantity Denote the set of all the possiblecoordinate times as Linfin

Proper time τ This notion of time is metric But it is very different from the notion of


time in special relativity for several reasons First it is determined by the grav-itational field Second we have a different time for each worldline or infinitesi-mally for every speed at every point For an infinitesimal timelike displacementdxμ at a point x the infinitesimal time interval is dτ =

radicminusgμν(x) dxμdxν This

notion of time is a radical departure from the notion of time used in special rela-tivity because it is determined by the dynamical fields in the theory A solutionof Einsteinrsquos equations defines a point in the phase space Γ of GR It assignsa metric structure to every worldline Therefore this notion of time is given bya function from the phase space Γ multiplied by the set of the worldlines wlinto the metric structures d wl times wl rarr R+ Denote this function as minfin Callldquointernalrdquo a notion of time affected by the dynamics

Before GR dynamics could be expressed as evolution in a single time vari-able which has metric properties and could be measured In general-relativisticphysics this concept of time splits into two distinct concepts we can still viewthe dynamics as evolution in a time variable x0 but this time has no metricproperties and is not observable alternatively there is a notion of time thathas metric properties τ but the dynamics of the theory cannot be expressed asevolution in τ Is there a way to go around this split and view GR as a dynam-ical theory in the sense of a theory expressing evolution in an observable metrictime

Clock time The dynamics of GR determines how observable quantities evolve withrespect to one another We can always choose one observable quantity tc de-clare it the independent one and describe how the other observables evolve asfunctions of it A typical example of this clock time is the radius of a spatiallycompact universe in relativistic cosmology R Formally clock time is a functionon the extended configuration space C of the theory (see Chapter 3) Denote thisnotion of time as the clock time τc C rarr R

Under this definition of time GR becomes similar to a standard hamiltoniandynamical theory A clock time however generally behaves as a clock only incertain states or for a limited amount of time The radius of the universe forinstance fails as a good time variable when the universe recollapses In gen-eral a clock time lacks temporal globality In fact several results are knownconcerning obstructions to defining a function tc that behaves as ldquoa good timerdquoglobally [66]

Notice that some of these relativistic notions of time are in a sense opposite to theprerelativistic case while in newtonian theory time evolution is captured by a functionfrom the metric line M (time) to the configuration or phase space now the notion oftime is captured by a function from the configuration or phase space to the metric lineThis inversion is the mathematical expression of the physical idea that the flow of timeis affected or determined by the dynamics of the system itself

Finally none of the ways of thinking of time in classical GR can be uncriticallyextended to the quantum regime Quantum fluctuations of physical clocks and quan-tum superposition of different metric structures make the very notion of time fuzzy atthe Planck scale As will be discussed in the second part of this book a fundamentalconcept of time may be absent in quantum gravity

Notions of time Notice that properties of time progressively disappear in going towardmore fundamental physical theories At the opposite end of the spectrum there are

24 Complements 85

properties associated with the notion of time used in the natural languages which arenot present in physical theories They play a role in other areas of natural investigationsI mention these properties for the sake of completeness These are for instance memoryexpectations and the psychological perception of free will

To summarize I have identified the following properties of the notion of time

1 Existence of memory and expectations2 Existence of a preferred instant of time the present the now3 Directionality the possibility of distinguishing the past from the future direction4 Uniqueness the feature that is lost in special and general relativity where we

cannot identify a preferred time variable5 The property of being external the independence of the notion of time from the

dynamical variables of the theory6 Spatial globality the possibility of defining the same time variable in all space

points7 Temporal globality the fact that every motion goes through every value of the

time variable once and only once8 Metricity the possibility of saying that two time intervals have equal duration9 One-dimensionality namely the possibility of arranging the time instants in a

one-dimensional manifold

This discussion suggests a sequence of notions of time which I list here in order ofdecreasing complexity

Time of natural language This is the notion of time of everyday language which in-cludes all the features just listed This notion of time is not necessarily nonscien-tific for instance any scientific approach to say the human brain should makeuse of this notion of time

Time-with-a-present This is the notion of time that has all the features just listedincluding the existence of a preferred instant the present but not the notionsof memory and expectations which are notions usually more related to complexsystems (brain) than to time itself The notion of present is generally considereda feature of time itself This notion of time is the one to which often people referwhen they refer to the ldquoflow of timerdquo or Eddingtonrsquos ldquovivid perception of theflow of timerdquo [67] This notion of time can be described by the structure of aparametrized line R

Thermodynamical time If we maintain the distinction between a future direction anda past direction but we give up the notion of present we obtain the notionof time typical of thermodynamics Since thermodynamics is the first physicalscience that appears in this list this is maybe a good place to emphasize that thenotion of present of the ldquonowrdquo is completely absent from the description of theworld in physical terms This notion of time can be described by the structureof an affine line A

Newtonian time In newtonian mechanics there is no preferred direction of time Noticethat in the absence of a preferred direction of time the notions of cause and effectare interchangeable This notion of time can be described by the structure of ametric line M

Special-relativistic time If we give up uniqueness we have the time used in specialrelativity different Lorentz observers have a different notion of time Special-relativistic time is still external spatially and temporally global metrical andone-dimensional but it is not unique There is a three-parameter set of quantitiesthat share the status of time This notion of time can be described by the three-parameter set of metric lines M3


Table 21 Notions of time

Time notion Property Example Form

natural language time memory brain time-with-a-present present biology Rthermodynamical time direction thermodynamics Anewtonian time unique newtonian mechanics Mspecial-relativistic time external special relativity M3

cosmological time spatially global cosmological time mproper time temporally global worldline proper time minfin

clock time metric clocks in GR cparameter time one-dimensional coordinate time Linfin

no-time none quantum gravity none

Cosmological time By this I indicate a time which is spatially and temporally globalmetrical and one-dimensional but it is not external namely it is dynamicallydetermined by the theory Proper time in cosmology is the typical example Itis the most structured notion of time that occurs in GR Denote it by m

Proper time By this I indicate a time which is temporally global metrical and one-dimensional but it is not spatially global as the notion of proper time alongworldlines in GR It can be represented by a function minfin defined on the carte-sian product of the phase space and the ensemble of the worldlines

Clock time By this I indicate a time which is metrical and one-dimensional but itis not temporally global A realistic matter clock in GR defines a time in thissense This notion of time can be described by a function c on the phase space

Parameter time By this we mean a notion of time which is not metric and not ob-servable The typical example is the coordinate-time in GR Another exampleof parameter time is the evolution parameter in the parametrized formulationof the dynamics of a relativistic particle Parameter time is described by anunparametrized line L or by an infinite set Linfin of unparametrized lines

No-time Finally this is the bottom level in the analysis it is not a time concept butrather I indicate by no-time the idea that a predictive physical theory can bewell defined also in the absence of any notion of time

The list must not be taken rigidly It is summarized in the Table 21There is a interesting feature that emerges from the above analysis the hierarchical

arrangement While some details of this arrangement may be artificial neverthelessthe analysis points to a general fact moving from theories of ldquospecialrdquo objects likethe brain or living beings toward more general theories that include larger portionsof Nature we make use of a physical notion of time that is less specific and has lessdeterminations If we observe Nature at progressively more fundamental levels andwe seek for laws that hold in more general contexts then we discover that these lawsrequire or admit an increasingly weaker notion of time

This observation suggests that ldquohigh levelrdquo features of time are not present at thefundamental level but ldquoemergerdquo as features of specific physical regimes like the notionof ldquowater surfacerdquo emerges in certain regimes of the dynamics of a combination of waterand air molecules (see for instance [68])

24 Complements 87

Notions of time with more attributes are high-level notions that have no meaning inmore general situations The uniqueness of newtonian time for instance makes senseonly in the special regime in which we consider an ensemble of bodies moving slowlywith respect to each other Thus the notion of a unique time is a high-level notionthat makes sense only for some regimes in Nature For general systems most featuresof time are genuinely meaningless

245 Nonrelativistic coordinates

The precise meaning of the coordinates x = (x t) in newtonian and special-relativisticphysics is far from obvious Let me recall it here in order to clarify the precise differencebetween these and the relativistic coordinates

Newton is well aware that the motions we observe are relative motions and stressesthis point in Principia His point is not that we can directly observe absolute motionHis point is that we can infer the absolute motion or ldquotrue motionsrdquo or motion withrespect to absolute space from its physical effects (such as the concavity of the waterin the bucket) starting from our observation of relative motions

For instance we observe and describe motions with respect to Earth but fromsubtle effects such as Foucaultrsquos pendulum we infer that these are not true motionsThe experiment of the bucket is an example of the possibility of revealing true motion(rotation of the water with respect to space) disentangling it from relative motion(rotation with respect to the bucket) by means of an observable effect (the concavityof the water surface)26

For Newton the coordinates x that enter his main equation

F = md2x(t)

dt2(2152)

are the coordinates of absolute space However since we cannot directly observe spacethe only way we can coordinatize space points is by using physical objects The co-ordinates x of the object A moving along the trajectory x(t) are therefore definedas distances from a chosen system O of objects which we call a ldquoreference framerdquoBut then x are not the coordinates of absolute space So how can equation (2152)work

The solution of the difficulty is to use the capacity of unveiling ldquotrue motionrdquo thatNewton has pointed out in order to select the objects forming the reference frame Owisely There are ldquogoodrdquo and ldquobadrdquo reference frames The good ones are the ones inwhich no effect such as the concavity of the water surface of Newtonrsquos bucket can be

26Newton accords deep significance to the fact that we can unveil true motion Hedescribes relative motion as the way reality is observed by us and true motion as theway reality might be directly ldquoperceivedrdquo or ldquosensedrdquo by God This is why Newtoncalls space ndash the entity with respect to which true motion happens ndash the ldquoSensoriumof Godrdquo true motion is motion ldquowith respect to Godrdquo or ldquoas perceived by GodrdquoThere is a platonic tone in this idea that reason finds the way to the veiled divinetruth beyond appearances I wouldnrsquot read this as so removed from modernity asit is often portrayed There isnrsquot all that much difference between Newtonrsquos inquiryinto Godrsquos way of ldquosensing the worldrdquo and the modern search for the most effectiveway of conceptualizing reality Newtonrsquos God plays a mere linguistical role herethe role of denoting a major enterprise upgrading our own conceptual structure forunderstanding reality


observed within a desired accuracy Equation (2152) is correct to the desired accuracyif we use coordinates defined with respect to these good frames In other words thephysical content of (2152) is actually quite subtle

There exist reference objects O with respect to which the motion ofany other object A is correctly described by (2152)

This is a statement that begins to be meaningful only when a sufficiently large numberof moving objects is involved

Notice also that for this construction to work it is important that the objects Oforming the reference frame are not affected by the motion of the object A Thereshouldnrsquot be any dynamical interaction between A and O

Special relativity does not change much of this picture Since absolute simultaneitymakes no sense if the event A is distant from the clock in the origin its time t is illdefined Einsteinrsquos idea is to define a procedure for assigning a t to distant events usingclocks moving inertially

At clock time te send a light signal that reaches the event Receivethe reflected signal back at tr The time coordinate of the event isdefined to be tA = 1

2 (te + tr)

It is important to emphasize that this is a useful definition not a metaphysical state-ment that the event A happens ldquoright at the time whenrdquo the observer clock displaystA

Special relativity replaces Newtonrsquos absolute space and absolute time with a singleentity Minkowskirsquos absolute spacetime while the notion of inertial system and themeaning of the coordinates are the same as in newtonian mechanics

Summarizing these coordinates have the following properties

(i) Coordinates describe position with respect to physical reference objects (referenceframes)

(ii) Space coordinates are defined by the distance from the reference bodies Timecoordinates are defined with respect to isochronous clocks

(iii) Reference objects are appropriately chosen they are such that the reference systemthey define is inertial

(iv) Inertial frames reveal the structure of absolute spacetime itself

(v) The object A whose dynamics is described by the coordinates does not interactwith the reference objects O There is no dynamical coupling between A and O

Relativistic coordinates do not have any of these properties The fact that the two areindicated with the same notation xμ is only an unfortunate historical accident

246 Physical coordinates and GPS observables

Instead of working with arbitrary unphysical coordinates xμ we can choose to coordina-tize spacetime events with coordinates Xμ having an assigned physical interpretationFor instance we can describe the Universe by giving a name X to each galaxy andchoosing X0 as the proper time from the Big Bang along the galaxy worldline If wedo so the defining properties of the coordinates X must be added to the formalismWe must add a certain number of equations for the gravitational field the equations ofmotions of the objects used to fix the coordinates (the galaxies in the example) Theseadditional equations gauge-fix general covariance

24 Complements 89

The gauge-fixing can also be partial For instance a common choice is

e00(X) = 1 ei0(X) = 0 e0

a(X) = 0 (2153)

where i = 1 2 3 and a = 1 2 3 This corresponds to partially fixing the coordinates byrequiring that X0 measures proper time that equal X0 surfaces are locally instantaneitysurfaces in the sense of Einstein for the constant X lines and that the local Lorentzframes are chosen so that these lines are still

If the coordinates are fully specified the set formed by these physical gauge-fixingequations and the equations of motion has no residual gauge invariance that is ini-tial data determine evolution uniquely This procedure can be implemented in manypossible ways since there are arbitrarily many ways of fixing physical coordinates andnone is a priori better than any other In spite of this arbitrariness this procedure isoften convenient when the physical situation suggests a natural coordinate choice asin the cosmological context mentioned

Physical coordinates Xμ defined by matter filling space can only be effectively usedin the cosmological context because it is only at the cosmological scale that matter fillsspace In a system in which there are empty regions such as the Solar System thesephysical coordinates are not available An interesting alternative choice is provided bythe GPS coordinates described below

The physical coordinates Xμ are partial observables and we can associate measuringdevices with them

Undetermined physical coordinates Finally there is a third interpretation of the co-ordinates of GR which is intermediate between arbitrary coordinates xμ and physicalcoordinates Xμ Imagine that a region of the universe is filled with certain light objectswhich may not be in free fall We can use these objects to define physical coordinatesXμ but also choose to ignore the equations of motion of these objects We obtain asystem of equations for the gravitational field and other matter expressed in termsof coordinates Xμ that are interpreted as the spacetime location of reference objectswhose dynamics we have chosen to ignore

This set of equations is under-determined the same initial conditions can evolve intodifferent solutions However the interpretation of such under-determination is simplythat we have chosen to neglect part of the equations of motion Different solutions withthe same initial conditions represent the same physical configuration of the fields butexpressed say in one case with respect to free-falling reference objects in the othercase with respect to reference objects on which a force has acted at a certain momentand so on This procedure has the disadvantage of being useless in quantum theorywhere we cannot assume that something is observable and at the same time neglect itsdynamics

In conclusion one should always be careful in talking about general-relativistic co-ordinates whether one is referring to

(i) arbitrary mathematical coordinates x

(ii) physical coordinates X with an interpretation as positions with respect to objectswhose equations of motion are taken into account

(iii) physical coordinates with an interpretation as positions with respect to objectswhose equations of motion are ignored

The system of equations of motion is nondeterministic in (i) and (iii) deterministic in(ii) The coordinates are partial observables in (ii) and (iii) but not in (i) Confusionabout observability in GR follows from confusing these three different interpretationsof the coordinates The following is an example of physical coordinates


GPS observables In the literature there are many attempts to define useful physicalcoordinates It is easier to define physical coordinates in the presence of matter thanin the context of pure GR Ideally we can consider GR interacting with four scalarmatter fields Assume that the configuration of these fields is sufficiently nondegener-ate Then the components of the gravitational field at points defined by given valuesof the matter fields are gauge-invariant observables This idea has been developed ina number of variants such as dust-carrying clocks and others (see [69ndash71] and refer-ences therein) The extent to which the result is realistic or useful is questionable Itis rather unsatisfactory to understand the theory in terms of fields that do not existor phenomenological objects such as dust and it is questionable whether these pro-cedures could make sense in the quantum theory where the aim is to describe Planckscale dynamics Earlier attempts to write a complete set of gauge-invariant observ-ables are in the context of pure GR [72] The idea is to construct four scalar functionsof the gravitational field (say scalar polynomials of the curvature) and use these tolocalize points The value of a fifth scalar function in a point where the four scalarfunctions have a given value is a gauge-invariant observable This works but the resultis mathematically very intricate and physically very unrealistic It is certainly possiblein principle to construct detectors of such observables but I doubt any experimenterwould get funded for a proposal to build such an apparatus

There is a simple way out based on GR coupled with a minimal and very realisticamount of additional matter Indeed this way out is so realistic that it is in fact realit is essentially already implemented by existing technology the Global PositioningSystem (GPS) which is the first technological application of GR or the first large-scaletechnology that needs to take GR effects into account [73]

Consider a generally covariant system formed by GR coupled with four small bodiesThese are taken to have negligible mass they will be considered as point particlesfor simplicity and called ldquosatellitesrdquo Assume that the four satellites follow timelikegeodesics that these geodesics meet in a common (starting) point O and at O theyhave a given (fixed) speed ndash the same for all four ndash and directions as the four vertices ofa tetrahedron The theory might include any other matter Then (there is a region R ofspacetime for which) we can uniquely associate four numbers sα α = 1 2 3 4 to eachspacetime point p as follows Consider the past lightcone of p This will (generically)intersect the four geodesics in four points pα The numbers sα are defined as thedistance between pα and O (That is the proper time along the satellitesrsquo geodesic)We can use the sα as physically defined coordinates for p The components gαβ(s) of themetric tensor in these coordinates are gauge-invariant quantities They are invariantunder four-dimensional diffeomorphisms (because these deform the metric as well asthe satellitesrsquo worldlines) They define a complete set of gauge-invariant observables forthe region R

The physical picture is simple and its realism is transparent Imagine that the fourldquosatellitesrdquo are in fact satellites each carrying a clock that measures the proper timealong its trajectory starting at the meeting point O Imagine also that each satellitebroadcasts its local time with a radio signal Suppose I am at the point p and have anelectronic device that simply receives the four signals and displays the four readings seeFigure 25 These four numbers are precisely the four physical coordinates sα definedabove Current technology permits us to perform these measurements with an accu-racy well within the relativistic regime [73 74] If we then use a rod and a clock andmeasure the physical 4-distances between sα coordinates we are directly measuring thecomponents of the metric tensor in the physical coordinate system In the terminologyof Chapter 3 the sα are partial observables while the gαβ(s) are complete observables

24 Complements 91

O

p

Σ

t

x

s2s1

Fig 25 s1 and s2 are the GPS coordinates of the point p Σ is a Cauchy surfacewith p in its future domain of dependence

As shown below the physical coordinates sα have nice geometrical properties theyare characterized by

gαα(s) = 0 α = 1 4 (2154)

Surprisingly in spite of the fact that they are defined by what looks like a rather non-local procedure the evolution equations for gαβ(s) are local These evolution equationscan be written explicitly using the ArnowittndashDeserndashMisner (ADM) variables (see [131]of Chapter 3 for details) Lapse and Shift turn out to be fixed local functions of thethree metric

In what follows I first introduce the GPS coordinates sα in Minkowski space ThenI consider a general spacetime I assume the Einstein summation convention only forcouples of repeated indices that are one up and one down Thus in (2154) α is notsummed While dealing with Minkowski spacetime the spacetime indices μ ν are raisedand lowered with the Minkowski metric Here I write an arrow over three- as well asfour-dimensional vectors Also here I use the signature [+minusminusminus] in order to havethe same expressions as in the original article on the subject

Consider a tetrahedron in three-dimensional euclidean space Let its center be atthe origin and its four vertices vα where the vectors vα have unit length |vα|2 = 1 andvα middot vβ = minus13 for α = β Here α = 1 2 3 4 is an index that distinguishes the fourvertices and should not be confused with vector indices With a convenient orientationthese vertices have cartesian coordinates (a = 1 2 3)

v1a = (0 0 1) v2a = (2radic

23 0 minus13) (2155)

v3a = (minusradic

23radic

23 minus13) v4a = (minusradic

23 minusradic

23 minus13) (2156)

Let us now go to a four-dimensional Minkowski space Consider four timelike 4-vectorsWα of length unity | Wα|2 = 1 representing the normalized 4-velocities of four par-ticles moving away from the origin in the directions vα at a common speed v Their


Minkowski coordinates (μ = 0 1 2 3) are

Wαμ =1radic

1 minus v2(1 v vαa) (2157)

Fix the velocity v by requiring the determinant of the matrix Wαμ to be unity (Thischoice fixes v at about one-half the speed of light a different choice changes only a fewnormalization factors in what follows) The four by four matrix Wαμ plays an importantrole in what follows Notice that it is a fixed matrix whose entries are certain givennumbers

Consider one of the four 4-vectors say W = W 1 Consider a free particle inMinkowski space that starts from the origin with 4-velocity W Call it a ldquosatelliterdquo Itsworldline l is x(s) = s W Since W is normalized s is precisely the proper time alongthe worldline Consider now an arbitrary point p in Minkowski spacetime with coordi-nates X Compute the value of s at the intersection between l and the past lightconeof p This is a simple exercise giving

s = X middot W minusradic

( X middot W )2 minus | X|2 (2158)

Now consider four satellites moving out of the origin at 4-velocity Wα If they radiobroadcast their position an observer at the point p with Minkowski coordinates Xreceives the four signals sα

sα = X middot Wα minusradic

( X middot Wα)2 minus | X|2 (2159)

Introduce (nonlorentzian) general coordinates sα on Minkowski space defined by thechange of variables (2159) These are the coordinates read out by a GPS device inMinkowski space The jacobian matrix of the change of coordinates is given by

partsα

partxμ= Wα

μ minus Wαμ ( X middot Wα) minusXμradic( X middot Wα)2 minus | X|2

(2160)

where Wαμ and Xμ are Wαμ and Xμ with the spacetime index lowered with the

Minkowski metric This defines the tetrad field eαμ(s)

eαμ(s(X)) =partsα

partxμ(X) (2161)

The contravariant metric tensor is given by gαβ(s) = eαμ(s)eμβ(s) Using the relation

| Wα|2 = 1 a straightforward calculation shows that

gαα(s) = 0 α = 1 4 (2162)

This equation has the following nice geometrical interpretation Fix α and considerthe one-form field ωα =dsα In sα coordinates this one-form has components ωα

β = δαβ

and therefore ldquolengthrdquo |ωα|2 = gβγωαβω

αγ = gαα But the ldquolengthrdquo of a one-form is

proportional to the volume of the (infinitesimal now) 3-surface defined by the formThe 3-surface defined by dsα is the surface sα = constant But sα = constant is the setof points that read the GPS coordinate sα namely that receive a radio broadcastingfrom a same event pα of the satellite α namely that are on the future lightcone of pαTherefore sα = constant is a portion of this lightcone it is a null surface and thereforeits volume is zero And so |ωα|2 = 0 and gαα = 0

24 Complements 93

Since the sα coordinates define sα = constant surfaces that are null we denote themas ldquonull GPS coordinatesrdquo It is useful to introduce another set of GPS coordinates aswell which have the traditional timelike and spacelike character We denote these assμ call them ldquotimelike GPS coordinatesrdquo and define them by

sα = Wαμ sμ (2163)

This is a simple algebraic relabeling of the names of the four GPS coordinates suchthat sμ=0 is timelike and sμ=a is spacelike In these coordinates the gauge condition(2162) reads

Wαμ Wα

ν gμν(s) = 0 (2164)

This can be interpreted geometrically as follows The (timelike) GPS coordinates arecoordinates sμ such that the four one-form fields

ωα = Wαμ dsμ (2165)

are nullLet us now jump from Minkowski space to full GR Consider GR coupled with four

satellites of negligible mass that move geodesically and whose worldlines emerge froma point O with directions and velocity as above Locally around O the metric can betaken to be minkowskian therefore the details of the initial conditions of the satellitesrsquoworldlines can be taken as above The phase space of this system is the one of pureGR plus ten parameters giving the location of O and the Lorentz orientation of theinitial tetrahedron of velocities The integration of the satellitesrsquo geodesics and of thelightcones can be arbitrarily complicated in an arbitrary metric However if the metricis sufficiently regular there will still be a region R in which the radio signals broadcastby the satellites are received (In the case of multiple reception the strongest one can beselected That is if the past lightcone of p intersects l more than once generically therewill be one intersection which is at shorter luminosity distance) Thus we still havewell-defined physical coordinates sα on R Equation (2162) holds in these coordinatesbecause it depends only on the properties of the light propagation around p We definealso timelike GPS coordinates sμ by (2163) and we get condition (2164) on the metrictensor

To study the evolution of the metric tensor in GPS coordinates it is easier to shiftto ADM variables NNa γab These are functions of the covariant components of themetric tensor defined in general by

ds2 = gμνdxμdxν = N2dt2 minus γab(dxa minusNadt)(dxb minusNbdt) (2166)

Equivalently they are related to the contravariant components of the metric tensor by

gμνvμvν = minusγabvavb + (nμvμ)2 (2167)

where γab is the inverse of γab and nμ = (1NNaN) Using these variables the gaugecondition (2164) reads

Wαa Wα

b γab = (Wαμ nμ)2 (2168)

Notice now that this can be solved for the Lapse and Shift as a function of the 3-metric(recall that Wα

μ are fixed numbers) obtaining

nμ = Wμα q

α (2169)


where Wμα is the inverse of the matrix Wα

μ and

qα =radic

Wαa Wα

b γab (2170)

Or explicitly

N =1

W 0αqα

Na =W a

αqα

W 0αqα

(2171)

The geometrical interpretation is as follows We want the one-form ωα defined in (2165)to be null namely its norm to vanish But in the ADM formalism this norm is the sumof two parts the norm of the pull-back of ωα on the constant time ADM surfacewhich is qα given in (2170) and depends on the three metric plus the square of theprojection of ωα on nμ We can thus obtain the vanishing of the norm by adjusting theLapse and Shift We have four conditions (one per α) and we can thus determine Lapseand Shift from the 3-metric In other words whatever the 3-metric we can alwaysadjust Lapse and Shift so that the gauge condition (2164) is satisfied But in theADM formalism the arbitrariness of the evolution in the Einstein equations is entirelycaptured by the freedom in choosing Lapse and Shift Since here Lapse and Shift areuniquely determined by the 3-metric evolution is determined uniquely if the initialdata on a Cauchy surface are known Therefore the evolution in the GPS coordinates0 of the GPS components of the metric tensor gμν(s) is governed by deterministicequations the ADM evolution equation with Lapse and Shift determined by (2170)ndash(2171) Notice also that evolution is local since the ADM evolution equations as wellas the (2170)ndash(2171) are local27

How can the evolution of the quantities gμν(s) be local The conditions on thenull surfaces described in the previous paragraph are nonlocal Coordinate distancestypically yield nonlocality imagine we define physical coordinates in the Solar Systemusing the cosmological time tc and the spatial distances xS xE xJ (at fixed tc) from saythe Sun the Earth and Jupiter The metric tensor gμν(tc xS xE xJ) in these coordinatesis a gauge-invariant observable but its evolution is highly nonlocal To see this imaginethat in this moment (in cosmological time) Jupiter is swept away by a huge cometThen the value of gμν(tc xS xE xJ) here changes instantaneously without any localcause the value of the coordinate xJ has changed because of an event happening faraway Whatrsquos special about the GPS coordinates that avoids this nonlocality Theanswer is that the value of a GPS coordinate at a point p does in fact depend onwhat happens ldquofar awayrdquo as well Indeed it depends on what happens to the satelliteHowever it only depends on what happened to the satellite when it was broadcastingthe signal received in p and this is in the past of p If p is in the past domain ofdependence of a partial Cauchy surface Σ then the value of gμν(s) in p is completelydetermined by the metric and its derivative on Σ namely evolution is causal becausethe entire information needed to set up the GPS coordinates is in the data in Σ seeFigure 25 Explicitly the sα = constant surfaces around Σ can be uniquely integratedahead all the way to p They certainly can as they represent just the evolution of alight front This is how local evolution is achieved by these coordinates

Summarizing I have introduced a set of physical coordinates determined by certainmaterial bodies Geometrical quantities such as the components of the metric tensorexpressed in physical coordinates are of gauge-invariant observables There is no needto introduce a large unrealistic amount of matter or to construct complicated andunrealistic physical quantities out of the metric tensor Four particles are sufficient to

27This does not imply that the full set of equations satisfied by gμν(s) must be localsince initial conditions on s0 = 0 satisfy four other constraints besides the ADM ones

24 Complements 95

Fig 26 A simple apparatus to measure the gravitational field Two GPS de-vices reading sμL and sμR respectively connected by a 1 meter rod If for instancesμR = sμL for μ = 0 2 3 then the local value of g11(s) is g11(s) = (s1

R minus s1L)minus2m2

coordinatize a (region of a) four-geometry Furthermore the coordinatization procedureis not artificial it is the real one utilized by existing technology

The components of the metric tensor in (timelike) GPS coordinates can be measuredas follows (see Figure 26) Take a rod of physical length L (small with respect to thedistance along which the gravitational field changes significantly) with two GPS de-vices at its ends (reading timelike GPS coordinates) Orient the rod (or search amongrecorded readings) so that the two GPS devices have the same reading s of all coor-dinates except for s1 Let δs1 be the difference in the two s1 readings Then we havealong the rod

ds2 = g11(s)δs1δs1 = L2 (2172)

Therefore

g11(s) =

(L

δs1

)2

(2173)

Nondiagonal components of gab(s) can be measured by simple generalizations of thisprocedure The g0b(s) are then algebraically determined by the gauge conditions Ina thought experiment data from a spaceship traveling in a spacetime region could beused to produce a map of values of the GPS components of the metric tensor Insteadof using a rod which is a rather crude device for measuring distances one could senda light pulse forward and back between the two GPS devices kept at fixed spatial sμ

coordinates If T is the (physical) time for flying back and forward measured by aprecise clock on one device then g11(s) = (cT2δs1)2 This is valid so long as T andL are small compared to the distances over which the gab(s) change by amounts of theorder of the experimental errors

The individual components of the metric tensor expressed in physical coordinatesare measurable The statement that ldquothe curvature is measurable but the metric is notmeasurablerdquo which is often heard is incorrect Both metric and curvature in physicalcoordinates are measurable and predictable Neither metric nor curvature in arbitrarynonphysical coordinates are measurable

The GPS coordinates are partial observables (see Chapter 3) The complete ob-servables are the quantities gμν(s) for any given value of the coordinates sμ Thesequantities are diffeomorphism invariant are uniquely determined by the initial dataand in a canonical formulation are represented by functions on the phase space thatcommute with all constraints

The GPS observables are a straightforward generalization of Einsteinrsquos ldquospacetimecoincidencesrdquo In a sense they are precisely Einsteinrsquos point coincidences Einsteinrsquosldquomaterial pointsrdquo are just replaced by photons (light pulses) the spacetime point sα is


characterized as the meeting point of four photons designated by the fact of carryingthe radio signals sα

mdashmdash


There are many beautiful classic textbooks on GR Two among the bestoffering remarkably different points of view on the theory are Weinberg[75] and Wald [76] The first stresses the similarity between GR and flat-space field theory the second on the contrary emphasizes the geometricreading of GR Here I have followed a third path I place emphasis onthe change of the notions of space and time needed for general-relativisticphysics (which affects quantization dramatically) but I put little emphasison the geometric interpretation of the gravitational field (which is goingto be largely lost in the quantum theory)

Relevant mathematics is nicely presented for instance in the text byChoquet-Bruhat DeWitt-Morette and Dillard-Bleick [77] and in [16] Onthe large empirical evidence in favor of GR piled up in the recent yearssee Ciufolini and Wheeler [78]

The tetrad formalism and its introduction into quantum gravity aremainly due to Cartan to Weyl [80] and to Schwinger [80] the first-orderformalism to Palatini The Plebanski two-form was introduced in [81] Theselfdual connection which is at the root of Ashtekarrsquos canonical theory(see Chapter 4) was introduced by Amitaba Sen [82] The lagrangianformulation for the selfdual connections was given in [83] A formulationof GR based on the sole connection is discussed in [84]

Interesting reconstructions of Einsteinrsquos path towards GR are in [8586]Kretschmannrsquos objection to the significance of general covariance ap-peared in [87] On this see also Andersonrsquos book [88] An account ofthe historical debate on the interpretation of space and motion is JulianBarbourrsquos [89] a wonderful historical book In the philosophy of sciencethe debate was reopened by a 1987 paper on the hole argument by JohnEarman and John Norton [90] On the contemporary version of this de-bate see [65 91ndash93] On the physical side of the discussion of what isldquoobservablerdquo in GR see [71]

The discussion of the different notions of time follows [94] A surprisingand inspiring book on the subject is Fraser [95] a book that will convincethe reader that the notion of time is far from being a monolithic conceptThe literature on the problem of time in quantum gravity is vast I list onlya few pointers here distinguishing various problems origin of the ldquoarrowof timerdquo and the cosmological time asymmetry [96] disappearance of thecoordinate-time variable in canonical quantum gravity [97] possibility of


a consistent interpretation of quantum mechanics for systems withoutglobal time [269899] problems in choosing an ldquointernal timerdquo in generalrelativity and the properties that such an internal time should have [66]see also [100] The presentation of the GPS observables follows [101] seealso [102103]

3Mechanics

In its conventional formulation mechanics describes the evolution of states and ob-servables in time This evolution is governed by a hamiltonian This is also true forspecial-relativistic theories where evolution is governed by a representation of thePoincare group which includes a hamiltonian This conventional formulation is notsufficiently broad because general-relativistic systems ndash in fact the world in which welive ndash do not fit into this conceptual scheme Therefore we need a more general formula-tion of mechanics than the conventional one This formulation must be based on notionsof ldquoobservablerdquo and ldquostaterdquo that maintain a clear meaning in a general-relativistic con-text A formulation of this kind is described in this chapter

The conventional structure of conventional nonrelativistic mechanics already pointsrather directly to the relativistic formulation described here Indeed many aspects ofthis formulation are already utilized by many authors For instance Arnold [104] iden-tifies the (presymplectic) space with coordinates (t qi pi) (time lagrangian variablesand their momenta) as the natural home for mechanics Souriau has developed a beau-tiful and little-known relativistic formalism [105] Probably the first to consider thepoint of view used here was Lagrange himself in pointing out that the most convenientdefinition of ldquophase spacerdquo is the space of the physical motions [106] Many of the toolsused below are also used in hamiltonian treatments of generally covariant theories asconstrained systems although generally within a rather obscure interpretative cloud

31 Nonrelativistic mechanics mechanics is about timeevolution

I begin with a brief review of conventional mechanics This is useful tofix notations and introduce some notions that will play a role in the rela-tivistic formalism I give no derivations here they are standard and theycan be obtained as a special case of the derivations in the next section

Lagrangian A dynamical system with m degrees of freedom describes theevolution in time t of m lagrangian variables qi where i = 1 m Thespace in which the variables qi take value is the m-dimensional (nonrela-tivistic) configuration space C0 The dynamics of the system is determined

98

31 Nonrelativistic mechanics 99

by a single function of 2m variables L(qi vi) the lagrangian Given twotimes t1 and t2 and two points qi1 and qi2 in C0 physical motions are suchthat the action

S[q] =int t2

t1

dt L(qi(t)

dqi(t)dt

)(31)

is an extremum in the space of the motions qi(t) such that qi(t1) = qi1and qi(t2) = qi2 A dynamical system is therefore specified by the couple(C0 L) Physical motions satisfy the Lagrange equations

ddt

pi

(qi(t)

dqi(t)dt

)= Fi

(qi(t)

dqi(t)dt

) (32)

where momenta and forces are defined by

pi(qi vi) =partL(qi vi)

partvi Fi(qi vi) =

partL(qi vi)partqi

(33)

Hamiltonian The Lagrange equations can be cast in first-order form byusing the lagrangian coordinates qi and the momenta pi as variablesInverting the function pi(qi vi) yields the function vi(qi pi) inserting thisin the function Fi(qi vi) defines the force fi(qi pi) equiv Fi(qi vi(qi pi)) asfunctions of coordinates and momenta The equations of motion (32)become

dqi(t)dt

= vi(qi(t) pi(t))dpi(t)

dt= fi(qi(t) pi(t)) (34)

These equations are determined by the function H0(qi pi) the nonrela-tivistic hamiltonian defined by H0(qi pi) = piv

i(qi pi) minus L(qi vi(qi pi))Indeed (34) is equivalent to (32) with

vi(qi pi) =partH0(qi pi)

partpi fi(qi pi) = minuspartH0(qi pi)

partqi (35)

Symplectic The Hamilton equations (34)ndash(35) can be written in a use-ful and compact geometric language The 2m-dimensional space coordi-natized by the coordinates qi and the momenta pi is the nonrelativisticphase space Γ0 (The reason for the subscript 0 will be clear below) Timeevolution is a flow (qi(t) pi(t)) in this space the vector field on Γ0 tangentto this flow is

X0 = vi(qi pi)part

partqi+ fi(qi pi)

part

partpi (36)

100 Mechanics

Therefore the dynamics is specified by assigning the vector field X0 on Γ0Now Γ0 can be interpreted as the cotangent space T lowastC0 Any cotangentspace carries a natural1 one-form θ0 = pidqi where dθ0 is nondegenerateA space equipped with such a one-form has the remarkable property thatevery function f determines a vector field Xf via the relation (dθ0)(Xf ) =minusdf A straightforward calculation shows that the flow defined by H0 isprecisely the time evolution vector field (36) Therefore the equations ofmotion (34)ndash(35) can be written simply2 as

(dθ0)(X0) = minusdH0 (37)

The two-form ω0 = dθ0 entering (37) is symplectic3 A dynamical systemis determined by a triple (Γ0 ω0 H0) where Γ0 is a manifold ω0 is asymplectic two-form and H0 is a function on Γ0

Presymplectic A very elegant formulation of mechanics and a crucialstep in the direction of the relativistic theory is provided by the presym-plectic formalism This formalism is based on the idea of describingmotions by using the graph of the function (qi(t) pi(t)) instead of thefunctions themselves The graph of the function (qi(t) pi(t)) is an un-parametrized curve γ in the (2m + 1)-dimensional space Σ = R times Γ0with coordinates (t qi pi) it is formed by all the points (t qi(t) pi(t)) inthis space The vector field

X =part

partt+ vi(qi pi)

part

partqi+ f i(qi pi)

part

partpi(38)

is tangent to all these curves (So is any other vector field obtained byscaling X namely any vector field X prime = fX where f is a scalar functionon Σ) Now consider the Poincare one-form

θ = pidqi minusH0(qi pi)dt (39)

on Σ The two-form ω = dθ is closed but it is degenerate (every two-formis degenerate in odd dimensions) that is there is a vector field X (calledthe null vector field of ω) satisfying

(dθ)(X) = 0 (310)

1It is defined intrinsically by θ0(X)(s) = s(πX) where X is a vector field on T lowastC0 s apoint in T lowastC0 and π the bundle projection

2The contraction between a two-form and a vector is defined by (αandβ)(X) = α(X)βminusβ(X)α

3That is closed and nondegenerate Closed means dω0 = 0 nondegenerate means thatω0(X) = 0 implies X = 0


The integral curves4 of the null vector field of a two-form ω are calledthe ldquoorbitsrdquo of ω It is easy to see that X given in (38) satisfies (310)Therefore the graphs of the motions are simply the orbits of dθ In otherwords (310) is a rewriting of the equations of motion

A space Σ equipped with a closed degenerate two-form ω is calledpresymplectic A dynamical system is thus completely defined by apresymplectic space (Σ ω) We use also the notation (Σ θ) where ω = dθ

Notice that (310) is homogeneous and therefore it determines X onlyup to scaling This is consistent with the fact that the vector field tangentto the motions is defined only up to scaling That is consistent with thefact that motions are represented by unparametrized curves in Σ

Finally it is easy to see that the action (31) is simply the line integral ofthe Poincare one-form (39) along the orbits if γ is an orbit (t qi(t) pi(t))of ω then the action of the motion qi(t) is

S[q] =int

γθ (311)

Extended Finally let me come to a formulation of dynamics that ex-tends naturally to general-relativistic systems In light of the presymplec-tic formulation described above it is natural to consider the relativisticconfiguration space

C = Rtimes C0 (312)

coordinatized by the m + 1 variables (t qi) and to describe motions withthe graphs of the functions qi(t) which are unparametrized curves in CConsider the cotangent space T lowastC with coordinates (t qi pt pi) and thefunction

H(t qi pt pi) = pt + H0(qi pi) (313)

on this space Let Σ be the surface in T lowastC defined by

H(qi t pi pt) = 0 (314)

We can coordinatize Σ with the coordinates (t qi pi) Since it is a cotan-gent space T lowastC carries a natural one-form which is

θ = pidqi + ptdt (315)

The restriction of this one-form to the surface (314) is precisely (39)Therefore the surface (314) is the presymplectic space that defines thedynamics

4An integral curve of a vector field is a curve everywhere tangent to the field

102 Mechanics

In other words the dynamics is completely defined by the couple (C H)a relativistic configuration space C and a function H on T lowastC The graphsof the motions are simply the orbits of dθ on the surface (314)5 I call Hthe relativistic hamiltonian

Remarkably the dynamics can be directly expressed in terms of a varia-tional principle based on (C H) An unparametrized curve γ in C describesa physical motion if γ extremizes the integral

S[γ] =int

γθ (316)

in the class of the curves γ in T lowastC satisfying (314) whose restriction γ toC connects two given points (t1 qi1) and (t2 qi2)

The relativistic configuration space C has the structure (312) and therelativistic hamiltonian H has the form (313) As we shall see the struc-ture (312)ndash(313) does not survive in the relativistic formulation of me-chanics

Relativistic phase space Denote Γ the space of the orbits of dθ in ΣThere is a natural projection π Σ rarr Γ that sends each point of Σ to thecurve to which it belongs It is not hard to show that there is one andonly one symplectic two-form ωph on Γ such that its pull-back to Σ is dθnamely πlowastωph = dθ Therefore Γ is a symplectic space Γ is the space ofthe physical motions I shall call it the relativistic phase space

The relation between the relativistic phase space Γ and the nonrela-tivistic phase space Γ0 = T lowastC0 is the following Γ0 is the space of theinstantaneous states the states that the system can have at a fixed timet = t0 On the other hand Γ is the space of all solutions of the equationsof motion Now fix a time say t = t0 If at t = t0 the system is in an ini-tial state in Γ0 it will then evolve in a well-defined motion The other wayaround each motion determines an instantaneous state at t = t0 There-fore there is a one-to-one mapping between Γ and Γ0 The identificationbetween Γ and Γ0 depends on the t0 chosen

HamiltonndashJacobi The HamiltonndashJacobi equation is

partS(qi t)partt

+ H0

(qi

partS(qi t)partqi

)= 0 (317)

If a family of solutions S(qi Qi t) depending on m parameters Qi is foundthen we can compute the function

Pi(qi Qi t) = minuspartS(qi Qi t)partQi

(318)

5More precisely the projections of these orbits on C


by simple derivation Inverting this function we obtain

qi(t) = qi(Qi Pi t) (319)

which are physical motions namely the general solution of the equationsof motion where the quantities (Qi Pi) are the 2m integration constants

Solutions of (317) can be found in the form S(qi Qi t) = EtminusW (qi Qi)where E is a constant and W satisfies

H0

(qi

partW (qi Qi)partqi

)= E (320)

S is called the principal HamiltonndashJacobi function W is called the char-acteristic HamiltonndashJacobi function

The HamiltonndashJacobi equation (317) can be obtained from the classicallimit of the Schrodinger equation

The Hamilton function Consider two points (t1 qi1) and (t2 qi2) in C Thefunction on G = C times C

S(t1 qi1 t2 qi2) =

int t2

t1

dt L(qi(t) qi(t)) (321)

where qi(t) is the physical motion from qi1(t1) to qi2(t2) (that minimizesthe action) is called the Hamilton function Equivalently

S(t1 qi1 t2 qi2) =

int

γθ (322)

where γ is the orbit into Σ that projects to qi(t) Notice the differencebetween the action (31) and the Hamilton function (321) the first is afunctional of the motion the second is a function of the end points It isnot hard to see that the Hamilton function solves the HamiltonndashJacobiequation (in both sets of variables) The Hamilton function is thereforea preferred solution of the HamiltonndashJacobi equation If we know theHamilton function we have solved the equations of motion because weobtain the general solution of the equations of motion in the form qi =qi(t Qi Pi T ) by simply inverting the function

Pi(t qi TQi) =partS(t qi TQi)

partQi(323)

with respect to qi The resulting function qi(t TQi Pi) is the generalsolution of the equations of motion where the integration constants arethe initial coordinate and momenta Qi Pi at time T

104 Mechanics

Thus the action defines a dynamical system the Hamilton function di-rectly gives all the motions6 The Hamilton function (321) is the classicallimit of the quantum mechanical propagator

Example a pendulum Let α be the lagrangian variable describing the elongation of asimple harmonic oscillator which I call ldquopendulumrdquo for simplicity The lagrangianis L(α v) = (mv22) minus (mω2α22) the nonrelativistic hamiltonian is H0(α p) =(p22m) + (mω2α22) The extended configuration space has coordinates (t α) andthe relativistic hamiltonian is

H(t α pt p) = pt +p2

2m+

mω2α2

2 (324)

Choose coordinates (t α p) on the constraint surface H = 0 which is therefore definedby pt = minusH0(α p) The restriction of the one-form θ = pt dt + pdα to this surface is

θ = pdαminus(

p2

2m+

mω2α2

2

)dt (325)

The presymplectic two-form is therefore

ω = dθ = dp and dαminus p

mdp and dtminusmω2α dα and dt (326)

The orbits are obtained by integrating the vector field

X = Xtpart

partt+ Xα

part

partα+ Xp

part

partp(327)

satisfying ω(X) = 0 Inserting (326) and (327) in ω(X) = 0 we get

ω(X) = Xt

(minus p

mdpminusmω2α dα

)+ Xα

(dp + mω2α dt

)+ Xp

(minusdα +

p

mdt

)

=(minus p

mXt + Xα

)dp +

(minusmω2αXt minusXp

)dα +

(mω2αXα +

p

mXp

)dt

= 0 (328)

Writing dt(τ)dτ = Xt dα(τ)dτ = Xα dp(τ)dτ = Xp equation (328) reads

dα(τ)

dτminus p

m

dt(τ)

dτ= 0 minusdp(τ)

dτminusmω2α

dt(τ)

dτ= 0 (329)

together with a third equation dependent on the first two Equation (329) can bewritten as

dα(t)

dt=

p

m

dp(t)

dt= minusmω2α (330)

which are the Hamilton equations of the pendulum We can write its general solutionin the form

α(t) = a eiωt + a eminusiωt (331)

The Hamilton function S(α1 t1 α2 t2) is the preferred solution of the HamiltonndashJacobiequation

partS(α t)

partt+

1

2m

(partS(α t)

partα

)2

+mω2α2

2= 0 (332)

6Hamilton (talking about himself in the third person) ldquoMr Lagrangersquos function statesthe problem Mr Hamiltonrsquos function solves itrdquo [107]

32 Relativistic mechanics 105

obtained by computing the action of the physical motion α(t) that goes from α(t1) = α1

to α(t2) = α2 This motion is given by (331) with

a =α1e

minusiωt2 + α2eminusiωt1

2i sin[(ω(t1 minus t2))] (333)

Inserting this in the action and integrating we obtain the Hamilton function

S(α1 t1 α2 t2) = mω2α1α2 minus (α2

1 + α22) cos[(ω(t1 minus t2))]

2 sin[(ω(t1 minus t2))] (334)

This concludes the short review of nonrelativistic mechanics I nowconsider the generalization of this formalism to relativistic systems

32 Relativistic mechanics

321 Structure of relativistic systems partial observablesrelativistic states

Is there a version of the notions of ldquostaterdquo and ldquoobservablerdquo broad enough to applynaturally to relativistic systems I begin by introducing the main notions and tools ofcovariant mechanics in the context of a simple system

The pendulum revisited Say we want to describe the small oscillations ofa pendulum To this aim we need two measuring devices a clock and adevice that reads the elongation of the pendulum Let t be the readingof the clock (in seconds) and α the reading of the device measuring theelongation of the pendulum (in centimeters) Call the variables t and α thepartial observables of the pendulum (I use also relativistic observables orsimply observables if there is no risk of confusion with the nonrelativisticnotion of observable which is different)

A useful observation is a reading of the time t and the elongation αtogether Thus an observation yields a pair (t α) Call a pair obtained inthis manner an event

Let C be the two-dimensional space with coordinates t and α CallC the event space of the pendulum (I use also relativistic configurationspace or simply configuration space if there is no risk of confusion withthe nonrelativistic configuration space C0 which is different)

Experience shows we can find mathematical laws characterizing se-quences of events This is the reason we can do science These laws havethe following form Call an unparametrized curve γ in C a motion of thesystem Perform a sequence of measurements of pairs (t α) and find thatthe points representing the measured pairs sit on a motion γ Then wesay that γ is a physical motion We express a motion as a relation in C

f(α t) = 0 (335)

Thus a motion γ is a relation or a correlation between partial observables

106 Mechanics

Then disturb the pendulum (push it with a finger) and repeat theentire experiment over At each repetition of the experiment a differentmotion γ is found That is a different mathematical relation of the form(335) is found Experience shows that the space of the physical motionsis very limited it is just a two-dimensional space Only a two-dimensionalspace of curves γ is realized in Nature

In the case of the small oscillations of a frictionless pendulum we cancoordinatize the physical motions by the two real numbers A ge 0 and0 le φ lt 2π and (335) is given by

f(α tA φ) = αminusA sin(ωt + φ) = 0 (336)

This equation gives a curve γ in C for each couple (A φ)Let Γ be the two-dimensional space of the physical motions coordina-

tized by A and φ Γ is the relativistic phase space of the pendulum (orthe space of the motions) A point in Γ is also called a relativistic state(Or a Heisenberg state or simply a state if there is no risk of confusionwith the nonrelativistic notion of state which is different)

Equation (336) is the mathematical law that captures the empiricalinformation we have on the pendulum This equation is the evolutionequation of the system The function f is the evolution function of thesystem

A relativistic state is determined by a couple (A φ) It determines acurve γ in the (t α) plane That is it determines a correlation betweenthe two partial observables t and α via (336) If we disturb the pendulumby interacting with it or if we start a new experiment over we have a newstate The state remains the same if we observe the pendulum and theclock without disturbing them (here we disregard quantum theory ofcourse)

Summarizing each state in the phase space Γ determines a correlationbetween the observables in the configuration space C The set of theserelations is captured by the evolution equation (336) namely by thevanishing of a function

f Γ times C rarr R (337)

The evolution equation f = 0 expresses all predictions that can be madeusing the theory Equivalently these predictions are captured by the sur-face f = 0 in the cartesian product of the phase space with the configu-ration space

General structure of the dynamical systems The (CΓ f) language de-scribed above is general It is sufficient to describe all predictions of con-ventional mechanics On the other hand it is broad enough to describe


general-relativistic systems All fundamental systems can be described (tothe accuracy at which quantum effects can be disregarded) by making useof these concepts

(i) The relativistic configuration space C of the partial observables

(ii) The relativistic phase space Γ of the relativistic states

(iii) The evolution equation f = 0 where f Γ times C rarr V

Here V is a linear space The state in the phase space Γ is fixed until thesystem is disturbed Each state in Γ determines (via f = 0) a motion γ ofthe system namely it describes a relation or a set of relations betweenthe observables in C

A motion is not necessarily a one-dimensional curve in C it can be asurface in C of any dimension k If k gt 1 we say that there is gaugeinvariance For a system with gauge invariance we call ldquomotionrdquo the mo-tion itself and any curve within it In this chapter we shall not deal muchwith systems with gauge invariance but we shall mention them whererelevant

Predictions are obtained as follows We first perform enough measure-ments to determine the state (In reality the state of a large system isoften ldquoguessedrdquo on the basis of incomplete observations and reasonableassumptions justified inductively) Once the state is so determined orguessed the evolution equation predicts all the possible events namelyall the allowed correlations between the observables in any subsequentmeasurement

In the example of the pendulum for instance the equation predicts thevalue of α that can be measured together with any given t or the valuesof t that can be measured together with any given α These predictionsare valid until the system is disturbed

The definitions of observable state configuration space and phase spacegiven here are different from the conventional definitions In particu-lar notions of instantaneous state evolution in time observable at afixed time play no role here These notions make no sense in a general-relativistic context For nonrelativistic systems the usual notions can berecovered from the definitions given The relation between the relativisticdefinitions considered here and the conventional nonrelativistic notions isdiscussed in Section 324

The task of mechanics is to find the (CΓ f) description for all phys-ical systems The first step kinematics consists in the specification ofthe observables that characterize the system Namely it consists in thespecification of the configuration space C and its physical interpretationPhysical interpretation means the association of coordinates on C with

108 Mechanics

measuring devices The second step dynamics consists in finding thephase space Γ and the function f that describe the physical motions ofthe system

In the next section I describe a relativistic hamiltonian formalism formechanics based on the relativistic notions of state and observable definedhere

322 Hamiltonian mechanics

Elementary physical systems can be described by hamiltonian mechanics7

Once the kinematics ndash that is the space C of the partial observables qa ndashis known the dynamics ndash that is Γ and f ndash is fully determined by givinga surface Σ in the space Ω of the observables qa and their momenta paThe surface Σ can be specified by giving a function H Ω rarr Rk Σ is thendefined by H = 08 Denote γ a curve in Ω (observables and momenta)and γ its restriction to C (observables alone) H determines the physicalmotions via the following

Variational principle A curve γ connecting the events qa1and qa2 is a physical motion if γ extremizes the action

S[γ] =int

γpa dqa (338)

in the class of the curves γ satisfying

H(qa pa) = 0 (339)

whose restriction γ to C connects qa1 and qa2

All (relativistic and nonrelativistic) hamiltonian systems can be formu-lated in this manner

If k = 1 H is a scalar function and is sometimes called the hamil-tonian constraint The case k gt 1 is the case in which there is gaugeinvariance In this case the system (339) is sometimes called the systemof the ldquoconstraint equationsrdquo I call H the relativistic hamiltonian or ifthere is no ambiguity simply the hamiltonian I denote the pair (C H)as a relativistic dynamical system The generalization to field theory isdiscussed in Section 33

The relativistic hamiltonian H is related to but should not be confusedwith the usual nonrelativistic hamiltonian denoted H0 in this book Halways exists while H0 exists only for nonrelativistic systems

7Perhaps because they are the classical limit of a quantum system8Different Hs that vanish on the same surface Σ define the same physical system


Indeed notice that this formulation of mechanics is similar to the ex-tended formulation of nonrelativistic mechanics defined in Section 31The novelty is that C and H do not have the structure (312)ndash(313) Thediscussion above shows that this structure is not necessary in order to havea well-defined physical interpretation of the formalism A nonrelativisticsystem is characterized by the fact that one of its partial observables qa

is singled out by having the special role of an independent variable tThis does not happen in a relativistic system The following simple ex-ample shows that the relativistic formulation of mechanics is a propergeneralization of standard mechanics

Timeless double pendulum I now introduce a genuinely timeless system which I willrepeatedly use as a simple model to illustrate the theory Consider a mechanical modelwith two partial observables say a and b whose dynamics is defined by the relativistichamiltonian

H(a b pa pb) = minus1

2

(p2a + p2

b + a2 + b2 minus 2E) (340)

where E is a constant The extended configuration space is C = R2 The constraintsurface has dimension 3 it is the sphere of radius

radic2E in T lowastC The phase space has

dimension 2 The motions are curves in the (a b) space For each state the theorypredicts the correlation between a and b

A straightforward calculation (see below) shows that the evolution equation deter-mined by H is an ellipse in the (a b) space

f(a bα β) =( a

sinα

)2

+( b

cosα

)2

+ 2a

sinα

b

cosαcosβ minus 2E2 sin2 β = 0 (341)

where α and β parametrize Γ Therefore motions are closed curves and in fact ellipsesin C The system does not admit a conventional hamiltonian formulation because fora nonrelativistic hamiltonian system motions in C = R times C0 are monotonic in t isin Rand therefore cannot be closed curves

The example is not artificial There exist cosmological models that have precisely thisstructure For instance we can identify a with the radius of a maximally symmetricuniverse and b with the spatially constant value of a field representing the mattercontent of that universe and adopt the approximation in which these are the only twovariables that govern the large-scale evolution of the universe Then the dynamics ofgeneral relativity reduces to a system with the structure (340)

The associated nonrelativistic system The system (340) can also be viewed as followsConsider a physical system which we denote the ldquoassociated nonrelativistic systemrdquoformed by two noninteracting harmonic oscillators Let me stress that the associatednonrelativistic system is a different physical system than the timeless double pendu-lum considered above The timeless double pendulum has one degree of freedom itsassociated nonrelativistic system has two degrees of freedom The partial observablesof the associated nonrelativistic system are the two elongations a and b and the timet The nonrelativistic hamiltonian that governs the evolution in t is

H0(a b pa pb) =1

2

(p2a + p2

b + a2 + b2 minus 2E) (342)

110 Mechanics

It it has the same form as the relativistic hamiltonian (340) of the timeless doublependulum9 The constant term 2E of course has no effect on the equations of motionit only redefines the energy Physically we can view the relation between the twosystems as follows Imagine that we take the associated nonrelativistic system but wedecide to ignore the clock that measures t we consider just measurements of the twoobservables a and b Furthermore assume that the energy of the double pendulum isconstrained to vanish namely

1

2

(p2a + p2

b + a2 + b2)

= E (344)

Then the observed relation between the measurements of a and b is described by therelativistic system (340)

Geometric formalism As for nonrelativistic hamiltonian mechanics theequations of motion can be expressed in an elegant geometric form Thevariables (qa pa) are coordinates on the cotangent space Ω = T lowastC Equa-tion (339) defines a surface Σ in this space The cotangent space carriesthe natural one-form

θ = padqa (345)

Denote θ the restriction of θ to the surface Σ The two-form ω = dθ onΣ is degenerate it has null directions The integral surfaces of these nulldirections are the orbits of ω on Σ Each such orbit projects from T lowastC toC to give a surface in C These surfaces are the motions

Consider the case k = 1 In this case Σ has dimension 2nminus1 the kernelof ω is generically one-dimensional and the motions are generically one-dimensional Let γ be a motion on Σ and X be a vector tangent to themotion then

ω(X) = 0 (346)

To find the motions we have just to integrate this equation Equation(346) is the equation of motion X is defined by the homogeneous equa-tion (346) only up to a multiplicative factor Therefore the tangent of theorbit is defined only up to a multiplicative factor and so the parametriza-tion of the orbit is not determined by (346)

The case k gt 1 is analogous In this case Σ has dimension 2n minus kthe kernel of ω is generically k-dimensional and the motions are generi-cally k-dimensional X is then a k-dimensional multi-tangent and it stillsatisfies (346)

Let π Σ rarr Γ be the projection map that associates with each pointof the constraint surface the motion to which the point belongs The

9The relativistic hamiltonian of the associated nonrelativistic system is

H(a b t pa pb pt) = pt +1

2

(p2a + p2

b + a2 + b2 minus 2E) (343)


projection π equips the phase space Γ with a symplectic two-form ωph

defined to be the two-form whose pull-back to Σ under π is ω Locally itexists and it is unique precisely because ω is degenerate along the orbits

Relation with the variational principle Let γ be an orbit of ω on Σ such that itsrestriction γ in C is bounded by the initial and final events q1 and q2 Let γprime be a curvein Σ infinitesimally close to γ such that its restriction γprime is also bounded by q1 and q2Let δs1 (and δs2) be the difference between the initial (and final) points of γ and γprimeThe four curves γ δs1 minusγprime and minusδs2 form a closed curve in Σ Consider the integral ofω over the infinitesimal surface bounded by this curve This integral vanishes becauseat every point of the surface one of the tangents is (to first order) a null directionof ω (the surface is a strip parallel to the motion γ) But ω = dθ and therefore byStokes theorem the integral of θ along the closed curve vanishes as well The integralof θ = padq

a along δs1 and δs2 is zero because qa is constant along these segmentsTherefore int

γ

θ +

int

minusγprimeθ = 0 (347)

or

δ

int

γ

θ = 0 (348)

for any variation in the class considered This is precisely the variational principlestated in Section 32

Hamilton equations Consider first the case k = 1 Motions are one-dimensional Parametrize the curve with an arbitrary parameter τ Thatis describe a motion (in Ω) with the functions (qa(τ) pa(τ)) These func-tions satisfy the Hamilton system

H(qa pa) = 0 (349)

dqa(τ)dτ

= N(τ) va(qa(τ) pa(τ))

dpa(τ)dτ

= N(τ) fa(qa(τ) pa(τ)) (350)

where

va(qa pa) =partH(qa pa)

partpa fa(qa pa) = minuspartH(qa pa)

partqa (351)

The function N(τ) is called the ldquoLapse functionrdquo It is arbitrary Differ-ent choices of N(τ) determine different parameters τ along the motionTo obtain a monotonic parametrization we need N(τ) gt 0 A preferredparametrization can be obtained by taking N(τ) = 1 that is replacing(350)ndash(351) by the equations (written in the usual compact form)

qa =partH

partpa pa = minuspartH

partqa (352)

112 Mechanics

where the dot indicates derivative with respect to τ This choice is calledthe Lapse = 1 gauge It is not preferred in a physical sense In particu-lar different but physically equivalent hamiltonians H defining the samesurface Σ determine different preferred parametrizations Nevertheless itis often the easiest gauge to compute with

If k gt 1 the function H has components Hj with j = 1 k and motions arek-dimensional surfaces We can parametrize a motion with k arbitrary parameters τ =τj Namely we can represent it using the 2n functions qa(τ) pa(τ) of k parametersτj These equations satisfy the system given by (349) and

partqa(τ)

partτj= Nj(τ)

partHj(qa pa)

partpa

partpa(τ)

partτj= minusNj(τ)

partHj(qa pa)

partqa (353)

A motion is determined by the full k-dimensional surface in C we can choose a particularcurve τ(τ) on this surface where τ is an arbitrary parameter and represent the motionby the one-dimensional curve qa(τ) = qa(τ(τ)) in C This satisfies the system formedby (349) and

dqa(τ)

dτ= Nj(τ)

partHj(qa pa)

partpa

dpa(τ)

dτ= minusNj(τ)

partHj(qa pa)

partqa(354)

for k arbitrary functions of one variable Nj(τ) Different choices of the functions Nj(τ)determine different curves on the single surface that defines a motion These are gauge-equivalent representations of the same motion

It is important to stress that the parameters τ or τj are an artifact ofthis technique They have no physical significance They are absent in thegeometric formalism as well as in the HamiltonndashJacobi formalism as weshall see below The physical content of the theory is in the motion in Cnot in the way the motion is parametrized That is the physical informa-tion is not in the functions qa(τ) it is in the image of these functions inC

Relation with the variational principle Parametrize the curve γ with a parameter τ The action (338) reads

S =

intdτ pa(τ)

dqa(τ)

dτ (355)

The constraint (339) can be implemented in the action with lagrange multipliers Ni(τ)This defines the action

S =

intdτ

(pa

dqa

dτminusNi H

i(pa qa)

) (356)

Varying this action with respect to Ni(τ) qa(τ) and pa(τ) gives the Hamilton equation(349) (354)

Example double pendulum Consider the system defined by the hamiltonian (340)The Hamilton equations (349) (352) in the Lapse = 1 gauge give

a = pa b = pb pa = minusa pb = minusb a2 + b2 + p2a + p2

b = 2E (357)


The general solution is

a(τ) = Aa sin(τ) b(τ) = Ab sin(τ + β) (358)

where Aa =radic

2E sinα and Ab =radic

2E cosα The motions are given by the image in Cof these curves These are the ellipses (341) The parametrization of the curves (358)has no physical significance The physics is in the unparametrized ellipses in C and inthe relation between a and b they determine

HamiltonndashJacobi HamiltonndashJacobi formalism is elegant general andpowerful it has a direct connection with quantum theory and is con-ceptually clear The relativistic formulation of HamiltonndashJacobi theory issimpler than the conventional nonrelativistic version indicating that therelativistic formulation unveils a natural and general structure of mechan-ical systems

The relativistic HamiltonndashJacobi formalism is given by the system of kpartial differential equations

H

(qa

partS(qa)partqa

)= 0 (359)

for the function S(qa) defined on the extended configuration space C LetS(qa Qi) be a family of solutions parametrized by the nminus k constants ofintegration Qi Pose

f i(qa Pi Qi) equiv partS(qa Qi)

partQi+ Pi = 0 (360)

for n minus k arbitrary constants Pi This is the evolution equation Theconstants Qi Pi coordinatize a 2(nminus k)-dimensional space Γ This is thephase space

The form of the relativistic HamiltonndashJacobi equation (359) is simplerthan the usual nonrelativistic HamiltonndashJacobi equation (317) Further-more there is no equation to invert as in the nonrelativistic formalismNotice also that the function S(qa Qi) can be identified with the principalHamiltonndashJacobi function S(t qi Qi) = Et+W (qi Qi) of the nonrelativis-tic formalism as well as with the characteristic HamiltonndashJacobi functionW (qi Qi) since (359) is formally like (320) with vanishing energy Thetwo functions are in fact identified in the relativistic formalism

Example double pendulum The HamiltonndashJacobi equation of the timeless system(340) is

(partS(a b)

parta

)2

+

(partS(a b)

partb

)2

+ a2 + b2 minus 2E = 0 (361)

114 Mechanics

A one-parameter family of solutions is given by

S(a b A) =a

2

radicA2 minus a2 +

A2

2arctan

(aradic

A2 minus a2

)

+b

2

radic2E minusA2 minus b2 +

2E minusA2

2arctan

(bradic

2E minusA2 minus b2

) (362)

The general solution (341) of the system is directly obtained by writing

partS(a b A)

partAminus φ = 0 (363)

where φ is an integration constant

Derivation of the HamiltonndashJacobi formalism Since the phase space Γ is a symplecticspace we can locally choose canonical coordinates (Qi Pi) over it These coordinatescan be pulled back to Σ where they are constant along the orbits In fact they labelthe orbits Let θph = PidQ

i therefore dθph = ω But ω = dθ = d(padqa) so on Σ we

have

d(θph minus θ) = d(PidQi minus padq

a) = 0 (364)

This implies that there should locally exist a function S on Σ such that

PidQi minus padq

a = minusdS (365)

Let us choose qa and Qi as independent coordinates on Σ Then (365) reads

dS(qa Qi) = pa(qa Qi)dqa minus Pi(q

a Qi)dQi (366)

that is

partS(qa Qi)

partqa= pa(q

a Qi) (367)

partS(qa Qi)

partQi= minusPi(q

a Qi) (368)

By the definition of Σ we have H(qa pa) = 0 which using (367) gives the HamiltonndashJacobi equation (359) Equation (368) is then immediately the evolution equation(360)

In other words S(qa Qi) is the generating function of a canonical transformationthat relates the observables and their momenta (qa pa) to new canonical variables(Qi Pi) satisfying Qi = 0 Pi = 0 These new variables are constants of motion andtherefore define Γ The relation between C and Γ given by the canonical transformationequations (367)ndash(368) is the evolution equation

323 Nonrelativistic systems as a special case

Here I discuss in more detail how the notions and the structures of conven-tional mechanics described in Section 31 are recovered from the relativis-tic formalism A nonrelativistic system is simply a relativistic dynamicalsystem in which one of the partial observables qa is denoted t and calledldquotimerdquo and the hamiltonian H has the form

H = pt + H0 (369)

partt+ X0 (374)


where H0 is independent from pt and is called the nonrelativistic hamil-tonian The quantity E = minuspt is called energy The device that measuresthe partial observable t is called a clock

The relativistic configuration space therefore has the structure

C = Rtimes C0 (370)

with coordinates qa = (t qi) where i = 1 nndash1 The space C0 is theusual nonrelativistic configuration space Accordingly the cotangent spaceΩ = T lowastC has coordinates (qa pa) = (t qi pt pi)

If H has the form (369) the relativistic HamiltonndashJacobi equation(359) becomes the conventional nonrelativistic HamiltonndashJacobi equation(317)

Given a state and a value t of the clock observable we can ask whatare the possible values of the observables qi such that (qi t) is a possibleevent That is we can ask what is the value of qi ldquowhenrdquo the time is t Thesolution is obtained by solving the evolution function f i(qi tQi Pi) = 0for the qi This gives

qi = qi(tQi Pi) (371)

which is interpreted as the evolution equation of the variables qi in thetime t The form (369) of the hamiltonian guarantees that we can solve fwith respect to the qi because the Hamilton equation for t (in the gaugeLapse = 1) is simply t = τ which can be inverted

In the parametrized hamiltonian formalism the evolution equationfor t(τ) is trivial and gives taking advantage of the freedom in rescal-ing τ just t = τ Using this equations (353) become the conventionalHamilton equations and (349) simply fixes the value of pt namely theenergy

In the presymplectic formalism the surface Σ turns out to be

Σ = Rtimes Γ0 (372)

where the coordinate on R is the time t and Γ0 = T lowastC0 is the nonrela-tivistic phase space The restriction of θ to this surface has the Cartanform

θ = pidqi minusH0dt = θ0 minusH0dt (373)

We can take the vector field X to have the form

X =part

116 Mechanics

where X0 is a vector field on Γ0 Then the equation of motion (346)reduces to the equation

(dθ0)(X0) = minusdH0 (375)

which is the geometric form of the conventional Hamilton equations ThusH determines how the variables in Γ0 are correlated to the variable t Thatis ldquohow the variables in Γ0 evolve in timerdquo In this sense the nonrelativis-tic hamiltonian H0 generates ldquoevolution in the time trdquo This evolution isgenerated in Γ0 by the hamiltonian flow X0 of H0 A point s = (qi pi) inΓ0 is taken to the point s(t) = (qi(t) pi(t)) where

ds(t)dt

= X0(s(t)) (376)

The evolution of an observable (not depending explicitly on time) de-fined by At(s) = A(s(t)) = A(s t) can be written introducing the Poissonbracket notation

AB = minusXA(B) = XB(A) =sum

i

(partA

partqipartB

partpiminus partA

partpi

partB

partqi

) (377)

asdAt

dt= At H0 (378)

Instantaneous states and relativistic states The nonrelativistic definitionof state refers to the properties of a system at a certain moment of timeDenote this conventional notion of state as the ldquoinstantaneous staterdquo Thespace of the instantaneous states is the conventional nonrelativistic phasespace Γ0 Letrsquos fix the value t = t0 of the time variable and characterizethe instantaneous state in terms of the initial data For the pendulumthese are position and momentum (α0 p0) at t = t0 Thus (α0 p0) arecoordinates on Γ0

On the other hand a relativistic state is a solution of the equations ofmotion (If there is gauge invariance a state is a gauge equivalence classof solutions of the equations of motion) The relativistic phase space Γ isthe space of the solutions of the equations of motion

Given a value t0 of the time there is a one-to-one correspondence be-tween initial data and solutions of the equations of motion each solutionof the equation of motion determines initial data at t = t0 and eachchoice of initial data at t0 determines uniquely a solution of the equationsof motion Therefore there is a one-to-one correspondence between in-stantaneous states and relativistic states Therefore the relativistic phasespace Γ is isomorphic to the nonrelativistic phase space Γ sim Γ0 How-ever the isomorphism depends on the time t0 chosen and the physical


interpretation of the two spaces is quite different One is a space of statesat a given time the other a space of motions

In the case of the pendulum the nonrelativistic phase space Γ0 can becoordinatized with (α0 p0) the relativistic phase space Γ can be coordina-tized with (A φ) The identification map (A φ) rarr (α0 p0) is given by

α0(A φ) = A sin(ωt0 + φ) (379)p0(A φ) = ωmA cos(ωt0 + φ) (380)

The nonrelativistic phase space Γ0 plays a double role in nonrelativistichamiltonian mechanics it is the space of the instantaneous states but itis also the arena of nonrelativistic hamiltonian mechanics over which H0

is defined In the relativistic context this double role is lost one mustdistinguish the cotangent space Ω = T lowastC over which H is defined fromthe phase space Γ which is the space of the motions This distinction willbecome important in field theory where Ω is finite-dimensional while Γ isinfinite-dimensional

In a nonrelativistic system X0 generates a one-parameter group oftransformations in Γ0 the hamiltonian flow of H0 on Γ0 Instead of havingthe observables in C0 depending on t one can shift perspective and viewthe observables in C0 as time-independent objects and the states in Γ0

as time-dependent objects This is a classical analog of the shift fromthe Heisenberg to the Schrodinger picture in quantum theory and can becalled the ldquoclassical Schrodinger picturerdquo

In the relativistic theory there is no special ldquotimerdquo variable C doesnot split naturally as C = R times C0 the constraints do not have the formH = pt +H0 and the description of the correlations in terms of ldquohow thevariables in C0 evolve in timerdquo is not available in general In a system thatdoes not admit a nonrelativistic formulation the classical Schrodingerpicture in which states evolve in time is not available only the relativisticnotions of state and observable make sense

Special-relativistic systems There are relativistic systems that do not ad-mit a nonrelativistic formulation such as the example of the double pen-dulum discussed above There are also systems that can be given a nonrel-ativistic formulation but their structure is far more clean in the relativis-tic formalism Lorentz-invariant systems are typical examples They canbe formulated in the conventional hamiltonian picture only at the priceof breaking Lorentz invariance The choice of a preferred Lorentz framespecifies a preferred Lorentz time variable t = x0 The predictions of thetheory are Lorentz invariant but the formalism is not This way of deal-ing with the mechanics of special-relativistic systems hides the simplicityand symmetry of its hamiltonian structure The relativistic hamiltonian

118 Mechanics

formalism exemplified below for the case of a free particle is manifestlyLorentz invariant

Example relativistic particle The configuration space C is a Minkowski space M withcoordinates xμ The dynamics is given by the hamiltonian H = pμpμ+m2 which definesthe mass-m Lorentz hyperboloid Km The constraint surface Σ is therefore given byΣ = T lowastM|H=0 = MtimesKm The null vectors of the restriction of dθ = dpμ and dxμ to Σare

X = pμpart

partxμ (381)

because ω(X) = pμdpμ = 2d(p2) = 0 on pμpμ = minusm2 The integral lines of X namelythe lines whose tangent is X are

xμ(τ) = Pμτ + Xμ pμ(τ) = Pμ (382)

which give the physical motions of the particle The space of these lines is six-dimensional (it is coordinatized by the eight numbers (Xμ Pμ) but PμPμ = minusm2

and (Pμ Xμ) defines the same line as (Pμ Xμ + Pμa) for any a) and represents thephase space The motions are thus the timelike straight lines in M

Notice that all notions used are completely Lorentz invariant A state is a time-like geodesic an observable is any Minkowski coordinate a correlation is a point inMinkowski space The theory is about correlations between Minkowski coordinatesthat is observations of the particle at certain spacetime points On the other handthe split M = RtimesR3 necessary to define the usual hamiltonian formalism is observerdependent

The relativistic formulation of mechanics is not only more general butalso more simple and elegant and better operationally founded than theconventional nonrelativistic formulation This is true whether one usesthe Hamilton equations the geometric language or the HamiltonndashJacobiformalism

324 Discussion mechanics is about relations between observables

The key difference between the relativistic formulation of mechanics dis-cussed in this chapter and the conventional one ndash and in particular be-tween the relativistic definitions of state and observable and the conven-tional ones ndash is the role played by time In the nonrelativistic context timeis a primary concept Mechanics is defined as the theory of the evolutionin time In the definition considered here on the other hand no specialpartial observable is singled out as the independent variable Mechanicsis defined as the theory of the correlations between partial observables

Technically C does not split naturally as C = R times C0 the constraintsdo not have the form H = pt + H0 and the Schrodinger-like descriptionof correlations in terms of ldquohow states and observables evolve in timerdquo isnot available in general


It is important to understand clearly the meaning of this shift of per-spective

The first point is that it is possible to formulate conventional mechanicsin this time-independent language In fact the formalism of mechanicsbecomes even more clean and symmetric (for instance Lorentz covariant)in this language This is a remarkable fact by itself What is remarkableis that the formal structure of mechanics doesnrsquot really treat the timevariable on a different footing than the other variables The structure ofmechanics is the formalization of what we have understood about thephysical structure of the world Therefore we can say that the physical(more precisely mechanical) structure of the world is quite blind to thefact that there is anything ldquospecialrdquo about the variable t

Historically the idea that in a relativistic context we need the time-independent notion of state has been advocated particularly by Dirac(see [148] in Chapter 5) and by Souriau [105] The advantages of therelativistic notion of state are multi-fold In special relativity for instancetime transforms with other variables and there is no covariant definitionof instantaneous state In a Lorentz-invariant field theory in particularthe notion of instantaneous state breaks explicit Lorentz covariance theinstantaneous state is the value of the field on a simultaneity surfacewhich is such for a certain observer only The relativistic notion of stateon the other hand is Lorentz invariant

The second point is that this shift in perspective is forced in gen-eral relativity where the notion of a special spacelike surface over whichinitial data are fixed conflicts with diffeomorphism invariance A gen-erally covariant notion of instantaneous state or a generally covariantnotion of observable ldquoat a given timerdquo makes little physical sense In-deed none of the various notions of time that appear in general rel-ativity (coordinate time proper time clock time) play the role that tplays in nonrelativistic mechanics A consistent definition of state andobservable in a generally covariant context cannot explicitly involvetime

The physical reason for this difference is discussed in Chapter 2 Innonrelativistic physics time and position are defined with respect to asystem of reference bodies and clocks that are implicitly assumed to ex-ist and not to interact with the physical system studied In gravitationalphysics one discovers that no body or clock exists which does not inter-act with the gravitational field the gravitational field affects directly themotion and the rate of any reference body or clock Therefore one cannotseparate reference bodies and clocks from the dynamical variables of thesystem General relativity ndash in fact any generally covariant theory ndash isalways a theory of interacting variables that necessarily include the phys-ical bodies and clocks used as references to characterize spacetime points

120 Mechanics

In the example of the pendulum discussed in Section 321 for instancewe can assume that the pendulum and the clock do not interact In ageneral-relativistic context the two always interact and C does not splitinto C0 and R

Summarizing it is only in the nonrelativistic limit that mechanics canbe seen as the theory of the evolution of the physical variables in time Ina fully relativistic context mechanics is a theory of correlations betweenpartial observables

325 Space of boundary data G and Hamilton function S

I describe here the relativistic version of a structure that plays an impor-tant role in the quantum theory

Hamilton function Notice that the Hamilton function defined in (321)is naturally a function on (two copies of) the relativistic configurationspace C In fact its definition extends to the relativistic context giventwo events qa and qa0 in C the Hamilton function is defined as

S(qa qa0) =int

γθ (383)

where γ is the orbit in Σ of the motion that goes from qa0 to qa This is alsothe value of the action along this motion For instance for a nonrelativisticsystem we can write

S(qa qa0) =int

γθ =

int

γpadqa (384)

=int 1

0pa(τ)qa(τ)dτ =

int 1

0

(pi(τ)qi(τ) + pt(τ)t(τ)

)dτ

=int 1

0

(pi(τ)qi(τ) minusH0(τ)t(τ)

)dτ

=int t

t0

(pi(t)

dqi(t)dt

minusH0(t))

dt

=int t

t0

L

(qi

dqi(t)dt

)dt (385)

where L is the lagrangian From the definition we have

partS(qa qa0)partqa

= pa(qa qa0) (386)

where pa(qa qa0) is the value of the momentum at the final event Notice


that this value depends on qa as well as on qa0 The derivation of thisequation is less obvious than appears at first sight I leave the details tothe acute reader

It follows from (386) that S(qa qa0) satisfies the HamiltonndashJacobi equa-tion (359) The quantities qa0 can be seen as the HamiltonndashJacobi inte-gration constants Notice that they are n not nminus1 Equations (360) nowread

fa(qa qa0 pa0) =partS(qa qa0)

partqa0+ pa0 = 0 (387)

Therefore the phase space is directly (over-)coordinatized by initial co-ordinates and momenta (qa0 pa0) These are not independent for tworeasons First they satisfy the equation H = 0 Second different sets(qa0(τ) pa0(τ)) along the same motion determine the same motion Fur-thermore one of the equations (387) turns out to be dependent on theothers

S(qa qa0) satisfies the HamiltonndashJacobi equation in both sets of vari-ables namely it satisfies also

H

(qa0 minus

partS(qa qa0)partqa0

)= 0 (388)

where the minus sign comes from the fact that the second set of variablesis in the lower integration boundary in (383)

If there is more than one physical motion γ connecting the boundarydata the Hamilton function is multivalued If γ1 γn are distinct so-lutions with the same boundary values we denote its different branchesas

Si(qa1 qa2) =

int

γi

θ (389)

The Hamilton function is strictly related to the quantum theory It isthe phase of the propagator W (qa qa0) which as we shall see in Chapter5 is the main object of the quantum theory If S is single valued we have

W (qa qa0) sim A(qa qa0) eiS(qaqa0 ) (390)

up to higher terms in If S is multivalued

W (qa qa0) simsum

i

Ai(qa qa0) eiSi(q

aqa0 ) (391)

122 Mechanics

Example free particle In the case of the free particle the value of the classical actionalong the motion is

S(x t x0 t0) =

int 1

0

(pt t + px)dt = pt

int t

t0

dt + p

int x

x0

dx

= minusm(xminus x0)2

2(tminus t0)+ m

(xminus x0)2

tminus t0

=m(xminus x0)

2

2(tminus t0) (392)

It is easy to check that S solves the HamiltonndashJacobi equation of the free particle Thefirst of the two equations (387) gives the evolution equation

partS(x t x0 t0)

partx0+ p0 = minusm

xminus x0

tminus t0+ p0 = 0 (393)

The second equation constrains the pt integration constant

partS(x t x0 t0)

partt0+ pt0 = minus 1

2mp20 + pt0 = 0 (394)

Recall that the propagator of the Schrodinger equation of the free particle is

W (x t x0 t0) =1

radici(tminus t0)

ei

m(xminusx0)2

2(tminust0) =1

radici(tminus t0)

eiS(xtx0t0) (395)

Example double pendulum The Hamilton function of the timeless system (340) canbe computed directly from its definition This gives

S(a b aprime bprime) = S(a b aprime bprime A(a b aprime bprime)

) (396)

where

S(a b aprime bprime A) = S(a b A) minus S(aprime bprime A) (397)

S(a b A) is given in (362) and A(a b aprime bprime) is the value of A of the ellipse (341) thatcrosses (a b) and (aprime bprime) This value can be obtained by noticing that (358) imply withlittle algebra that

A2 =a2 + aprime2 minus 2aaprime cos τ

sin2 τ(398)

and

E =(a2 + b2 + aprime2 + bprime2) minus 2(aaprime + bbprime) cos τ

sin2 τ (399)

The second equation can be solved for τ(a b aprime bprime) and inserting this in the first givesA(a b aprime bprime) It is not complicated to check that the derivative of partS(a b aprime bprime A)partAvanishes when A = A(a b aprime bprime) Using this it is easy to see that (396) solves theHamiltonndashJacobi equation in both sets of variables

Notice that for given (a b aprime bprime) equation (398) gives A as a function of τ We cantherefore consider also the function

S(a b aprime bprime τ) = S(a b A(τ)) minus S(aprime bprime A(τ)) (3100)

which is the value of the action of the nonrelativistic system formed by two harmonicoscillators evolving in a physical time τ with a nonrelativistic hamiltonian H that is


it is the Hamilton function of this system With some algebra this can be written alsoas

S(a b aprime bprime τ) = Mτ +(a2 + b2 + aprime2 + bprime2) cos τ minus 2(aaprime + bbprime)

sin τ (3101)

As for A we have immediately

partS(a b aprime bprime τ)

partτ

∣∣∣∣τ=τ(abaprimebprime)

= 0 (3102)

This means that the Hamilton function of the timeless system is numerically equalto the Hamilton function of the two oscillators for the ldquocorrectrdquo time τ needed to gofrom (aprime bprime) to (a b) staying on a motion of total energy E And that this ldquocorrectrdquotime τ = τ(a b aprime bprime) is the one that minimizes the Hamilton function of the twooscillators

More precisely for given (a b aprime bprime) there are two paths connecting (aprime bprime) with (a b)these are the two paths in which the ellipse that goes through (aprime bprime) and (a b) is cutby these two points Denote S1 and S2 the two values of the action along these pathsTheir relation is easily obtained by noticing that the action along the entire ellipse iseasily computed as

S1 + S2 = 2πE (3103)

The space of the boundary data G The Hamilton function is a functionon the space G = C times C An element α isin G is an ordered pair of elementsof the extended configuration space C α = (qa qa0) Notice that α isthe ensemble of the boundary conditions for a physical motion For anonrelativistic system α = (t qi t0 qi0) the motion begins at qi0 at timet0 and ends at qi at time t

The space G carries a natural symplectic structure In fact let i G rarr Γbe the map that sends each pair to the orbit that the pair defines Thenwe can define the two-form ωG = ilowastωph where ωph is the symplecticform of the phase space defined in Section 322 In other words α =(qa qa0) can be taken as a natural over-coordinatization of the phase spaceInstead of coordinatizing a motion with initial positions and momentawe coordinatize it with initial and final positions In these coordinatesthe symplectic form is given by ωG

The two-form ωG can be computed without having first to computeΓ and ωph Denote γα the orbit in Σ with boundary data α and γα itsprojection to C Then α is the boundary of γα We write α = partγα Denotes and s0 the initial and final points of γα in Σ That is s = (qa pa) ands0 = (qa0 p0a) where in general both pa and p0a depend on qa and onqa0 Let δα = (δqa δqa0) be a vector (an infinitesimal displacement) at αThen the following is true

ωG(α)(δ1α δ2α) = ωG(qa qa0)((δ1qa δ1qa0) (δ2qa δ2qa0))= ω(s)(δ1s δ2s) minus ω(s0)(δ1s0 δ2s0) (3104)

124 Mechanics

Notice that δ1s the variation of s is determined by δ1q as well as byδ1q0 and so on This equation expresses ωG directly in terms of ω As weshall see this equation admits an immediate generalization in the fieldtheoretical framework where ω will be a five-form and ωG is a two-form

Now fix a pair α = (qa qa0) and consider a small variation of only oneof its elements say

δα = (δqa 0) (3105)

This defines a vector δα at α on G which can be pushed forward toΓ If the variation is along the direction of the motion then the pushforward vanishes that is ilowastδα = 0 because α and α + δα define the samemotion It follows that if the variation is along the direction of the motionωG(δα) = 0 Therefore the equation

ωG(X) = 0 (3106)

gives the solutions of the equations of motionThus the pair (G ωG) contains all the relevant information of the sys-

tem The null directions of ωG define the physical motions and if we divideG by these null directions the factor space is the physical phase spaceequipped with the physical symplectic structure

Example free particle The space G has coordinates α = (t x t0 x0) Given this pointin G there is one motion that goes from (t0 x0) to (t x) which is

t(τ) = t0 + (tminus t0)τ (3107)

x(τ) = x0 + (xminus x0)τ (3108)

Along this motion

p = mxminus x0

tminus t0 (3109)

pt = minusm(xminus x0)2

2(tminus t0)2 (3110)

The map i G rarr Γ is thus given by

P = p = mxminus x0

tminus t0 (3111)

Q = xminus p

mt = xminus xminus x0

tminus t0t (3112)

and therefore the two-form ωG is

ωG = ilowastωΓ = dP (t x t0 x0) and dQ(t x t0 x0)

= m dxminus x0

tminus t0and d

(xminus xminus x0

tminus t0t

)

=m

tminus t0

(dxminus xminus x0

tminus t0dt

)and

(dx0 minus xminus x0

tminus t0dt0

) (3113)


Immediately we see that a variation δα = (δt δx 0 0) (at constant (x0 t0)) such thatωG(δα) = 0 must satisfy

δx =xminus x0

tminus t0δt (3114)

This is precisely a variation of x and t along the physical motion (determined by(x0 t0)) Therefore ωG(δα) = 0 gives again the equations of motion The two nulldirections of ωG are thus given by the two vector fields

X =xminus x0

tminus t0partx + partt (3115)

X0 =xminus x0

tminus t0partx0 + partt0 (3116)

which are in involution (their Lie bracket vanishes) and therefore define a foliation ofG with two-dimensional surfaces These surfaces are parametrized by P and Q givenin (3111) (3112) and in fact

X(P ) = X(Q) = X0(P ) = X0(Q) = 0 (3117)

We have simply recovered in this way the physical phase space the space of thesesurfaces is the phase space Γ and the restriction of ωG to it is the physical symplecticform ωph

Physical predictions from S There are several different ways of derivingphysical predictions from the Hamilton function S(qa qa0)

bull Equation (387) gives the evolution function f in terms of the Hamil-ton function

bull If we can measure the partial observables qa as well as their momentapa then the Hamilton function can be used for making predictionsas follows Let

p1a(q

a1 q

a2) =

partS(qa1 qa2)

partqa1

p2a(q

a1 q

a2) =

partS(qa1 qa2)

partqa2 (3118)

The two equations

p1a = p1

a(qa1 q

a2)

p2a = p2

a(qa1 q

a2) (3119)

relate the four partial observables of the quadruplet (qa1 p1a q

a2 p

2a)

The theory predicts that it is possible to observe the quadruplet(qa1 p

1a q

a2 p

2a) only if this satisfies (3119) In this way the classical

theory determines which combinations of values of partial observ-ables can be observed

126 Mechanics

bull Alternatively we can fix two points qai and qaf in C and ask whethera third point qa is on the motion determined by qai and qaf That isask whether or not we could observe the correlation qa given thatthe correlations qai and qaf are observed A moment of reflection willconvince the reader that if the answer to this question is positivethen

S(qaf qa) + S(qa qai ) = S(qaf q

ai ) (3120)

because the action is additive along the motion Furthermore theincoming momentum at qa and the outgoing one must be equaltherefore

partS(qaf qa)

partqa= minus

partS(qa qaf )partqa

(3121)

326 Evolution parameters

A physical system is often defined by an action which is the integral of alagrangian in an evolution parameter But there are two different physicalmeanings that the evolution parameter may have

We have seen that the variational principle governing any hamiltoniansystem can be written in the form (here k = 1)

S =int

dτ(pa

dqa

dτminusNH(pa qa)

) (3122)

The action is invariant under reparametrizations of the evolution param-eter τ The evolution parameter τ has no physical meaning there is nomeasuring device associated with it

On the other hand consider a nonrelativistic system where qa = (t qi)and H = pt + H0 The action (3122) becomes

S =int

dτ(pt

dtdτ

+ pidqi

dτminusN(pt + H0(pi qi))

) (3123)

Varying N we obtain the equation of motion

pt = minusH0 (3124)

Inserting this relation back into the action we obtain

S =int

dτ(minusH0

dtdτ

+ pidqi

dτ

) (3125)


We can now change the integration variable from τ to t(τ) Defining (inbad physicistsrsquo notation) qi(t) equiv qi(τ(t)) and so on we can write

S =int

dτdtdτ

(minusH0 + pi

dqi

dt

)=

intdt

(pi

dqi(t)dt

minusH0

) (3126)

The evolution parameter in the action is no longer an arbitrary unphysicalparameter τ It is one of the partial observables the time observable t

If we are given an action we must understand whether the evolutionparameter in the action is a partial observable such as t or an unphysicalparameter such as τ If the action is invariant under reparametrizationsof its evolution parameter then the evolution parameter is unphysical Ifit is not then the evolution parameter is a partial observable

The same is true if the action is given in lagrangian form In performingthe Legendre transform from the lagrangian to the hamiltonian formalismthe consequence of the invariance of the action under reparametrizationsis doublefold First the relation between velocities and momenta cannotbe inverted The map from the space of the coordinates and velocities(qa qa) to the space of coordinates and momenta (qa pa) is not invertibleThe image of this map is a subspace Σ of Ω and we can characterize Σby means of an equation H = 0 for a suitable hamiltonian H Secondthe canonical hamiltonian computed via the Legendre transform vanisheson Σ In the language of constrained system theory this is because thecanonical hamiltonian generates evolution in the parameter of the actionsince this is unphysical this evolution is gauge the generator of a gaugeis a constraint and therefore vanishes on Σ

The evolution parameter in the action is often denoted t whether it isa partial observable or an unphysical parameter One should not confusethe t in the first case with the t in the second case They have verydifferent physical interpretations The time coordinate t in Maxwell theoryis a partial observable The time coordinate t in GR is an unphysicalparameter The fact that the two are generally denoted with the sameletter and with the same name is a very unfortunate historical accident

Example relativistic particle As we have seen the hamiltonian dynamics of a relativis-tic particle is defined by the relativistic hamiltonian H = pμp

μ + m2 namely by theaction principle

S =

intdτ

(pμx

μ minus N

2(pμp

μ + m2)

) (3127)

The relation between velocities and momenta obtained by varying pμ is xμ = NpμThe inverse Legendre transform therefore gives

S =1

2

intdτ

(xμx

μ

NminusNm2

) (3128)

128 Mechanics

We can also get rid of the Lagrange multiplier N from this action by writing its equationof motion

minus xμxμ

N2minusm2 = 0 (3129)

which is solved by

N =

radicminusxμxμ

m (3130)

and inserting this relation back into the action This gives

S = m

intdτ

radicminusxμxμ (3131)

which is the best known reparametrization invariant action for the relativistic particle

327 Complex variables and reality conditions

In GR it is often convenient to use complex dynamical variables sincethese simplify the form of the dynamical equations A particularly conve-nient choice is a mixture of complex and real variables where one canoni-cal variable is complex while the conjugate one is real As we shall see theselfdual connection (219) which is complex naturally leads to canonicalvariables of this type To exemplify how the use of such variables affectsdynamics consider a free particle with coordinate x momentum p andhamiltonian H0(x p) = p22m and assume we want to describe its dy-namics in terms of the variables (x z) where

z = xminus ip (3132)

In terms of these variables the nonrelativistic hamiltonian reads

H0(x z) = minus 12m

(xminus z)2 (3133)

Consider z as a configuration variable and ix as its momentum variableThe HamiltonndashJacobi equation becomes

partS(z t)partt

= minusH0

(minusi

partS(z t)partz

z

)=

12m

(ipartS(z t)

partz+ z

)2

(3134)

This is solved by

S(z t k) = kz +i2z2 minus k2

2mt (3135)

Equating the derivative of S with respect to the parameter k to a constantwe obtain the solution

C =partS(z t k)

partk= z minus k

mt (3136)

33 Field theory 129

that is

z(t) =k

mt + C (3137)

This is not the end of the story since so far k and C can be arbitrarycomplex constants To find the good solutions corresponding to real x andp we have to remind ourselves that z and x are not truly independentsince x is the real part of z

z + z = 2x (3138)

that is

z + z = minus2ipartS

partz (3139)

Inserting the solutions (3137) in the lhs we get

Im [k]t

m+ Im[C] = minusk (3140)

Therefore k is real and the imaginary part of C is minusk This immediatelygives the correct solution

Equation (3138) is called the reality condition The example illustratesthat in the HamiltonndashJacobi formalism the reality condition restricts thevalues of the HamiltonndashJacobi constants once the solutions of the evolu-tion equations are inserted

33 Field theory

There are several ways in which a field theory can be cast in hamiltonian form Onepossibility is to take the space of the fields at fixed time as the nonrelativistic configu-ration space Q This strategy badly breaks special- and general-relativistic invarianceLorentz covariance is broken by the fact that one has to choose a Lorentz frame for thet variable Far more disturbing is the conflict with general covariance The very founda-tion of generally covariant physics is the idea that the notion of a simultaneity surfaceover all the Universe is devoid of physical meaning It is better to found hamiltonianmechanics on a notion not devoid of physical significance

A second alternative is to formulate mechanics on the space of the solutions of theequations of motion The idea goes back to Lagrange In the generally covariant con-text a symplectic structure can be defined over this space using a spacelike surface butone can show that the definition is surface independent and therefore it is well definedThis strategy has been explored by several authors [108] The structure is viable inprinciple and has the merit of showing that the hamiltonian formalism is intrinsicallycovariant In practice it is difficult to work with the space of solutions to the field equa-tions in the case of an interacting theory Therefore we must either work over a spacethat we canrsquot even coordinatize or coordinatize the space with initial data on someinstantaneity surface and therefore effectively go back to the conventional fixed-timeformulation

130 Mechanics

The third possibility which I consider here is to use a covariant finite-dimensionalspace for formulating hamiltonian mechanics I noted above that in the relativisticcontext the double role of the phase space as the arena of mechanics and the spaceof the states is lost The space of the states namely the phase space Γ is infinite-dimensional in field theory essentially by definition of field theory But this does notimply that the arena of hamiltonian mechanics has to be infinite-dimensional as wellThe natural arena for relativistic mechanics is the extended configuration space C ofthe partial observables Is the space of the partial observables of a field theory finite-or infinite-dimensional

331 Partial observables in field theory

Consider a field theory for a field φ(x) with N components The fieldis defined over spacetime M with coordinates x and takes values in anN -dimensional target space T

φ M minusrarr T

x minusrarr φ(x) (3141)

For instance this could be Maxwell theory for the electric and magneticfields φ = ( E B) where N = 6 In order to make physical measurementson the field described by this theory we need N measuring devices to mea-sure the components of the field φ and four devices (one clock and threedevices giving us the distance from three reference objects) to determinethe spacetime position x Field values φ and positions x are therefore thepartial observables of a field theory Therefore the operationally motivatedrelativistic configuration space for a field theory is the finite-dimensionalspace

C = M times T (3142)

which has dimension 4 + N A correlation is a point (x φ) in C It repre-sents a certain value (φ) of the fields at a certain spacetime point (x) Thisis the obvious generalization of the (t α) correlations of the pendulum ofthe example in Section 321

A physical motion γ is a physically realizable ensemble of correlationsA motion is determined by a solution φ(x) of the field equations Such asolution determines a 4-dimensional surface in the ((4 +N)-dimensional)space C the surface is the graph of the function (3141) Namely theensemble of the points (x φ(x)) The space of the solutions of the fieldequations namely the phase space Γ is therefore an (infinite-dimensional)space of 4d surfaces γ in the (4 + N)-dimensional configuration space CEach state in Γ determines a surface γ in C

Hamiltonian formulations of field theory defined directly on C = MtimesTare possible and have been studied The main reason is that in a local field

33 Field theory 131

theory the equations of motion are local and therefore what happens at apoint depends only on the neighborhood of that point There is no needtherefore to consider full spacetime to find the hamiltonian structure ofthe field equations I refer the reader to the beautiful and detailed paper[109] and the ample references therein for a discussion of this kind ofapproach I give a simple and self-contained illustration of the formalismbelow with the emphasis on its general covariance

332 Relativistic hamiltonian mechanics

Consider a field theory on Minkowski space M Call φA(xμ) the fieldwhere A = 1 N The field is a function φ M rarr T where T = RN

is the target space namely it is the space in which the field takes valuesThe extended configuration space of this theory is the finite-dimensionalspace C = M times T with coordinates qa = (xμ φA) The coordinates qa

are the (4 + N) partial observables whose relations are described by thetheory A solution of the equations of motion defines a four-dimensionalsurface γ in C If we coordinatize this surface using the coordinates xμthen this surface is given by [xμ φA(xμ)] where φA(xμ) is a solution of thefield equations If alternatively we use an arbitrary parametrization withparameters τρ ρ = 0 1 2 3 then the surface is given by [xμ(τρ) φA(τρ)]where φA(xμ(τρ)) = φA(τρ)

In the case of a finite number of degrees of freedom (and no gauges)motions are given by one-dimensional curves At each point of the curvethere is one tangent vector and momenta coordinatize the one-forms Infield theory motions are four-dimensional surfaces and have four inde-pendent tangents Xμ or a ldquoquadritangentrdquo X = εμνρσXμotimesXνotimesXρotimesXσ

at each point Accordingly momenta coordinatize the four-forms LetΩ = Λ4T lowastC be the bundle of the four-forms pabcddqa and dqb and dqc and dqd

over C A point in Ω is thus a pair (qa pabcd) The space Ω carries thecanonical four-form

θ = pabcd dqa and dqb and dqc and dqd (3143)

In general given the finite-dimensional space C of the partial observ-ables qa dynamics is defined by a relativistic hamiltonian H Ω rarr V where Ω = Λ4T lowastC and V is a vector space Denote γ a four-dimensionalsurface in Ω and γ the projection of this surface on C The physical mo-tions γ are determined by the following

Variational principle A surface γ with a boundary α is aphysical motion if γ extremizes the integral

S[γ] =int

γpabcd dqa and dqb and dqc and dqd (3144)

132 Mechanics

in the class of the surfaces γ satisfying

H(qa pabcd) = 0 (3145)

and whose restriction γ to C is bounded by α

This is a completely straightforward generalization of the variational prin-ciple of Section 32 Equation (3145) defines a surface Σ in Ω As beforewe denote θ the restriction of θ to Σ and ω = dθ

For a field theory on Minkowski space without gauges the system(3145) is given by

pABCD = pABCμ = pABμν = 0 (3146)

H = π + H0(xμ φA pμA) = 0 (3147)

where H0 is DeDonderrsquos covariant hamiltonian [110] (see below for anexample) It is convenient to use the notation pμνρσ = πεμνρσ and pAνρσ =pμAεμνρσ for the nonvanishing momenta and to use coordinates (xμ φA pμA)on Σ On the surface defined by (3146)

θ = π d4x + pμA dφA and d3xμ (3148)

where we have introduced the notation d4x = dx0 and dx1 and dx2 and dx3 andd3xμ = d4x(partμ) = 1

3εμνρσdxν and dxρ and dxσ On Σ defined by (3146) and(3147)

θ = θ|Σ = minusH0(xμ φA pμA)d4x + pμA dφA and d3xμ (3149)

and ω is the five-form

ω = minusdH0(xμ φA pμA) and d4x + dpμA and dφA and d3xμ (3150)

An orbit of ω is a four-dimensional surface m immersed in Σ such thatat each of its points a quadruplet X of tangents to the surface satisfies

ω(X) = 0 (3151)

I leave to the reader the exercise of showing that the projection of anorbit on C is a physical motion

In more detail let (partμ partA partAμ ) be the basis in the tangent space of

Σ determined by the coordinates (xμ φA pμA) Parametrize the surfacewith arbitrary parameters τρ The surface is then given by the points[xμ(τρ) φA(τρ) pμA(τρ)] Let partρ = partpartτρ Then let

Xρ = partρxμ(τρ) partμ + partρφ

A(τρ) partA + partρpμA(τρ) partA

μ (3152)

33 Field theory 133

Then X = X0 otimesX1 otimesX2 otimesX3 is a rank four tensor on Σ If ω(X) = 0then φA(xμ) determined by φA(xμ(τρ)) = φA(τρ) is a physical motion

Summarizing the canonical formalism of field theory is completely de-fined by the couple (C H) where C is the finite-dimensional space ofthe partial observables (field values and spacetime coordinates) and H ahamiltonian on the finite-dimensional space Ω = Λ4T lowastC Equivalently itis completely defined by the finite-dimensional presymplectic space (Σ θ)The formalism as well as its interpretation make sense even in the case inwhich the coordinates of C do not split into xμ and φA and the relativistichamiltonian does not have the particular form (3146)ndash(3147)

Example scalar field As an example consider a scalar field φ(xμ) on Minkowski spacesatisfying the field equations

partμpartμφ(xμ) + m2φ(xμ) + V prime(φ(xμ)) = 0 (3153)

Here the Minkowski metric has signature [+minusminusminus] and V prime(φ) = dV (φ)dφ The fieldis a function φ M rarr T where here T = R The relativistic configuration space of thistheory is the five-dimensional space C = M times T with coordinates (xμ φ) The spaceΩ has coordinates (xμ φ π pμ) (equation (3146) is trivially satisfied) and carries thecanonical four-form

θ = π d4x + pμ dφ and d3xμ (3154)

The dynamics is defined on this space by the DeDonder relativistic hamiltonian

H = π + H0 = 0 (3155)

H0 =1

2

(pμpμ + m2φ2 + 2V (φ)

) (3156)

Therefore we can use coordinates (xμ φ pμ) on the surface Σ defined by these equationsand (3149) gives

θ = minus1

2

(pμpμ + m2φ2 + 2V (φ)

)d4x + pμ dφ and d3xμ (3157)

The couple (Σ θ) defines the presymplectic formulation of the system ω is the five-form

ω = dθ = minus(pμdpμ + m2φdφ + V prime(φ)dφ

)and d4x + dpμ and dφ and d3xμ (3158)

A tangent vector has the form

V = Xμpartxμ + Xφpartφ + Y μpartpμ (3159)

If we coordinatize the orbits of ω with the coordinates xμ at every point we have thefour independent tangent vectors

Xμ = partxμ + (partμφ)partφ + (partμpρ)partpρ (3160)

and the quadritangent X = εμνρσXμ otimesXν otimesXρ otimesXσ Inserting (3160) and (3158) inω(X) = 0 a straightforward calculation yields

partμφ(x) = pμ(x) (3161)

partμpμ(x) = minusm2φ(x) minus V prime(φ(x)) (3162)

and therefore precisely the field equations (3153) Notice that the canonical formalismis manifestly Lorentz covariant and no equal-time initial data surface has to be chosen

134 Mechanics

A state is a 4d surface (x φ(x)) in the extended configurations space C It representsa set of combinations of measurements of partial observables that can be realized inNature The phase space Γ is the infinite-dimensional space of these states A statedetermines whether or not a certain correlation (x φ) or a certain set of correlations(x1 φ1) (xn φn) can be observed They can be observed if the points (xi φi) lie onthe 4d surface that represents the state Conversely the observation of a certain setof correlations gives us information on the state the surface has to pass through theobserved points

333 The space of boundary data G and the Hamilton function S

The space of boundary data G described in Section 325 plays a key role inquantum theory In the finite-dimensional case G is the cartesian productof the extended configuration space with itself but the same is not true inthe field theoretical context where we need an infinite number of bound-ary data to characterize solutions Recall that in the finite-dimensionalcase G is the space of the possible boundaries of a motion in C In fieldtheory a motion is a 4d surface in C Its boundary is a three-dimensionalsurface α without boundaries in C Let us therefore define G in field theoryas a space of oriented three-dimensional surfaces α without boundaries inC As C = M times T the boundary data α includes a 3d boundary surfaceσ in spacetime as well as the value ϕ of the field on this surface

More precisely let xμ be spacetime coordinates in M and φA coordi-nates in the target space Coordinatize the 3d surface α with 3d coordi-nates τ = (τ1 τ2 τ3) Then α is given by the functions

α = [σ ϕ] (3163)σ τ rarr xμ(τ) (3164)ϕ τ rarr ϕA(τ) (3165)

The functions xμ(τ) define the 3d surface σ without boundaries in space-time The functions ϕA(τ) define the value of the field φ(x) on this surface

φA(x(τ)) equiv ϕA(τ) (3166)

Say σ is the boundary of a connected region R of M Then genericallyϕ determines a solution φ(x) of the equations of motion in the interior Rsuch that φ|σ = ϕ Imagine that σ is a cylinder in Minkowski space Todetermine a solution in the interior we need the initial value of the fieldon the bottom of the cylinder its final value on the top of the cylinder aswell as spatial boundary conditions on the side of the cylinder The dataα determine all these field values as well as the spacetime location of thecylinder itself These data form the field theoretical generalization of theset (t qi tprime qprimei) which form the argument of the Hamilton function and ofthe quantum propagator in finite-dimensional mechanics Alternatively

33 Field theory 135

the surface α need not be connected For instance it can be formed bytwo components which we can view as initial and final configurations

The Hamilton function S[α] = S[σ ϕ] is defined as the action of thesolution of the equations of motion φ(x) such that φ|σ = ϕ in R We shallsee below that S[α] satisfies a functional HamiltonndashJacobi equation andcan be seen as the classical limit of a quantum mechanical propagator10

We can give a more formal definition of S[α] analogous to the definition(383) Let γ be the motion in C bounded by α Let γ be the lift of γ toΣ That is let γ be the orbit of ω that projects down to γ Then

S[α] =int

γθ (3167)

Example scalar field For a scalar field for instance

S[α] =

int

γ

θ =

int

γ

(πd4x + pμdφ and d3xμ) =

int

R(π + pμpartμφ) d4x

=

int

R

(minus1

2pμpμ minus 1

2m2φ2 minus V (φ) + pμpartμφ

)d4x (3168)

=

int

R

(1

2partμφpart

μφminus 1

2m2φ2 minus V (φ)

)d4x

=

int

RL(φ partμφ) d4x (3169)

where L is the lagrangian density and we have used the equation of motion pμ = partμφIt is not hard to compute the Hamilton function for a free scalar field in the special

case in which α is formed by the two spacelike parallel hypersurfaces xμ(τ) = (t1 τ)and xμ(τ) = (t2 τ) and by the values φ1(x) and φ2(x) of the field on these surfacesThe calculation is simplified by the fact that a free field is essentially a collection of

oscillator with modes of wavelength k and frequency ω(k) =

radic|k|2 + m2 Using this

fact and (334) it is straightforward to compute the field for given boundary values andits action This gives

S(φ1 t1 φ2 t2) =

intd3k ω(k)

2φ1(k)φ2(k) minus (|φ1|2(k) + |φ2|2(k)) cos[ω(k)(t1 minus t2)]

2 sin[ω(t1 minus t2)]

(3170)

where φ(k) are the Fourier components of φ(x)

The symplectic structure on G As in the finite-dimensional case wecan define a symplectic structure on G Let s be the 3d surface in Σ thatbounds γ That is s = [xμ(τ) ϕA(τ) pμA(τ)] where the momenta pμA(τ)are determined by the solution of the field equations determined by theentire α

10S[α] is only defined on the regions of G where this solution exists and it is multivaluedwhere there is more than one solution

136 Mechanics

Define a two-form on G as follows

ωG [α] =int

sω (3171)

The form ωG is a two-form it is the integral of a five-form over a 3dsurface More precisely let δα be a small variation of α This variationcan be seen as a vector field δα(τ) defined on α This variation determinesa corresponding small variation δs which in turn is a vector field δs(τ)over s Then

ωG [α](δ1α δ2α) =int

sα

ω(δ1s δ2s) (3172)

Thus the five-form ω on the finite-dimensional space Σ defines the two-form ωG on the infinite-dimensional space G

Consider a small local variation δα of α This means varying the surfaceαM in Minkowski space as well as varying the value of the field over itAssume that this variation satisfies the field equations that is the vari-ation of the field is the correct one for the solution of the field equationsdetermined by α We have

ωG [α](δα) =int

sα

ω(δs) (3173)

But the variation δs is by construction along the orbit namely in the nulldirection of ω and therefore the right-hand side of this equation vanishesIt follows that if δα is an infinitesimal physical motion then

ωG(δα) = 0 (3174)

The pair (G ωG) contains all the relevant information on the systemThe null directions of ωG determine the variations of the 3-surface α alongthe physical motions The space G divided by these null directions namelythe space of the orbits of these variations is the physical phase space Γand the ωG restricted to this space is the physical symplectic two-formof the system

Example scalar field Letrsquos compute ωG in a slightly more explicit form for the exampleof the scalar field From the definition (3171)

ωG [α] =

int

s

ω =

int

s

dπ and d4x + dpμ and dφ and d3xμ

=

int

s

(pνdpν + m2φdφ + V primedφ) and d4x + dpμ and dφ and d3xμ

=

int

αM

d3xν

[(pμminuspartμφ)dpμ and dxν + (m2φ + V prime + partμp

μ)dφ and dxν + dpν and dφ]

=

int

αM

d3xν dpν and dφ (3175)

33 Field theory 137

where we have used the xμ coordinates themselves as integration variables and there-fore the integrand fields are the functions of the xμ Notice that since the integral is ons the pμ in the integrand is the one given by the solution of the field equation deter-mined by the data on α Therefore it satisfies the equations of motion (3161)ndash(3162)which we have used above Using (3161) again we have

ωG [α] =

int

αM

d3x nν d(nablaνφ) and dφ (3176)

In particular if we consider variations δα that do not move the surface and such thatthe change of the field on the surface is δφ(x) we have

ωG [α](δ1α δ2α) =

int

αM

d3x nν

(δ1φnablaνδ2φminus δ2φnablaνδ1φ

) (3177)

This formula can be directly compared with the expression of the symplectic two-formgiven on the space of the solutions of the field equations in [108] The expression is thesame but with a nuance in the interpretation ωG is not defined on the space of thesolutions of the field equations it is defined on the space of the lagrangian data G andthe normal derivative nνnablaνφ of these data is determined by the data themselves viathe field equations

334 HamiltonndashJacobi

A HamiltonndashJacobi equation for the field theory can be written as a localequation on the boundary satisfied by the Hamilton function I illustratehere the derivation of the HamiltonndashJacobi equation in the case of thescalar field leaving the generalization to the interested reader From thedefinition

S[α] =int

γθ =

int

γ(πd4x + pμdφ and d3xμ) (3178)

we can write

δS[α]δxμ(τ)

= π(τ) nμ(τ) + εμνρσ pν(τ) partiφ(τ) partjxρ(τ) partkx

σ(τ) εijk (3179)

where

nμ(τ) =13εμνρσpart1x

ν(τ)part2xρ(τ)part3x

σ(τ) (3180)

is the normal to the 3-surface σ The momentum π depends on the fullα Contracting this equation with nμ we obtain

π(τ) = nμ(τ)δS[α]δxμ(τ)

+ pi(τ) partiφ(τ) (3181)

Using the equation of motion pμ = partμφ this becomes


+ partiφ(τ)partiφ(τ) (3182)

138 Mechanics

Also

δS[α]δϕ(τ)

= pμ(τ)nμ(τ) (3183)

The derivation of these two equations requires steps analogous to the oneswe used to derive (386)

Now from (3155) and (3156) we have that the scalar field dynamicsis governed by the equation

π +12

(pμpμ + m2φ2 + 2V (φ)

)= 0 (3184)

We split pμ into its normal (p = pμnμ) and tangential (pi) components(so that pμ = pipartix

μ + pnμ) obtaining

π +12

(p2 minus pipi + m2φ2 + 2V (φ)

)= 0 (3185)

Inserting (3182) and (3183) we obtain

δS[α]δxμ(τ)

nμ(τ) +12

[(δS[α]δϕ(τ)

)2

+partjϕ(τ)partjϕ(τ) + m2ϕ2(τ) + 2V (ϕ(τ))

]

= 0

(3186)

This is the HamiltonndashJacobi equation Notice that the function

S[xμ(τ) ϕ(τ)] = S[σ ϕ] = S[α] (3187)

is a function of the surface not the way the surface is parametrizedTherefore it is invariant under a change of parametrization It followsthat

δS[α]δxμ(τ)

partjxμ(τ) +

δS[α]δϕ(τ)

partjϕ(τ) = 0 (3188)

(This equation can be obtained also from the tangential component of(3179)) The two equations (3186) and (3188) govern the HamiltonndashJacobi function S[α]

The connection with the nonrelativistic field theoretical HamiltonndashJacobi formalism is the following We can restrict the formalism to apreferred choice of parameters τ Choosing τ j = xj we obtain S in theform S[t(x) φ(x)] and the HamiltonndashJacobi equation (3186) becomes

δS

δt(x)+

12

[(δS[α]δφ(x)

)2

+ partjφpartjφ + m2φ2 + 2V (φ)

]

= 0 (3189)

33 Field theory 139

Further restricting the surfaces to the ones of constant t gives the func-tional S[t φ(x)] satisfying the HamiltonndashJacobi equation

partS

partt+

12

intd3x

[(δS

δφ(x)

)2

+ |nablaφ|2 + m2φ2 + 2V (φ)

]

= 0 (3190)

which is the usual nonrelativistic HamiltonndashJacobi equation

partS

partt+ H

(φ nablaφ

δS[α]δφ(x)

)= 0 (3191)

where H(φ nablaφ parttφ) is the nonrelativistic hamiltonian

Canonical formulation on G We can write a hamiltonian density functionH(τ) directly for the infinite-dimensional space G H(τ) is a function onthe cotangent space T lowastG We coordinatize this cotangent space with thefunctions (xμ(τ) ϕ(τ)) and their momenta (πμ(τ) p(τ)) The hamiltonianis then

H[xμ ϕ πμ p](τ) = πμ(τ)nμ(τ) +12

[p2(τ) + partjϕ(τ)partjϕ(τ)

+m2ϕ2(τ) + 2V (ϕ(τ))] (3192)

and the HamiltonndashJacobi equation (3186) reads

H

[xμ ϕ

δS[α]δxμ

δS[α]δϕ

](τ) = 0 (3193)

If we restrict the surface xμ(τ) to the case xμ(τ) = (t τ) then H(τ)becomes

H[xμ ϕ πμ p](x) = π0(x) + H0(x) (3194)

where H0(x) is the conventional nonrelativistic hamiltonian density

H0[φ p] =12

[p2+partjϕpart

jϕ + m2ϕ2 + 2V (ϕ)] (3195)

Physical predictions from S The complete physical predictions of thetheory can be obtained directly from the Hamilton function S[α] = S[σ ϕ]as follows Let p(τ) be a function on the surface σ Define

F [σ ϕ p](τ) =δS[σ ϕ]δϕ(τ)

minus p(τ) (3196)

140 Mechanics

Given a closed surface σ in spacetime we can observe field boundaryvalues φ(x(τ)) = ϕ(τ) together with momenta nμpartμφ(x(τ)) = p(τ) if andonly if

F [σ ϕ p](τ) = 0 (3197)

This equation is equivalent to the equations of motion and expresses di-rectly the physical content of the theory as a restriction on the partialobservables that can be observed on a boundary surface

As in the case of finite-dimensional systems the general solution ofthe equations of motion can be obtained by derivations For instance letα be formed by two connected components that we denote α = [σ ϕ]and α0 = [σ0 ϕ0] parametrized by τ and τ0 respectively Consider theequation for α

f [α](τ) =δS[α cup α0]δϕ0(τ0)

minus p0(τ0) = 0 (3198)

where p0(τ) is an arbitrary initial value momentum This is the evolutionequation that determines all surfaces α compatible with the initial dataϕ0 p0 on σ0

34 Thermal time hypothesis

Earth lay with Sky and after them was born TimeThe wily youngest and most terrible of her children

Hesiod Theogony [111]

In the macroscopic world the physical variable t measured by a clockhas peculiar properties It is not easy to pinpoint these properties withprecision without referring to a presupposed notion of time but it is alsodifficult to deny that they exist From the point of view developed inthis book at the fundamental level the variable t measured by a clockis on the same footing as any other partial observable If we accept thisidea we have then to reconcile the fact that time is not a special variableat the fundamental level with its peculiar properties at the macroscopiclevel What is so special about time An interesting possibility is that itis statistical mechanics and therefore thermodynamics that singles outt and gives it its special properties I briefly illustrate this idea in thissection

The world around us is made up of systems with a large number ofdegrees of freedom such as fields We never measure the totality of thesedegrees of freedom Rather we measure certain macroscopic parametersand make predictions on the basis of assumptions on the state of the other

34 Thermal time hypothesis 141

degrees of freedom The viability of our choice of macroscopic parametersand our assumptions about the state of the others is justified a posterioriif the system of prediction works We represent our incomplete knowledgeand assumptions in terms of a statistical state ρ The state ρ can berepresented as a normalized positive function on the phase space Γ

ρ Γ rarr R+ (3199)int

Γds ρ(s) = 1 (3200)

ρ(s) represents the assumed probability density of the state s in Γ Thenthe expectation value of any observable A Γ rarr R in the state ρ is

ρ[A] =int

Γds A(s) ρ(s) (3201)

The fundamental postulate of statistical mechanics is that a system leftfree to thermalize reaches a time-independent equilibrium state that canbe represented by means of the Gibbs statistical state

ρ0(s) = NeminusβH0(s) (3202)

where β = 1T is a constant ndash the inverse temperature ndash and H0 isthe nonrelativistic hamiltonian Classical thermodynamics follows fromthis postulate Time evolution At = αt(A) of A is determined by (378)Equivalently At(s) = A(t(s)) where s(t) is the hamiltonian flow of H0 onΓ The correlation probability between At and B is given by

WAB(t) = ρ0[αt(A)B] =int

Σds A(s(t)) B(s) eminusβH0(s) (3203)

In this chapter we have seen that the formulas of mechanics do notsingle out a preferred variable because all mechanical predictions canbe obtained using the relativistic hamiltonian H which treats all vari-ables on an equal footing instead of using the nonrelativistic hamiltonianH0 which singles t out Is this true also for statistical mechanics andthermodynamics Equations (3200)ndash(3201) are meaningful also in therelativistic context where Γ is the space of the solutions of the equationsof motion But this is not true for (3202) and (3203) These dependon the nonrelativistic hamiltonian They depend on the fact that tis a variable different from the others Equations (3202) and (3203) defi-nitely single out t as a special variable This observation indicates that thepeculiar properties of the t variable have to do with statistical mechanicsand thermodynamics rather than with mechanics With purely mechani-cal measurements we cannot recognize the time variable With statisticalor thermal measurements we can

142 Mechanics

Indeed notice that if we try to pinpoint what is special about thevariable t we generally find features connected to thermodynamics irre-versibility convergence to equilibrium memory feeling of ldquoflowrdquo and soon

Indeed there is an intriguing fact about (3202) and (3203) Imaginethat we study a system which is in equilibrium at inverse temperature βand we do not know its nonrelativistic hamiltonian H0 In principle wecan figure out H0 simply by repeated microscopic measurements on copiesof the system without any need of observing time evolution Indeed ifwe find out the distribution of microstates ρ0 then up to an irrelevantadditive constant we have

H0 = minus 1β

ln ρ0 (3204)

Therefore in a statistical context we have in principle an operationalprocedure for determining which one is the time variable First measureρ0 second compute H0 from (3204) third compute the hamiltonianflow s(t) of H0 on Σ The time variable t is the parameter of this flowA ldquoclockrdquo is any measuring apparatus whose reading grows linearly withthis flow The multiplicative constant in front of H0 just sets the unit inwhich time is measured Up to this unit we can find out which one is thetime variable just by measuring ρ0 This is in strident contrast with thepurely mechanical context where no operational procedure for singlingout the time variable is available

Now let me come to the main observation Imagine that we have atruly relativistic system where no partial observable is singled out as thetime variable Imagine that we make measurements on many copies of thesystem and find that the statistical state describing the system is givenby a certain arbitrary11 state ρ Define the quantity

Hρ = minus ln ρ (3205)

Let s(tρ) be the hamiltonian flow of Hρ Call tρ ldquothermal timerdquo Callldquothermal clockrdquo any measuring device whose reading grows linearly withthis flow Given an observable A consider the one-parameter family ofobservables Atρ defined by Atρ(s) = A(tρ(s)) Then it follows that thecorrelation probability between the observables Atρ and B is given by

WAB(tρ) =int

Σds A(tρ(s)) B(s) eminusHρ(s) (3206)

What is the difference between the physics described by (3202)ndash(3203)and that described by (3205)ndash(3206) None That is whatever the

11For (3205) to make sense assume that ρ nowhere vanishes on Σ


statistical state ρ there exists always a variable tρ measured by the ther-mal clock with respect to which the system is in equilibrium and whosephysics is the same as in the conventional nonrelativistic statistical caseThis key observation naturally leads us to the following hypothesis

The thermal time hypothesis In Nature there is no preferredphysical time variable t There are no equilibrium states ρ0 preferreda priori Rather all variables are equivalent we can find the systemin an arbitrary state ρ if the system is in a state ρ then a preferredvariable is singled out by the state of the system This variable iswhat we call time

In other words it is the statistical state that determines which variable isphysical time and not any a priori hypothetical ldquoflowrdquo that drives the sys-tem to a preferred statistical state All variables are physically equivalentat the mechanical level But if we restrict our observations to macroscopicparameters and assume the other dynamical variables are distributed ac-cording to a statistical state ρ then a preferred variable is singled outby this procedure This variable has the property that correlations withrespect to it are described precisely by ordinary statistical mechanics Inother words it has precisely the properties that characterize our macro-scopic time parameter

In other words when we say that a certain variable is ldquothe timerdquo weare not making a statement concerning the fundamental mechanical struc-ture of reality12 Rather we are making a statement about the statisticaldistribution we use to describe the macroscopic properties of the systemthat we describe macroscopically

The hamiltonian Hρ determined by a state ρ is called the thermalhamiltonian The ldquothermal time hypothesisrdquo is the idea that what wecall ldquotimerdquo is simply the thermal time of the statistical state in whichthe world happens to be when described in terms of the macroscopicparameters we have chosen

Let the system be in the mechanical microstate s Describe it with macroscopicobservables Ai In general (but not always) there exists a statistical state ρ whose meanvalues give the correct predictions for the Ai that is Ai(s) sim ρ[Ai] Assuming it exitsρ codes in a sense our ignorance of the microscopic details of the state Intuitivelywe can therefore say that the existence of time is the result of this ignorance of oursTime is the expression of our ignorance of the microstate

The thermal time hypothesis works surprisingly well in a number ofcases For example if we start from a radiation-filled covariant cosmo-logical model having no preferred time variable and write a statistical

12Time Kρoνoς comes after matter (Earth Γαια and Sky Ovρανoς) also in Greek

mythology See Hesiodrsquos quote [111] at the beginning of this section

144 Mechanics

state representing the cosmological background radiation then the ther-mal time of this state turns out to be precisely the Friedmann time [112]Furthermore we will see in Section 551 that this hypothesis extends inan extremely natural way to the quantum context and even more nat-urally to the quantum field theoretical context where it leads also to ageneral abstract state-independent notion of time flow

mdashmdash


The hamiltonian theory of systems with constraints is one of Diracrsquos manymasterpieces The theory is not just a technical complication of standardhamiltonian mechanics it is a powerful generalization of mechanics whichremains valid in the general-relativistic context The title of Diracrsquos initialwork on the subject was ldquoGeneralized Hamiltonian dynamicsrdquo [113] Thetheory is synthesized in [114] For modern accounts and developments see[115] For the notion of partial observable I have followed [116] On thegeneral structure of mechanics I have followed [117 118] For a nontrivialexample of relational evolution treated in detail see [119]

The canonical treatment of field theory on finite-dimensional spaces de-rives from the Weyl and DeDonderrsquos calculus of variations [110 120] Abeautiful comprehensive and mathematically precise discussion of covari-ant hamiltonian field theory is in [109] which contains complete referencesto the literature on the subject See also [121]

The idea of the thermal origin of time was introduced in [112 122] inthe context of classical field theory and was independently suggested byAlain Connes It is developed in quantum field theory (see Section 551)in [125] see also [124] For a related Boltzmann-like approach see [123]

4Hamiltonian general relativity

I begin this chapter by presenting the HamiltonndashJacobi formulation of GR This is thebasis of the quantum theory

In the remainder of the chapter I present formulations of hamiltonian GR on afinite-dimensional configuration space along the lines illustrated at the end of theprevious chapter

This order of presentation is inverse to the logical order which should start from thefinite-dimensional configuration space of the partial observables But I do not want toforce the hurried reader to navigate through the entire chapter before finding the fewsimple equations that are the basis of the quantum theory

I take the cosmological constant to be zero and ignore matter fields leaving to thereader the generally easy exercise of adding the cosmological and matter terms to therelevant equations

41 EinsteinndashHamiltonndashJacobi

GR can be expressed in terms of a complex field Aia(τ) and a 3d real

momentum field Eai (τ) defined on a three-dimensional space σ without

boundaries satisfying the reality conditions

Aia + Ai

a = Γia[E] (41)

where Γ is defined below in (423)ndash(424) The theory is defined by thehamiltonian system

DaEai = 0 (42)

Eai F

iab = 0 (43)

F ijabE

ai E

bj = 0 (44)

145

146 Hamiltonian general relativity

where F ijab = εijk F

kab (see pg xxii) and Da and F i

ab are the covariantderivative and the curvature of Ai

a defined by

Davi = partavi + εijkAjav

k (45)

F iab = partaA

ib minus partbA

ia + εijkA

jaA

kb (46)

I sketch the derivation of these equations from the lagrangian formalismbelow An indirect derivation via a finite-dimensional canonical formula-tion is given at the end of this chapter

The HamiltonndashJacobi system is given in terms of the functional S[A]by writing

Eai (τ) =

δS[A]δAi

a(τ)(47)

in the hamiltonian system The first two equations that we obtain

DaδS[A]δAi

a(τ)= 0 F i

ab(τ)δS[A]δAi

a(τ)= 0 (48)

require that S[A] is invariant under 3d diffeomorphisms (diffs) and localSO(3) transformations as I will show in a moment The last reads

F ijab(τ)

δS[A]δAi

a(τ)δS[A]

δAjb(τ)

= 0 (49)

This is the HamiltonndashJacobi equation of GR It defines the dynamics ofGR

Smeared form Equivalently we can integrate equations (42)ndash(44)against suitable ldquotestrdquo functions and demand the integral to vanish forany such function For the first two we get

G[λ] = minusint

d3τλi DaEai =

intd3τDaλ

iEai = 0 (410)

C[f ] = minusint

d3τfaF iabE

bi = 0 (411)

The quantities Daλi and faF i

ab that appear in these equations are theinfinitesimal transformations of the connection under an internal gaugetransformation with generator λi(τ) and under (the combination of aninternal gauge transformation and) an infinitesimal diffeomorphism gen-erated by the vector field fa(τ)

δλAia = Daλ

i δfAia = f bF i

ab (412)

41 EinsteinndashHamiltonndashJacobi 147

Therefore the smeared form of (48) readsint

d3τ δλAia(τ)

δS[A]δAi

a(τ)= 0

intd3τ δfA

ia(τ)

δS[A]δAi

a(τ)= 0 (413)

which is the requirement that S[A] is invariant under gauge and diffeo-morphisms

The quantity (44) on the other hand is a density of weight two Tobe able to integrate it against a scalar quantity and get a well-definedresult we need a density of weight one This can be obtained by dividingthe hamiltonian by the square roote of the determinant of E exploitingthe freedom in the definition of the hamiltonian The Poisson bracketderived below in (425) between the volume

V =int

d3xradic| detE(x)| (414)

and the connection is

V Aia(x) = (8πiG)

Ebj (x)Ec

k(x)εabcεijk

4radic

| detE(x)| (415)

Using this we can write (44) in the form

H[N ] =int

N tr(F and V A) = 0 (416)

This form of the hamiltonian will prove convenient in the quantum theoryEquations (410) (411) and (416) define GR

411 3d fields ldquoThe length of the electric field is the areardquo

What is the relation between the 4d fields used in Chapter 2 and the3d fields used above Consider a solution (eIμ(x) Ai

μ(x)) of the Einsteinequations (221) Choose a 3d surface σ τ = (τa) rarr xμ(τ) withoutboundaries in the coordinate space The four-dimensional forms Ai (theselfdual connection defined in (219)) Σi (the 4d Plebanski two-formdefined in (223)) and eI (the gravitational field introduced in (21))induce the three-dimensional forms

Ai(τ) = Aia(τ) dτa (417)

Σi(τ) = Σiab(τ) dτa and dτ b (418)

eI(τ) = eIa(τ) dτa (419)

on σ The 3d field E is defined as the vector density associated to Σi thatis

Eai(τ) = εabc Σibc(τ) (420)


Letrsquos write eI(τ) = (e0(τ) ei(τ)) Choose a gauge in which

e0(τ) = 0 (421)

(The extension of the formalism to a more general gauge deserves to beinvestigated See [127]) It is easy to see that in this gauge Ea

i (τ) is realand

Eai (τ) =

12εijk εabc ejb(τ) ekc (τ) (422)

The connection Γi[E](τ) = εijkΓj

k[E](τ) used in (41) is defined by

dei + Γij [E] and ej = 0 (423)

(this is the first Cartan structure equation for σ) which is solved by

Γjak =

12ebk(partae

jb minus partbe

ja + ecjealpartbe

lc) (424)

That is it is the spin connection of the triad eia It is also easy to verifythat in this gauge the two quantities Ai

a(τ) and Eai (τ) defined by (417)

and (420) satisfy the ldquoreality conditionrdquo (41)The quantity Ea

i (τ) is (8πiG times) the momentum conjugate to Aia(τ)

Hence we can write immediately the Poisson brackets

Aia(τ) Eb

j (τprime) = (8πiG) δbaδ

ijδ

3(τ τ prime) (425)

In Maxwell and YangndashMills theories the momentum conjugate to thethree-dimensional connection A is called electric field The field E istherefore called the gravitational electric field In the gauge (421) weare considering E is determined just by eia(τ) the triad field of σEquation (422) shows that E is the inverse matrix of the triad eia(τ)multiplied by its determinant

Eai = (det e)eai (426)

I sketch here the derivation of the basic equations of the hamiltonianformalism namely the Poisson brackets (425) and the constraint system(42)ndash(44) For a detailed discussion of this derivation see for instanceI [2 9 20 126] An indirect derivation via a finite-dimensional canonicalformulation is given at the end of this chapter We can start for instancefrom the action (227) without the cosmological constant and write it as


follows

S[Σ A] =minusi

16πG

intΣi and F i =

minusi16πG

intΣiμνF

iρσε

μνρσd4x

=minusi

8πG

int (ΣiabF

ic0 + Σi0aF

ibc

)εabcd4x

=minusi

8πG

int (Ec

i

(part0A

ic minus partcA

i0 + εijkA

j0A

kc

)+ PiIJe

Jae

J0F

ibcε

abc)d4x

=minusi

8πG

int (Ec

i Aic + Ai

0DcEci +

12(εijke

jae

k0 + ie0

0eia

)Fibcε

abc

)d4x

=minusi

8πG

int (Ec

i Aic + λi

0

(DcE

ci

)+ λb

(Ea

i Fiab

)+ λ

(Ea

jEbkF

jkab

))d4x

(427)

The dot over A indicates time derivative I have used the gauge conditione0i = 0 and the Lagrange multipliers are multiples of the nondynamical

variables Ai0 e

00 e

i0 The first term shows that Ec

i 8iπG is the momentumconjugate to Ai

c varying with respect to the Lagrange multipliers yieldsthe constraint system (42)ndash(44)

The geometry of the three surface In Section 214 we saw that the grav-itational field has a metric interpretation The metric structure inheritedby σ depends on the gravitational electric field E In particular considera two-dimensional surface S σ = (σ1 σ2) rarr τ(σi) embedded in thethree-dimensional surface σ What is the area of S From the definitionof the area equation (270) we have in a few steps

A(S) =int

Sd2σ |E| (428)

Here the norm is defined by |v| =radicδijvivj and

Ei(σ) = Eai (τ(σ)) na(σ) (429)

the normal to the surface being defined by

na(σ) = εabcpartτ b(σ)partσ1

partτ c(σ)partσ2

(430)

Equation (428) can be interpreted as the surface integral of the norm ofthe two-form

Ei = Eai εabc dxb and dxc (431)

and writtenA(S) =

int

S|E| (432)


Thus Eai or more precisely its norm or ldquolengthrdquo |E| defines the area

element We could therefore say that in gravity ldquothe length of the electricfield is the areardquo or more precisely the area of a surface is the flux of(the norm of) the gravitational electric field across the surface

Using (273) a similar calculation gives the volume of a 3d region R

V(R) =int

Rd3τ

radic| detE| (433)

If we know the area of any surface and the volume of any region we knowthe geometry

These expressions for area and volume in terms of the gravitationalelectric field E play a major role in quantum gravity The correspondingquantum operators have a discrete spectrum their eigenstates are knownand determine a convenient basis in the quantum state space

For later use notice that detE = det ((det e)eminus1)=(det e)3(det e)minus1 =(det e)2 Hence

radicn middot n = | det e| =

radic| detE| where n is defined in (430)

Phase space states and relation with the Einstein equations A state ofGR is an equivalence class of 4d field configurations eIμ(x) solving the Ein-stein equations under the two gauge transformations (2123) and (2124)The space Γ of these equivalence classes is the phase space of GR

Given one solution of the Einstein equations consider a 3d surface σwithout boundaries in coordinate space Let Ai

a(τ) be the connection on σinduced by the 4d selfdual connection Ai

μ(x) A state determines a familyof possible 3d fields Ai

a(τ) called compatible with the state obtainedby changing the representative in the equivalence class of solutions orequivalently changing σ A family of 3d fields Ai

a(τ) compatible with astate can be obtained in principle from a solution of the HamiltonndashJacobisystem as follows

In general to solve the system we need a solution S[Aα] of theHamiltonndashJacobi system depending on a sufficiently large number αn ofparameters A state is then determined by constants αn and βn as followsThe equation

F [Aα] =partS[Aα]partαn

minus βn = 0 (434)

determines the Aia(τ) compatible with the state In this sense a solution

S[Aα] of the HJ equation (49) contains the solution of the Einstein equa-tions In what follows I focus on the particular solution of the HamiltonndashJacobi system provided by the Hamilton function


412 Hamilton function of GR and its physical meaning

A preferred solution of the HamiltonndashJacobi equation is the Hamiltonfunction S[A] This is defined as the value of the action of the regionR bounded by σ = partR computed on a solution of the field equationsdetermined by the boundary value A

A boundary value A on σ determines a solution (eIμ(x) Aiμ(x)) of the

Einstein equations in the region R In turn this solution induces on σ the3d field E[A] The Hamilton function satisfies

δS[A]δAi

a(τ)= Ea

i (τ)[A] (435)

Notice that Eai (τ)[A] is the value of Ea

i (τ) which is determined via theEinstein equations by the value of A on the entire surface σ Define thefunctional

F [AE](τ) =δS[A]δAi

a(τ)minus Ea

i (τ) (436)

then the equation

F [AE] = 0 (437)

is equivalent to the Einstein equations It expresses the conditions thatthe Einstein equations put on the possibility of having fields A and E ona 3d surface

This can be viewed as an evolution problem in the special case in whichwe take σ to be formed by two connected components σin and σoutbounding a single connected region R For instance σin and σout could betwo spacelike surfaces of a spatially closed universe In this case a solutionis determined by the components [Ain Aout] of the connection on σin andσout and also by the components [Ain Ein] of the connection and electricfield on σin alone We can write

F [Aout Ain Ein](τ) =δS[Ain Aout]δAin

ia(τ)

minus Einai (τ) = 0 (438)

Taking Aout as the unknown and Ain and Ein as data this equation givesthe general solution of the Einstein equations for fixed Ain and Ein itis solved by all 3d connections Aout on σ that are compatible with asolution bounded by a 3d surface with ldquoinitial conditionsrdquo Ain and EinThat is (438) determines all fields Aout that can ldquoevolverdquo from the ldquoinitialconditionsrdquo Ain and Ein Therefore the Hamilton function S[A] containsthe full solution of the Einstein equations S[A] expresses the full dynamicsof GR


As we shall see the full dynamics of quantum GR is contained in thecorresponding quantum propagator W [A] To the first relevant order in W [A] will be related to eminus

iS[A]

I have made no request that the 3d surface σ be spacelike In particular I haveavoided the issue of whether arbitrary boundary values A admit or determine an inter-polating solution In general the function S[A] will be defined only on a region andcan be multivalued in some other region These issues are important and I refer theinterested reader to [109] and references therein for literature on the topic On theother hand I think that the insistence on spacelike surfaces might be more tied to ourprerelativistic thinking habits than to their relations with Cauchy problems In viewof the construction of the quantum theory these problems can perhaps be postponedIf needed the requirement that the 3d surface is spacelike can be implemented as arestriction on the momentum E

Experiments Suppose we knew explicitly the Hamilton function S[A]How could we compare the theory with experience The answer is simpleWe should measure the 3d fields A and E on a closed 3-surface σ Thetheory predicts that the only fields (AE) we could measure are the onesthat satisfy (436) and (437) Therefore the theory determines which 3dfields could be measured and which could not In turn this determinesrestrictions (namely predictions) on any other quantity depending onthese fields Several important observations are in order

First the prediction is local in the sense that it regards a finite regionof spacetime Observables that require the full spacetime or the full spaceto be observed are not realistic

Second and most important where is the surface σ located Whichsurface σ should we consider The remarkable answer is it doesnrsquot matterThis is a key point in the interpretation of GR and should be understoodin detail

Consider a concrete experimental situation Consider for instance ascattering experiment in a particle accelerator or the propagation andreception of waves (electromagnetic or gravitational) In a nonrelativisticsituation say on Minkowski spacetime we can view the situation as fol-lows We have a certain number of objects and detectors located in certainknown positions of spacetime We measure the initial or incoming dataWe measure the final or outgoing data Furthermore we specify spatialboundary values (that forbid for instance spurious incoming radiation)The initial final and boundary values of the fields can be represented bythe value of the fields on a compact 3d surface σ These data howeverare not sufficient to make theoretical predictions we also need to knowthe location of σ in spacetime To fix the ideas say that σ is a cylinderin Minkowski space The height of the cylinder for instance is the timelapse between the beginning and the end of the experiment

42 Euclidean GR and real connection 153

Notice that the only relevant aspects of the location of objects appara-tus and detectors are their relative distances and time lapses Thereforethe only relevant aspect of the location of σ is the value of the metricon the surface and in its interior Indeed if we displace the surface (thatis the full experiment) in such a way that the geometry of the experimentremains the same we expect that the outcome will not change Since thegeometry of the interior is dictated (on Minkowski) by the geometry of σwe actually need to know only the geometry of σ It is this geometry thatdetermines the relative distances and time lapses between emissions anddetections Thus the full data that we need in a prerelativistic situationare

(i) the value of the dynamical fields on σ and(ii) the geometry of σ

Consider now the general-relativistic situation The same data as aboveare needed but now the geometry of σ is determined by the value ofthe dynamical fields on σ because the geometry is determined by thegravitational field Therefore the data that we need is

(i) the value of the dynamical fields on σ

and nothing elseThe ldquolocationrdquo of σ in the coordinate manifold is irrelevant because it

only reflects the arbitrary choice of coordinatization of spacetime In otherwords the distances and the time lapses among the detectors are preciselypart of the boundary data (AE) on σ For instance if σ is a cylinder thetime lapse between the initial and final measurement is precisely codedin the value of the gravitational field on the vertical (timelike) side ofthe cylinder Asking what happens after a longer time means nothing butasking what happens for larger values of E on the side of the cylinder

42 Euclidean GR and real connection

421 Euclidean GR

In this section I describe a field theory different from GR but which playsan important role in quantum gravity This is often called ldquoeuclideanGRrdquo Usual physical GR is then denoted ldquolorentzianrdquo to emphasize itsdistinction from euclidean GR Euclidean GR can be defined by the sameequations as GR for instance the action (213) with the only differencethat indices I J in the internal space are raised and lowered with theeuclidean metric δIJ instead of the Minkowski metric ηIJ Accordinglythe euclidean spin connection ω is an SO(4) connection instead of anSO(3 1) connection


It is still convenient to define the selfdual connection A as in (219) butthe appropriate selfdual projector P is now defined without the imaginaryfactor that is

Ai = ωi + ω0i (439)

Therefore the selfdual connection A is real in the euclidean case Theabsence of the imaginary factor gives immediately the Poisson brackets

Aia(τ) Eb

j (τprime) = (8πG) δbaδ

ijδ

3(τ τ prime) (440)

instead of (425)There is an important difference between the lorentzian and euclidean

cases In the euclidean case the connection lives in the so(4) algebraThis algebra decomposes as so(4) = so(3) oplus so(3) The real connection(439) is simply one of the two components Therefore (439) has half theinformation of ω

In the lorentzian case on the other hand the Lorentz algebra so(3 1)does not decompose at all However its complexification so(3 1C) de-composes as so(3 1C) = so(3C) oplus so(3C) A real ω determines twocomplex components which are complex conjugate to each other andeach component contains the same information as ω itself In this caseindeed the connection (219) has three complex components which isprecisely the same information as the six real components of ω

Remarkably the canonical formalism for the euclidean theory parallelscompletely the one for the lorentzian theory The theory is defined bythe same HamiltonndashJacobi equations (42)ndash(44) with the only differencethat (in the gauge (421)) the reality conditions (41) are replaced by

Eai minus Eai = 0 Aia minusAi

a = 0 (441)

The world is described by lorentzian GR not by euclidean GR Why then is euclideanGR useful at all Because euclidean GR plays a role in the search for a physical quantumtheory of gravity in several ways These will be discussed in more detail in the secondpart of the book but it is appropriate to anticipate some of these reasons here

First the key difficulty of quantum gravity is to understand how to formulate anontrivial generally covariant quantum field theory Euclidean GR is an example of anontrivial generally covariant field theory which is simpler than lorentzian GR becausethe reality conditions are simpler Therefore a complete and consistent formulation ofeuclidean quantum GR is not yet a quantum theory of gravity but is probably a majorstep in that direction Euclidean GR is a highly nontrivial model of the true theory

Second it is well known that the euclidean version of flat-space quantum field the-ories is strictly connected to the physical lorentzian version Under wide assumptionsone can prove that physical n-point functions are analytical continuations of the onesof the euclidean theory Naively one can simply Wick-rotate the time coordinate in theimaginary plane More precisely solid theorems from axiomatic quantum field theory


assure us that Wightman distributions are indeed the analytic continuation of the mo-ments of an euclidean process (the Schwinger functions) under very general hypothe-ses Defining the euclidean quantum field theory is therefore equivalent to defining thephysical theory In fact calculations are routinely performed in the euclidean region instandard quantum field theory We cannot assume naively that the same remains truein quantum gravity There is no Wick rotation to consider (recall the coordinate t isirrelevant for observable amplitudes anyway) and we are outside the hypotheses ofthe axiomatic approach Therefore we cannot as we do on flat space content ourself todefine the euclidean quantum field theory and lazily be sure that a consistent physicaltheory will follow

Still the very strict connection between the euclidean and the lorentzian theorythat exists on flat space strongly suggests that some connection between euclideanand lorentzian quantum GR is likely to exist Stephen Hawking in particular has ex-plored the hypothesis that physical quantum gravity could be directly defined in termsof the quantization of the euclidean theory There are various indications for that Firstthe formal functional path integral of the euclidean theory solves the WheelerndashDeWittequation for the lorentzian theory as well Second there is a standard technique forobtaining the vacuum of a quantum theory by propagating for an infinite euclideantime the adaptation of this idea to gravity led Jim Hartle and Stephen Hawking to theidea that a quantum gravitational ldquovacuumrdquo is obtained from propagation in imaginarytime or equivalently from the quantum euclidean theory

Finally as I show in the next section the lorentzian theory admits a formulationthat has the same kinematics as the euclidean theory It is therefore reasonable toexpect that the kinematical features of the two theories are the same and thereforekinematical aspects of the physical theory can be studied in the euclidean context

422 Lorentzian GR with a real connection

Let us return to lorentzian GR In this context define the quantity

Ai = ωi + ω0i (442)

precisely as in (439) This quantity does not transform as a connectionunder a local Lorentz transformation (as it does in the euclidean case)but it is still a well-defined field If we fix the gauge (421) then thereduced local internal gauge invariance is SO(3) and A defined in (442)transforms as a connection under SO(3) transformations For this reasonit is denoted the ldquoreal connectionrdquo of the lorentzian theory

Remarkably we can take the real connection or more precisely itsthree-dimensional restriction to the boundary surface as a canonical co-ordinate Lorentzian GR in other words can be expressed in terms of areal SO(3) connection The reality conditions are trivial The only differ-ence with respect to the euclidean theory is the form of the hamiltonianwhich acquires another more complicated term with respect to (44)

H = (F ijab + 2Ki

[aKjb]) Ea

i Ebj (443)


where Kia = Ai

a minus Γia[E] (See for instance [20]) The connection (442)

and the hamiltonian (443) provide a second hamiltonian formalism forGR alternative to the one described at the beginning of this chapter

423 Barbero connection and Immirzi parameter

Finally there is a third possible formalism for lorentzian GR It consistsin using the connection

Ai = ωi + γω0i (444)

where γ is an arbitrary complex parameter This is called the Barberoconnection it derives naturally from the use of the Holst action (see Sec211) The case γ = i gives the selfdual connection When γ is real it iscalled the Immirzi parameter In this case the reality conditions are stilltrivial (that is A = A) and the hamiltonian is a small modification of(443) (see [20])

H = (F ijab + (γ2 + 1)Ki

[aKjb]) Ea

i Ebj (445)

We will use this formalism in the quantum theorySince γ scales the term ω0i which is the one that has nonvanishing

Poisson brackets with E it is easy to see that the Poisson brackets betweenthe Barbero connection and the electric field are

Aia(x) Eb

j (y) = (8πγG) δbaδijδ

3(x y) (446)

The fact that γ can be arbitrary is important because as we shall seethe quantum theories obtained starting with different values of γ leadto different physical predictions That is in pure gravity γ has no effectin the classical theory but has an effect in the quantum theory (In thepresence of minimally coupled fermions γ appears in the equation ofmotion [128]) Presumably the presence of this parameter reflects a one-parameter quantization ambiguity of the theory γ is a parameter of thequantum theory that is absent in the classical theory such as for instancethe θ parameter of the QCD θ-vacua In fact γ can also be introduced asthe constant in front of a topological term added to the action precisely asthe θ parameter in QCD Such terms do not affect the classical equationsof motion but affect the quantum theory

As we shall see in Chapter 8 γ enters in several key predictions of thequantum theory In particular it enters in the computation of the black-hole entropy Comparing the black-hole entropy with the one determinedthermodynamically then determines γ A calculation along these linessketched in Chapter 8 suggests the value

γ asymp 02375 (447)

43 Hamiltonian GR 157

It has also been repeatedly suggested that γ may determine the relationbetween the bare and renormalized Newton constant Nevertheless thephysical interpretation of this parameter is not yet clear

43 Hamiltonian GR

I give here a formulation of canonical GR on a finite-dimensional config-uration space along the lines described in Section 332

431 Version 1 real SO(3 1) connection

Let T be the space on which the fields e and ω take value This is a (16 +24)-dimensional space with coordinates (eIμ ω

IJμ ) Let Σ = M times T be the

(4 + 16 + 24)-dimensional space with coordinates (xμ eIμ ωIJμ ) Consider

the four-form

θ = εIJKL eIμ eJν DωKLρ and dxμ and dxν and dxρ (448)

defined on this space Here the covariant differential D is defined by

DωKLρ = dωKL

ρ + ωKσI ω

ILρ dxσ (449)

This structure defines GR as follows Consider a four-dimensional surfaceγ in Σ Recall from Section 332 that we say that γ is an orbit of ω if thequadritangent X to the orbit is in the kernel of the five-form ω = dθ

dθ(X) = 0 (450)

The orbits of ω are the solutions of the Einstein equations If we use thex as coordinates on the γ then γ is represented by

γ = (xμ eIμ(x) ωIJμ (x)) (451)

If γ is an orbit of ω then the functions eIμ(x) ωIJμ (x) solve the Einstein

equations The demonstration is a straightforward calculation along thelines sketched for the scalar field example in Section 332

432 Version 2 complex SO(3) connection

Consider the space Σ with coordinates (xμ Aiμ e

Iμ) where Ai

μ is complexand eIμ is real Define the gauge-covariant differential acting on all quan-tities with internal i indices as

Dvi = dvi + εijkAjμv

kdxμ (452)


andDAi

μ = dAiμ + εijkA

jνA

kμdxν (453)

GR is defined by the four-form

θ = PIJi eI and eJ and DAi (454)

where P iIJ is the selfdual projector defined in (217) Indeed the orbits

(xμ Aiμ(xμ) eIμ(xμ)) of ω = dθ satisfy the Einstein equations in the form

eI and (deJ + PJKi Ai and eK) = 0 (455)

PIJi and eI and F i = 0 (456)

where F i is the curvature of Ai The calculation is straightforward

433 Configuration space and hamiltonian

Above I have defined canonical GR directly as a presymplectic (Σ θ)system This form can be derived from a configuration space and a hamil-tonian namely from the (C H) formalism described in Section 332 asfollows

Consider the finite-dimensional space C with coordinates (xμ Aiμ) Here

Aiμ is a complex matrix Assuming immediately (3146) the corresponding

space Ω has coordinates (xμ Aiμ π p

μνi ) and carries the canonical four-

formθ = πd4x + pμνi dAi

ν and d3xμ (457)

Using D the canonical form (457) reads

θ = pd4x + pμνi DAiμ and d3xν (458)

where p = π minus pμνi AjνAk

μεijk Also define

Eiμν = εμνρσ δijpρσj (459)

and the forms Ai = AiμdxμDAi = dAi

μ and dxμ + AjνAk

μεijkdx

ν and dxν Ei =Ei

μνdxμ and dxν and so on on Ω

GR is defined by the hamiltonian system

p = 0 (460)

pμνi + pνμi = 0 (461)

Ei and Ej = 0 (462)

(δikδjl minus13δijδkl)Ei and Ej = 0 (463)


The key point is that the constraints (462) (463) imply that thereexists a real four by four matrix eIμ where I = 0 1 2 3 such that Ei

μν isthe selfdual part of eIμe

Jν In fact it is easy to check that (462) and (463)

are solved by

Ei = P iIJ eI and eJ (464)

and the counting of degrees of freedom indicates that this is the uniquesolution Therefore we can use the coordinates (xμ Ai

μ eIμ) on the con-

straint surface Σ (where Aiμ is complex and eIμ is real) and the induced

canonical four-form is (454) Thus we recover the above (Σ θ) structure

434 Derivation of the HamiltonndashJacobi formalism

Let α be a three-dimensional surface in C Thus α = [xμ(τ) Aiμ(τ)] where

τ = (τ1 τ2 τ3) = (τa) Define the functional

S[α] =int

γθ (465)

as in (3167) That is γ is the four-dimensional surface in Σ which is anorbit of dθ and therefore a solution of the field equations and is suchthat the projection of its boundary to C is α From the definition (454)

δS[α]δAi

μ(τ)= PiIJ εμνρσeJρ (τ)eIσ(τ)nν(τ) (466)

where nν is defined in (3180) Since from this equation we have immedi-ately

nμ(τ)δS[α]δAi

μ(τ)= 0 (467)

it follows that the dependence of S[α] on Aiμ(τ) is only through the restric-

tion of Ai(τ) to the 3-surface αM that is only through the components

Aia(τ) = partax

μ(τ)Aiμ(τ) (468)

Thus S[α] = S[xμ(τ) Aia(τ)] and

δS[α]δAi

a(τ)= PiJK εaνbc partbx

ρ(τ)partcxσ(τ)eJρ (τ)eKσ (τ)nν(τ) equiv Eai (τ) (469)

Therefore Eai is the conjugate momentum to the connection Ai

a Noticethat Ea

i is the dual of the restriction to the boundary surface σ of thePlebanski two-form Σi = Σi

μνdxμ and dxν defined in (223) Assume for


simplicity that the boundary surface is given by x0 = 0 and coordinatizedby x(τ) = τ and that we have chosen the gauge e0

b(τ) = 0 Then nμ =(1 0 0 0) and

Eai = εabc Σibc (470)

Its real part is the densitized inverse triad

ReEai = minusεijk εabc ejbe

kc = det(e) eai (471)

where eai is the matrix inverse to the ldquotriadrdquo one-form eia Its imaginarypart is

ImEai = εabc eibe

0c (472)

The projection of the field equations (456) on σ written in terms ofEa

i read DaEai = 0 F i

abEai = 0 and F i

abEaiEbkεijk = 0 where Da and F i

abare the covariant derivative and the curvature of Ai

a Using (469) thesegive the three HamiltonndashJacobi equations of GR

DaδS[α]δAi

a(τ)= 0 (473)

δS[α]δAi

a(τ)F iab = 0 (474)

F ijab(τ)

δS[α]δAi

a(τ)δS[α]

δAjb(τ)

= 0 (475)

Kinematical gauges Equation (473) could have been obtained by simplyobserving that S[α] is invariant under local SU(2) gauge transformationson the 3-surface Under one such transformation generated by a functionf i(τ) the variation of the connection is δfAi

a = Dafi Therefore S satisfies

0 = δfS =int

d3τ δfAia(τ)

δS[α]δAi

a(τ)=

intd3τ Daf

i(τ)δS[α]δAi

a(τ)

= minusint

d3τ f i(τ) DaδS[α]δAi

a(τ) (476)

This gives (473) Next the action is invariant under a change of coordi-nates on the 3-surface αM Under one such transformation generated by afunction fa(τ) the variation of the connection is δfAi

a = f bpartbAia+Ai

bpartafb

Integrating by parts as in (476) this gives

partbAia

δS[α]δAi

a(τ)+ (partbAi

a)δS[α]δAi

a(τ)= 0 (477)

which combined with (473) gives (474) Thus (473) and (474) aresimply the requirement that S[α] is invariant under internal gauge andchanges of coordinates on the 3-surface The three equations (473) (474)and (475) govern the dependence of S on Ai

a(τ)


Dropping the coordinates It is easy to see that S is independent fromxμ(τ) A change of coordinates xμ(τ) tangential to the surface cannotaffect the action which is independent of the coordinates used Moreformally the invariance under change of parameter τ implies

δS[α]δxμ(τ)

partjxμ(τ) =

δS[α]δAi

a(τ)δjA

ia(τ) (478)

and we have already seen that the right-hand side vanishes The variationof S under a change of xμ(τ) normal to the surface is governed by theHamiltonndashJacobi equation proper equation (3186) In the present casefollowing the same steps as for the scalar field we obtain

δS[α]δxμ(τ)

nμ(τ) + εijkFiab

δS[α]

δAja(τ)

δS[α]δAk

b (τ)= 0 (479)

But the second term vanishes because of (475) Therefore S[α] is inde-pendent of tangential as well as normal parts of xμ(τ) S depends onlyon [Ai

a(τ)]We can thus drop altogether the spacetime coordinates xμ from the

extended configuration space Define a smaller extended configurationspace C as the 9d complex space of the variables Ai

a Geometrically thiscan be viewed as the space of the linear mappings A D rarr sl(2C) whereD = R3 is a ldquospace of directionsrdquo and we have chosen the complex selfdualbasis in the sl(2C) algebra We then identify the space G as a space ofparametrized 3d surfaces A with components [Ai

a(τ)] and without bound-aries in C GR is defined on this space by the HamiltonndashJacobi system

DaδS[A]δAi

a(τ)= 0 (480)

δS[A]δAi

a(τ)F iab = 0 (481)

F ijab(τ)

δS[A]δAi

a(τ)δS[α]

δAjb(τ)

= 0 (482)

These are the equations presented at the beginning of this chapter onwhich we will base quantum gravity

Equivalently we can solve immediately (480) and (481) by definingthe space G0 of the equivalence classes of 3d SU(2) connections A undergauge and 3d diffeomorphisms (Aa

i(τ) = partτ primebpartτa A

primebi(τ prime(τ))) transformation

Then GR is defined by the sole equation (482) on this space (wherefunctions S[Ai

a(τ)] overcoordinatize G0) Accordingly we can interpretGR as the dynamical system defined by the extended configuration space


G0 and the relativistic hamiltonian

H(τ) = F ijab(τ) Ea

i (τ) Ebj (τ) (483)

435 Reality conditions

The two variables on which we have based the canonical formulation ofGR described above are a complex 3d connection Ai

a and its complex con-jugate momentum Eai They have 9 complex components each On theother hand the degrees of freedom of GR have (9 + 9) real componentsof which (2 + 2) are physical degrees of freedom 7 are constrained and 7are gauges The explanation of the apparent doubling of the componentsis that A and E are like the coordinates z = x + ip and z = x minus ip overthe phase space of a one-dimensional system That is they are not inde-pendent of each other

To find out these relations let us write the real and imaginary parts ofA and E From their definition we have

ReAia = ωi

a (484)ImAi

a = ω0ia (485)

ReEai = det(e) eai (486)

ImEai = εabc e0

b eic (487)

We have chosen a gauge in which e0a = 0 Then (487) implies that E

is real Recall that the tetrad and the connection ω are related by theequation deI = ωI

J and eJ Projecting this equation on the 3-surface weobtain

dei = ωij and ej + ωi

0 and e0 (488)

In the gauge chosen the last term vanishes and ωij is the spin connection

of the triad ei Hence (484) implies that the real part of A satisfies (41)Without fixing the gauge e0

a = 0 the reality conditions are a bit morecumbersome

mdashmdash


The hamiltonian formulation of GR was developed independently byPeter Bergmann and his group [129] and by Dirac [130] The long-termgoal of both was quantum gravity The main tool for this the hamiltoniantheory of constrained systems was developed for this purpose The greatalgebraic complexity of the hamiltonian formalism was dramatically re-duced by the introduction of the ADM variables by Arnowitt Deser and


Misner [131] and then by the selfdual connection variables systematizedby Ashtekar [132]

The conventional derivation of the fundamental equations (42ndash44)from the lagrangian formalism can be found in many books and arti-cles see for instance I [2 9 20 126] See also the original articles [132]The expression (416) of the hamiltonian which plays an important rolein the quantum theory was introduced by Thomas Thiemann [133] Theusefulness of the Barbero connection was pointed out in [134] on itsgeometrical interpretation see [136] The importance of the Immirzi pa-rameter for the quantum theory in [135] An (inconclusive) discussion onthe Immirzi parameter and its physical interpretation is in [137]

For the finite-dimensional formulation I have followed here [138 139]On other versions of this formalism see [140] For the covariant HamiltonndashJacobi formalism for GR see also [141]

5Quantum mechanics

Quantum mechanics (QM) is not just a theory of micro-objects it is our currentfundamental theory of motion It expresses a deeper understanding of Nature thanclassical mechanics Precisely as classical mechanics the conventional formulationof QM describes evolution of states and observables in time Precisely as classicalmechanics this is not sufficient to deal with general relativistic systems because thesesystems do not describe evolution in time they describe correlations between observ-ables Therefore a formulation of QM slightly more general than the conventional onendash or a quantum version of the relativistic classical mechanics discussed in the previouschapter ndash is needed In this chapter I discuss the possibility of such a formulationIn the last section I discuss the general physical interpretation of QM

QM can be formulated in a number of more or less equivalent formalisms canonical(Hilbert spaces and self-adjoint operators) covariant (Feynmanrsquos sum-over-histories)algebraic (states as linear functionals over an abstract algebra of observables) andothers Generally but not always we are able to translate these formalisms into oneanother but often what is easy in one formulation is difficult in another A general-relativistic sum-over-histories formalism has been developed by Jim Hartle [26] HereI focus on the canonical formalism because the canonical formalism has provided themathematical completeness and precision needed to explicitly construct the mathemat-ical apparatus of quantum gravity Later I will consider alternative formalisms

51 Nonrelativistic QM

Conventional QM can be formulated as follows

States The states of a system are represented by vectors ψ in a complexseparable Hilbert space H0

Observables Each observable quantity A is represented by a self-adjointoperator A on H0 The possible values that A can take are thenumbers in the spectrum of A

164

51 Nonrelativistic QM 165

Probability The average of the values that A takes over many equal statesrepresented by ψ is a = 〈ψ|A|ψ〉〈ψ|ψ〉

Projection If the observable A takes values in the spectral interval Ithe state ψ becomes then the state PIψ where PI is the spectralprojector on the interval I

Evolution States evolve in time according to the Schrodinger equation

iparttψ(t) = H0ψ(t) (51)

where H0 is the hamiltonian operator corresponding to the energyEquivalently states do not evolve in time but observables do andtheir evolution is governed by the Heisenberg equation

ddt

A(t) = minus i

[A(t) H0] (52)

A given quantum system is defined by a family (generally an algebra) ofoperators Ai including H0 defined over an Hilbert space H0

This scheme for describing Nature differs substantially from the newto-nian one Here are the main features of the physical content of the abovescheme

Probability Predictions are only probabilistic

Quantization Some physical quantities can take certain discrete valuesonly (are ldquoquantizedrdquo)

Superposition principle If a system can be in a state A where a physicalquantity q has value a as well as in state B where q has value bthen the system can also be in states (denoted ψ = caA+ cbB with|ca|2 + |cb|2 = 1) where q has value a with probability |ca|2 andvalue b with probability |cb|2

Uncertainty principle There are couples of (conjugate) variables thatcannot have determined values at the same time

Effect of observations on predictions The properties we expect the sys-tem to have at some time t2 are determined not only by the proper-ties we know the system had at time t0 but also by the propertieswe know the system has at the time t1 where t0 lt t1 lt t21

1Bohr expressed this fact by saying that observation affects the observed system Butformulations such as Bohmrsquos or consistent histories force us to express this physicalfact using more careful wording

166 Quantum mechanics

In Section 56 I discuss the physical content of QM in more depthIn general a quantum system (H0 Ai H0) has a classical limit which

is a mechanical system describing the results of observations made onthe system at scales and with accuracy larger than the Planck constantIn the classical limit Heisenberg uncertainty can be neglected and theobservables Ai can be taken as coordinates of a commutative phase spaceΓ0 Quantum commutators define classical Poisson brackets and (52)reduces to Hamilton equation (378)

If the classical limit is known the search for a quantum system fromwhich this limit may derive is called the quantization problem There isno reason for the quantization problem to have a unique solution Theexistence of distinct solutions is denoted ldquoquantization ambiguityrdquo Ex-perience shows that the simplest quantization of a given classical systemis very often the physically correct one If we are given a classical systemdefined by a nonrelativistic configuration space C0 with coordinates qi andby a nonrelativistic hamiltonian H0(qi pi) then a solution of the quan-tization problem can be obtained by interpreting the HamiltonndashJacobiequation (317) as the eikonal approximation of the wave function (51)that governs the quantum dynamics [142] This can be achieved by defin-ing multiplicative operators qi derivative operators pi = minusi part

partqiand the

hamiltonian operator

H0 = H0

(qiminusi

part

partqi

)(53)

on the Hilbert space H0 = L2[C0] the space of the square integrablefunctions on the nonrelativistic configuration space [143]

In a special-relativistic context this structure remains the same but theEvolution postulate above is extended to the requirement that H0 carriesa unitary representation of the Poincare group and H0 is the generator ofthe time translations of this representation

This structure is not generally relativistic In particular the notions ofldquostaterdquo and ldquoobservablerdquo used above are the nonrelativistic ones Can thestructure of QM be extended to the relativistic framework In Section52 I discuss such an extension As a preliminary step however in therest of this section I introduce and illustrate some tools needed for thisreformulation in the context of a very simple system ndash as I did for classicalmechanics

511 Propagator and spacetime states

Nonrelativistic formulation The quantum theory of the pendulum canbe written on the Hilbert space H0 = L2[R] of wave functions ψ0(α) in


terms of the multiplicative position operator α the momentum operatorpα = minusi part

partα and the hamiltonian

H0 = minus 2

2mpart2

partα2+

mω2

2α2 (54)

More precisely the theory is defined on a rigged Hilbert space or Gelfand triple AGelfand triple S sub H sub S prime is formed by a Hilbert space H a proper subset S densein H and equipped with a weak topology and the dual S prime of S with their naturalidentifications A manifold M with a measure dx determines a rigged Hilbert spaceSM sub HM sub S prime

M where SM is the space of smooth functions on M with fast decrease(Schwarz space) HM = L2[M dx] and S prime

M is the space of the tempered distributionson M This setting allows us in particular to deal with eigenstates of observables withcontinuous spectrum and Fourier transforms

The operators (here h = 1)

α(t) = eitH0αeminusitH0 (55)

which solve (52) are the Heisenberg position operators that give the posi-tion at any time t Denote |α t〉 the generalized eigenstate of the operatorα(t) with eigenvalue α (which are in S prime)

α(t)|α t〉 = α|α t〉 (56)

and |α〉 = |α 0〉 Clearly |α t〉 = eitH0 |α〉 Given a state |ψ〉 theSchrodinger wave function

ψ(α t) = 〈α t|ψ〉 = 〈α|eminusitH0 |ψ〉 (57)

satisfies the Schrodinger equation (51) Conversely each solution of theSchrodinger equation restricted to t = 0 defines a state in H0 Thereforethere is a one-to-one correspondence between states at fixed time ψ0(α)and solutions of the Schrodinger equation ψ(α t) I call H the space ofthe solutions of the Schrodinger equation Thanks to the identificationjust mentioned H is a Hilbert space isomorphic to the Hilbert space H0

of the states at fixed time I call

R0 H rarr H0 (58)ψ(α t) rarr ψ0(α) = ψ(α 0) (59)

the identification map The relation between H and H0 is analogous tothe relation between the spaces Γ and Γ0 in classical mechanics discussedin Chapter 3

The propagator is defined as

W (α t αprime tprime) = 〈α t|αprime tprime〉 = 〈α|eminusi(tminustprime)H0 |αprime〉=

sum

n

Hn(α) eminusiEn(tminustprime) Hn(αprime) (510)


where Hn(α) is the eigenfunction of H0 with eigenvalue En Explicitly astraightforward calculation that can be found in many books gives

W (α t αprime tprime) =radic

mω

ih sin[ω(tminus tprime)]e

iωm2h

[(α2+αprime2) cos[ω(tminustprime)]minus2ααprime

sin2[ω(tminustprime)]

]

(511)

where h = 2πh The propagator satisfies the Schrodinger equation in thevariables (α t) (and the conjugate equation in the variables (αprime tprime))

Spacetime states It is convenient to consider the following states Givenany compact support complex function f(α t) the state

|f〉 =int

dα dt f(α t) |α t〉 (512)

is in H0 and is called the ldquospacetime smeared staterdquo or simply the ldquospace-time staterdquo of the function f(α t) Since standard normalizable statesare dimensionless (for 〈ψ|ψ〉 = 1 to make sense) and the states |α t〉have dimension Lminus12 the function f must have dimensions Tminus1Lminus12These states generalize the conventional wave packets for which f(α t) =f(α)δ(t) Conventional wave packets can be thought of as being associatedwith results of instantaneous position measurements with finite resolutionin space as I will illustrate later on spacetime states can be associatedwith realistic measurements where the measuring apparatus has finiteresolution in space as well as in time The Schrodinger wave function of|f〉 is

ψf (α t) = 〈α t|f〉

= 〈α t|int

dαprimedtprime f(αprime tprime) |αprime tprime〉

=int

dαprimedtprime W (α t αprime tprime) f(αprime tprime) (513)

and satisfies the Schrodinger equation The scalar product of two space-time states is

〈f |f prime〉 =int

dα dt dαprimedtprime f(α t) W (α t αprime tprime) f prime(αprime tprime) (514)

In particular we can associate a normalized state |R〉 to each spacetimeregion R

|R〉 = CR

int

Rdα dt |α t〉 (515)


where the factor

Cminus2R =

intdα dt dαprimedtprime W (α t αprime tprime) (516)

fixes the normalization 〈R|R〉 = 1 as well as giving the state the rightdimensions

512 Kinematical state space K and ldquoprojectorrdquo P

As discussed in Chapter 3 the kinematics of a pendulum is describedby two partial observables time t and elongation α These coordinatizethe relativistic configuration space C The classical relativistic formalismtreats α and t on an equal footing The quantum relativistic formalismas well treats α and t on an equal footing and therefore it is based onfunctions f(α t) on C

To be precise let S sub K sub S prime be the Gelfand triple defined by C and the measuredαdt That is S is the space of the smooth functions f(α t) on C with fast decreaseK = L2[C dαdt] and S prime is formed by the tempered distributions over C

I call S the ldquokinematical state spacerdquo and its elements f(α t) ldquokine-matical statesrdquo

In the relativistic formalism the dynamics of the system is defined bythe relativistic hamiltonian H(α t p pt) given in (324) The quantumdynamics is defined by the ldquoWheelerndashDeWittrdquo (WdW) equation

H ψ(α t) = 0 (517)

where

H = H

(α tminusih

part

partαminusih

part

partt

)

= minusihpart

partt+ H0

= minusihpart

parttminus h2

2mpart2

partα2+

mω2

2α2 (518)

and H0 is given in (54) In the case of the pendulum (517) reducesto the Schrodinger equation (51) but (517) is more general than theSchrodinger equation because in general H does not have the nonrela-tivistic form H = pt +H0 Solutions ψ(α t) of this equation form a linearspace H which carries a natural scalar product that I will construct in amoment The key object for the relativistic quantum theory is the oper-ator

P =int

dτ eminusiτH (519)


defined on S prime This operator maps arbitrary functions f(α t) into solu-tions of the WdW equation (517) namely into H

To see this expand a function f(α t) as

f(α t) =sum

n

intdE fn(E) Hn(α) eminusiEt (520)

Acting with P on this function we obtain

[Pf ](α t) =

intdτ eminusiτH

sum

n

intdE fn(E) Hn(α) eminusiEt

=

intdτ

sum

n

intdE eminusiτ(minusE+En) fn(E) Hn(α) eminusiEt

=sum

n

intdE δ(E minus En) fn(E) Hn(α) eminusiEt

=sum

n

ψn Hn(α) eminusiEnt (521)

where ψn = fn(En) which is the general solution of (517) Therefore P sends arbitraryfunctions into solutions of the WdW equation Intuitively P sim δ(H)

The integral kernel of P is the propagator (510) Indeed the inverseof (520) gives

ψn = fn(En) =int

dαdt Hn(α) eiEnt f(α t) (522)

Inserting this in (521) we have

[Pf ](α t) =sum

n

intdαprimedtprime Hn(αprime) eiEntprimeHn(α) eminusiEnt f(αprime tprime)

=int


P is often called ldquothe projectorrdquo although improperly so Intuitively it ldquoprojectsrdquoon the space of the solutions of the WdW equation In some systems (when 0 is aneigenvalue in the discrete spectrum of H) P is indeed a projector But generically andin particular for the nonrelativistic systems (where 0 is in the continuum spectrum ofH) P is not a projector because its domain is smaller than the full S prime In particularit does not contain the solutions of the WdW equation namely P rsquos codomain Thedomain of P contains on the other hand S

The matrix elements of P

〈f |P |f prime〉K =int



define a degenerate inner product in S Dividing S by the kernel of thisinner product that is identifying f and f prime if Pf = Pf prime and completingin norm we obtain a Hilbert space But if Pf = Pf prime then f and f prime definethe same solution of the WdW equation In fact they define the solutionthat corresponds to the spacetime state |f〉 defined above Therefore anelement of this Hilbert space corresponds to a solution of the WdW equa-tion the Hilbert space can be identified with the space of the solutions ofthe WdW equation H Therefore

P S rarr Hf rarr |f〉 (525)

It follows that P directly equips the space H of the solutions with aHilbert space structure if ψ = Pf and ψprime = Pf prime are two solutions of theWdW equation (517) their scalar product is defined by

〈ψ|ψprime〉 equiv 〈f |P |f prime〉K (526)

where the right-hand side is the scalar product in K and is explicitlygiven in (524)

Notice that the scalar product on the space of the solutions of the WdWequation can be defined just by using the relativistic operator P withoutany need of picking out t as a preferred variable

For all nonrelativistic systems the configuration space has the structureC = C0timesR where t isin R and a function ψ(α t) in H is uniquely determinedby its restriction ψt = Rtψ on C0 for a fixed t

ψt(α) equiv ψ(α t) (527)

For each t denote Ht the space of the L2[C0] functions ψt(α) so thatRt H rarr Ht The spaces H and Ht are in one-to-one correspondence theinverse map Rminus1

t is the evolution determined by equation (517) In par-ticular H0 is the Hilbert space used in the nonrelativistic formulation ofthe quantum theory Under the identification between H and H0 given byR0 the scalar product defined above is precisely the usual scalar productof the nonrelativistic Hilbert space

This can be directly seen by noticing that the right-hand side of (524) is precisely(514) More explicitly let ψ(α t) =

sumn ψnHn(α)eminusiEnt be a function in H namely a

solution of the WdW equation Its restriction to t = 0 is ψ0(α) =sum

n ψnHn(α) and itsnorm in H0 is ||ψ0||2 =

intdα |ψ0(α)|2 =

sumn |ψn|2 A function f such that Pf = ψ is

for instance simply f(α t) = ψ0(α)δ(t) =sum

n ψnHn(α)int

dEeminusiEt (This is actuallynot in S0 but we could take a sequence of functions in S0 converging to f But f isin the domain of P and such a procedure would not give anything new) The norm


of ψ is

||ψ||2 = 〈f |f〉H = 〈f |P |f〉K

=

intdτ

intdα

intdt f(α t) eminusiτH f(α t)

=

intdτ

intdα

intdt

sum

n

ψn Hn(α)

intdEeiEteminusiτH

sum

m

ψm Hm(α)

intdEprimeeminusiEprimet

=sum

n

intdτ

intdt

intdE

intdEprime |ψn|2eiEteminusiτ(EprimeminusEn) eminusiEprimet

=sum

n

|ψn|2 = ||ψ0||2 (528)

513 Partial observables and probabilities

Consider two events (α t) and (αprime tprime) in the extended configuration spaceSuppose we have observed the event (αprime tprime) What is the probability ofobserving the event (α t)

To measure this probability we need measuring apparata for α and fort In general these apparata will have a certain resolution say Δα andΔt The proper question is therefore what is the probability of observingan event included in the region R = (αplusmn Δα tplusmn Δt) It is important toremark that no realistic measuring device or detector can have Δα = 0nor Δt = 0 Most QM textbooks put much emphasis on the fact thatΔα gt 0 and completely ignore the fact that Δt gt 0 Consider thus tworegions R and Rprime If a detector at Rprime has detected the pendulum what isthe probability PRRprime that a detector at R detects the pendulum

If the regions R and Rprime are much smaller than any other physical quan-tity in the problem including the spatial and temporal separation of Rand Rprime a direct application of perturbation theory shows that

PRRprime = γ2 |〈R|Rprime〉|2 (529)

where γ2 is a dimensionless constant related to the efficiency of the de-tector (We may assume that a ldquoperfectrdquo detector is defined by γ = 1)The reader can repeat the calculation himself or find it for instance in[144] Explicitly we can write this probability as the modulus square ofthe amplitude PRRprime = |ARRprime |2

ARRprime = γ〈R|Rprime〉

radic〈R|R〉

radic〈Rprime|Rprime〉

(530)

〈R|Rprime〉 =int

Rdαdt

int

Rprimedαprimedtprime W (α t αprime tprime) (531)

Therefore the propagator has all the information about transition proba-bilities


Assume that R is sufficiently small so that the wave function ψ(α t) =〈α t|Rprime〉 is constant within R and has the value ψ(α t) Then we can writethe probability of the pendulum being detected in R as

PR = γ (VRCR)2 |ψ(α t)|2 (532)

where VR is the volume of the region R Now assume the region R hassides ΔαΔt A direct calculation (see [144]) shows that if Δt mΔα2hthen (VRCR)2 is proportional to Δα therefore

PR sim Δα |ψ(α t)|2 (533)

So for small regions we have the two important results that (i) the tem-poral resolution of the detector drops out from the detection probabilityand (ii) the probability is proportional to the spacial resolution of thedetector Because of (i) we can forget the temporal resolution of the de-tector and take the idealized limit of an instantaneous detector Becauseof (ii) we can associate a probability density in α to each infinitesimalinterval dα in α Fixing the overall normalization by requiring that anidealized perfect detector covering all values of α detects with certaintythis yields the results that |ψ(α t)|2 is the probability density in α todetect the system at (α t) with an instantaneous detector That is werecover the conventional probabilistic interpretation of the wave functionfrom (529)

In the opposite limit when Δt mΔα2h (VRCR)2 is proportionalto (Δt)minus12 Therefore

PR sim (Δt)minus12 |ψ(α t)|2 (534)

and we cannot associate a probability density in t with this detectorbecause the detection probability does not scale linearly with Δt Thedifferent behavior of the probability in α and t is a consequence of thespecific form of the dynamics

Partial observables in quantum theory Recall that α and t are partial ob-servables They determine commuting self-adjoint operators in K Theseact simply by multiplication Their common generalized eigenstates |α t〉are in S The states |α t〉 satisfy

〈α t|P |αprime tprime〉 = W (α t αprime tprime) (535)

We can view the states |α t〉 as ldquokinematical statesrdquo that do not know any-thing about dynamics They correspond to a single quantum event Theldquokinematicalrdquo scalar product of these states in K given below in (536)expresses only their independence while the ldquophysicalrdquo scalar product of


these states in H given in (535) expresses the physical relation betweenthe two events it determines the probability that one event happens giventhat the other happened

Do not confuse |α t〉 with |α t〉 The first is an eigenstate of α and t the secondis an eigenstate of α(t) They both determine (generalized) functions on C The state|α t〉 determines a delta distribution at the point (α t)

〈αprime tprime|α t〉 = δ(αprime α)δ(tprime t) (536)

while the state |α t〉 determines a solution of the Schrodinger equation This solutionhas support all over C and is such that on the line t = constant it is a delta functionin α

〈αprime t|α t〉 = δ(αprime α) (537)

while for different trsquos〈α t|αprime tprime〉 = W (α t αprime tprime) (538)

The relation between the two is simply

|α t〉 = P |α t〉 (539)

Notice that (538) and (539) give

W (α t αprime tprime) = 〈α t|P daggerP |α t〉H (540)

which is consistent with (535) because the definition of the scalar product in H (indi-

cated in (540) by 〈middot|middot〉H) is (526)

514 Boundary state space K and covariant vacuum |0〉In this subsection I introduce some notions that play an important role inthe field theoretical context Fix two times t = 0 and t Let H0 = L2[Rdα]be the space of the instantaneous quantum states ψ0 at t = 0 Let Ht sim H0

be the space of the instantaneous states ψt at t The probability amplitudeof measuring a state ψt at t if the state ψ0 was measured at t = 0 is

A = 〈ψt|eminusiH0t|ψ0〉 (541)

Consider the boundary state space

Kt = Hlowastt otimesH0 = L2[R2 dαdαprime] (542)

The linear functional ρt defined by

ρt(ψt otimes ψ0) = 〈ψt|eminusiH0t|ψ0〉 (543)

is well defined on Kt This functional captures the entire dynamical infor-mation about the system A linear functional on a Hilbert space definesa state I denote |0t〉 the state defined by ρt

ρt(ψ) = 〈0t|ψ〉Kt (544)

and call it the ldquodynamical vacuumrdquo state in boundary state space Kt


These definitions can be given the following physical interpretationWe make a measurement on the system at t = 0 and a measurement att We can measure the positions (α αprime) or the momenta or other com-binations The outcomes of the two measurements are not independentbecause of the dynamics but to start with letrsquos ignore the dynamicsAll possible outcomes of measurements at t = 0 (with their kinematicalrelations) are described by instantaneous states at t = 0 namely by thenonrelativistic Hilbert space H0 Similarly for t If we ignore the dynam-ical correlations we can view the two measurements as if they were doneon two independent systems and therefore we can describe the outcomesof the two measurements using the Hilbert space Kt Dynamics is a cor-relation between the two measurements These correlations are describedby a probability amplitude associated with any given couple of statesNamely to any state in Kt

It is a simple exercise that I leave to the reader to show that in therepresentation Kt = L2[R2dαdαprime] the state |0t〉 is precisely the propaga-tor

〈0t |α αprime〉 = W (α t αprime 0) (545)

Dynamical vacuum versus Minkowski vacuum Denote |0M〉 the lowesteigenstate of H0 in H0

〈α|0M〉 = H0(α) =1radic2π

eminus12α2

(546)

and call it the ldquoMinkowskirdquo vacuum because of its analogy with thevacuum state of the quantum field theories on Minkowski space Considerthe analytic continuation in imaginary time of the propagator (510)

W (αminusit αprime 0) = 〈α|eminusH0t|αprime〉 =sum

n

Hn(α) eminusEnt Hn(αprime) (547)

For large t only the lowest-energy state survives in the sum and we have

W (αminusit αprime 0) minusrarrtrarrinfin H0(α) eminusE0t H0(αprime) (548)

Using the definitions of the previous section this can be written as

limtrarrinfin

eE0t |0minusit〉 = |0M〉 otimes 〈0M| (549)

(The ket and bra in the right-hand side are in H0 while the ket in theleft-hand side is in K = Hlowast

0 otimesH0) This expression relates the dynamicalvacuum |0t〉 and the Minkowski vacuum |0M〉 We will use this equationto find the quantum states corresponding to Minkowski spacetime fromthe spinfoam formulation of quantum gravity


The boundary state space K and covariant vacuum |0〉 The constructionabove can be given a more covariant formulation as follows Consider theHilbert space

K = Klowast otimesK = L2[R4dα dt dαprimedtprime] = L2[G] (550)

I call this space the ldquototalrdquo quantum space The propagator defines apreferred state |0〉 in K

〈α t αprime tprime|0〉 = W (α t αprime tprime) (551)

I call this state the covariant vacuum stateTo run a complete experiment in a one-dimensional quantum system

we need to measure two events a ldquopreparationrdquo and a ldquomeasurementrdquoThe space K describes all possible (a priori equal) outcomes of the mea-surements of these two events Any couple of measurements is representedby operators on K and any outcome is represented by a state ψ isin K whichis an eigenstate of these operators The dynamics is given by the bra 〈0|The probability amplitude of the given outcome is determined by

A = 〈0|ψ〉 (552)

This is a compact and fully covariant formulation of quantum dynamics

515 Evolving constants of motion

The interpretation of the theory is already entirely contained in (529)Still to make the connection with the nonrelativistic formalism moredirect we can also consider operators related to observable quantitieswhose probability distribution can be predicted by the theory

In the classical theory if we know the (relativistic) state of the pen-dulum we can predict the value of α when t has value say t = T Inthe quantum theory there is an operator that corresponds to this physi-cal prediction It is of course the Heisenberg position operator (55) fort = T that is α(T ) (For clarity it is convenient to distinguish the par-ticular numerical value T from the argument of the wave function t) Inow define and characterize this operator in a relativistic language

First of all notice that the operator α(T ) defined on H0 in (55) is infact well defined on H as

α(T ) = Rminus10 α(T )R0 = Rminus1

0 eiTH0 α eminusiTH0R0 = Rminus1T α RT (553)

The operator α(T ) can be directly defined on H without referring to H0as follows Consider the operator

a(T ) = eminusiω(Tminust)(α + i

pαmω

)(554)

52 Relativistic QM 177

and its real part

α(T ) = Re [a(T )] =a(T ) + adagger(T )

2 (555)

defined on S These operators commute with H for any T Therefore theyare well defined on the space of the solutions of (517) namely on H Therestriction of the operator (555) to H is precisely the operator (553)

The operator α(T ) is characterized by two properties First the factthat it commutes with the hamiltonian

[α(T ) H] = 0 (556)

Second if we put T = t in the expressions (554) (555) we obtain αThat is α(T ) is defined as an operator function α(T )(α pα t) such that

α(T )(α pα T ) = α (557)

Intuitively these two equations determine α(T ) since the second fixes itat t = T and the first evolves it for all t Operators of this kind arecalled ldquoevolving constants of motionrdquo They are ldquoevolvingrdquo because theydescribe the evolution (here the evolution of α with respect to t) theyare ldquoconstants of motionrdquo because they commute with the hamiltonianIn GR the operators of this kind are independent from the temporalcoordinate

52 Relativistic QM

In the previous section I used the example of a pendulum to introducea certain number of notions on which a relativistic hamiltonian formula-tion of QM can be based It is now time to attempt a general theory ofrelativistic QM

521 General structure

Kinematical states Kinematical states form a space S in a rigged Hilbertspace S sub K sub S prime

Partial observables A partial observable is represented by a self-adjointoperator in K Common eigenstates |s〉 of a complete set of com-muting partial observables are denoted quantum events

Dynamics Dynamics is defined by a self-adjoint operator H in K the(relativistic) hamiltonian The operator from S to S prime

P =int



is (sometimes improperly) called the projector (The integrationrange in this integral depends on the system) Its matrix elements

W (s sprime) = 〈s|P |sprime〉 (559)

are called transition amplitudes

Probability Discrete spectrum the probability of the quantum event sgiven the quantum event sprime is

Pssprime = |W (s sprime)|2 (560)

where |s〉 is normalized by 〈s|P |s〉 = 1 Continuous spectrum theprobability of a quantum event in a small spectral region R given aquantum event in a small spectral region Rprime is

PRRprime =

∣∣∣∣∣

W (RRprime)radic

W (RR)radicW (Rprime Rprime)

∣∣∣∣∣

2

(561)

whereW (RRprime) =

int

Rds

int

Rprimedsprime W (s sprime) (562)

To this we may add

Boundary quantum space and covariant vacuum For a finite number ofdegrees of freedom the boundary Hilbert space K = Klowast otimes K rep-resents any observations of pairs of quantum events The covariantvacuum state |0〉 isin K defined by

〈0|(ψ otimes ψprime)〉K = 〈ψ|P |ψprime〉K (563)

expresses the dynamics It determines the correlation probabilityamplitude of any such observation The extension to QFT is con-sidered in Section 535

States A physical state is a solution of the WheelerndashDeWitt equation

Hψ = 0 (564)

Equivalently it is an element of the Hilbert space H defined by thequadratic form 〈 middot |P | middot 〉 on S (Elements of K are called kinematicalstates and elements of K are called boundary states)

Complete observables A complete observable A is represented by a self-adjoint operator on H A self-adjoint operator A in K defines acomplete observable if

[AH] = 0 (565)


Projection If the value of the observable A is restricted to the spectral in-terval I the state ψ becomes the state PIψ where PI is the spectralprojector on the interval I If an event corresponding to a sufficientlysmall region R is detected the state becomes |R〉

A relativistic quantum system is defined by a rigged Hilbert space ofkinematical states K and a set of partial observables Ai including a rela-tivistic hamiltonian operator H Alternatively it is defined by giving theprojector P

Axiomatizations are meant to be clarifying not prescriptive The struc-ture defined above is still tentative and perhaps incomplete There are as-pects of this structure that deserve to be better understood clarified andspecified Among these is the precise meaning of the ldquosmallnessrdquo of the re-gion R in the case of the continuum spectrum and the correct treatmentof repeated measurements On the other hand the conventional structureof QM is certainly physically incomplete in the light of GR The aboveis an attempt to complete it making it general relativistic

522 Quantization and classical limit

In general a quantum system (K Ai H) has a classical limit which isa relativistic mechanical system (C H) describing the results of observa-tions on the system at scales and with accuracy larger than the Planckconstant In the classical limit Heisenberg uncertainty can be neglectedand a commuting set of partial observables Ai can be taken as coordinatesof a commutative relativistic configuration space C

If we are given a classical system defined by a nonrelativistic config-uration space C with coordinates qa and by a relativistic hamiltonianH(qa pa) a solution of the quantization problem is provided by the mul-tiplicative operators qa the derivative operators

pa = minusihpart

partqa (566)

and the hamiltonian operator

H = H

(qaminusih

part

partqa

)(567)

on the Hilbert space K = L2[Cdqa] or more precisely the Gelfand tripledetermined by C and the measure dqa The physics is entirely containedin the transition amplitudes

W (qa qprimea) = 〈qa|P |qprimea〉 (568)


where the states |qa〉 are the eigenstates of the multiplicative operatorsqa

In turn the space K has the structure

K = L2[G] (569)

As we shall see this remains true in field theory and in quantum gravityThe space G was defined in Section 325 for finite-dimensional systems inSection 333 for field theories and in Section 434 in the case of gravity

In the limit h rarr 0 the WheelerndashDeWitt equation becomes the rela-tivistic HamiltonndashJacobi equation (359) and the propagator has the form(writing q equiv (qa))

W (q qprime) simsum

i

Ai(q qprime) eihSi(qq

prime) (570)

where Si(q qprime) are the different branches of the Hamilton function as in(389) Now the reverse of each path is still a path The Hamilton functionand the amplitude of a reversed path acquires a minus giving

W (q qprime) simsum

i

Ai(q qprime) sin[

1h Si(q qprime)

] (571)

and W is real Assuming only one path matters

W (q qprime) sim A(q qprime) sin[

1h S(q qprime)

](572)

and we can write for instance

limhrarr0

1W

ihpart

partqaih

part

partqbW (q qprime) =

partS(q qprime)partqa

partS(q qprime)partqb

(573)

This equation provides a precise relation between a quantum theory (en-tirely defined by the propagator W (q qprime)) and a classical theory (entirelydefined by the Hamilton function S(q qprime)) Using (386) and (566) thisequation can be written in the suggestive form

limhrarr0

1W

papbW (q qprime) = pa(q qprime) pb(q qprime) (574)

523 Examples pendulum and timeless double pendulum

Pendulum An example of relativistic formalism is provided by the quan-tization of the pendulum described in the previous section the kinematicalstate space is K = L2[R2dαdt] The partial observable operators are the


multiplicative operators α and t acting on the functions ψ(α t) in K Dy-namics is defined by the operator H given in (518) The WheelerndashDeWittequation is therefore

(minusih

part

parttminus h2

2mpart2

partα2+

mω2

2α2

)Ψ(α t) = 0 (575)

H is a space of solutions of this equation The ldquoprojectorrdquo operator P K rarr H defined by H is given in (523) and defines the scalar product inH Its matrix elements W (α t αprime tprime) between the common eigenstates ofα and t are given by the propagator (511) They express all predictionsof the theory Because of the specific form of H these define a probabilitydensity in α but not in t as explained in Section 513

Equivalently the quantum theory can be defined by the boundarystate space K = L2[G] where G is the boundary space of the classi-cal theory with coordinates (α t αprime tprime) and the covariant vacuum state〈α t αprime tprime|0〉 = W (α t αprime tprime) which determines the amplitude A = 〈0|ψ〉of any possible outcome ψ isin K of a preparationmeasurement experiment

Timeless double pendulum An example of a relativistic quantum sys-tem which cannot be expressed in terms of conventional relativistic quan-tum mechanics is provided by the quantum theory of the timeless system(340) The kinematical Hilbert space K is L2[R2 dadb] and the WheelerndashDeWitt equation is

12

(minush2 part2

parta2minus h2 part2

partb2+ a2 + b2 minus 2E

)Ψ(a b) = 0 (576)

Below I describe this system in some detail

States Since H = Ha + Hb minus E where Ha (resp Hb) is the harmonicoscillator hamiltonian in the variable a (resp b) this equation is easy tosolve by using the basis that diagonalizes the harmonic oscillator Let

ψn(a) = 〈a|n〉 =1radicn

Hn(a) eminusa22h (577)

be the normalized nth eigenfunction of the harmonic oscillator with eigen-value En = h(n+12) Here Hn(a) is the nth Hermite polynomial Thenclearly

Ψnanb(a b) = ψna(a)ψnb

(b) equiv 〈a b|na nb〉 (578)

solves (576) ifh(na + nb + 1) = E (579)


Therefore the quantum theory exists (with this ordering) only if Eh =N + 1 is an integer which we assume from now on The general solutionof (576) is

Ψ(a b) =sum

n= 0N

cn ψn(a) ψNminusn(b) (580)

Therefore H is an (N+1)-dimensional proper subspace of K An orthonor-mal basis is formed by the N + 1 states |nN minus n〉 with n = 0 N

Projector The projector P S rarr H is in fact a true projector and canbe written explicitly as

P =sum

n= 0N

|nN minus n〉〈nN minus n| (581)

This can be obtained from (558) by taking the integration range to be2π determined by the range of τ in the classical hamiltonian evolutionor by the fact that H is the generator of an U(1) unitary action on Kwith period 2π Indeed

int 2π

0dτ eminus

ihτH =

int 2π

0dτ

sum

nanb

|na nb〉eminusihτ(h(na+nb+1)minusE)〈na nb|

=sum

nanb

|na nb〉δ(na + nb + 1 minus Eh)〈na nb|

= P (582)

Transition amplitudes The transition amplitudes are the matrix elementsof P In the basis that diagonalizes a and b

W (a b aprime bprime) = 〈a b|P |aprime bprime〉 =sum

n=0N

〈a b|nN minus n〉〈nN minus n|aprime bprime〉

(583)Explicitly this is

W (a b aprime bprime) =sum

n=0N

1radic

n(N minus n)Hn(a)HNminusn(b)

timesHn(aprime)HNminusn(bprime) eminus(a2+b2+aprime2+bprime2)2h (584)

This function codes all the properties of the quantum system Roughlyit determines the probability density of measuring (a b) if (aprime bprime) wasmeasured Let us study its properties


Semiclassical limit of the projector Notice that by inserting (582) into(583) we can write the projector as

W (a b aprime bprime) =int 2π

0dτ 〈a b|eminus i

hHτ |aprime bprime〉

=int 2π

0dτ e

ihEτ 〈a|eminus i

hHaτ |aprime〉〈b|eminus i

hHbτ |bprime〉 (585)


0dτ e

ihEτW (a aprime τ) W (b bprime τ) (586)

where W (a aprime τ) is the propagator of the harmonic oscillator in a physicaltime τ given in (511) Inserting (511) in (586) we obtain


0dτ

1sin τ

eminusihS(abaprimebprimeτ) (587)

where S(a b aprime bprime τ) is given in (3101) We can evaluate this integral ina saddle-point approximation This gives

W (a b aprime bprime) simsum

i

1sin τi

eminusihS(aaprimebbprimeτi) (588)

where the τi are determined by

partS(a b aprime bprime τ)partτ

∣∣∣∣τ=τi(abaprimebprime)

= 0 (589)

But this is precisely (3102) that defines the value of τ giving the Hamil-ton function of the timeless system This equation has two solutions cor-responding to the two portions into which the ellipse is cut The relationbetween the two actions is given in (3103) Recalling that Eh is aninteger this gives

W (a b aprime bprime) sim 1sin τ(a b aprime bprime)

(eminus

ihS(aaprimebbprime) minus e

ihS(aaprimebbprime)

) (590)

that is


sin[

1hS(a aprime b bprime)

] (591)

as in (572) Here sim indicates equality in the lowest order in h Thisequation expresses the precise relation between the quantum theory andthe classical theory


Propagation ldquoforward and backward in timerdquo Notice that the two termsin (590) have two natural interpretations One is that they representthe two classical paths going from (aprime bprime) to (a b) in C The other moreinteresting interpretation is that they correspond to a trajectory goingfrom (aprime bprime) to (a b) and a ldquotime reversedrdquo trajectory going from (a b)to (aprime bprime) In fact the projector (which recall is real) can be naturallyinterpreted as the sum of two propagators one going forward and onegoing backward in the parameter time τ

The distinction between forward and backward in the parameter timeτ has no physical significance in the classical theory because the physicsis only in the ellipses in C not in the orientation of the ellipses

However in the quantum theory we can identify in H ldquoclockwise-movingrdquo and ldquoanticlockwise-movingrdquo components These components arethe eigenspaces of the positive and negative eigenvalues of the angular mo-mentum operator L = apartbminusbparta (or L = partφ where a = r sinφ b = r cosφ)Thus we can write wave packets ldquotraveling along the ellipses purely for-ward or purely backward in the parameter timerdquo If we consider only alocal evolution in a small region of C and we interpret say b as the in-dependent time variable and a as the dynamical variable then these twocomponents have respectively positive and negative energy In a sensethey can be viewed as particles and antiparticles

53 Quantum field theory

I assume the reader is familiar with standard quantum field theory (QFT) Here Iillustrate the connection between QFT and the relativistic formalism developed aboveand I recall a few techniques that will be used in Part II and are not widely knownOf particular importance are the distinction between Minkowski vacuum and covariantvacuum the functional representation of a field theory and the construction of thephysical Hilbert space of lattice YangndashMills theory

In Chapter 3 we have seen that a classical field theory can be definedcovariantly by the boundary space G of closed surfaces α in a finite-dimensional space C and a relativistic hamiltonian H on T lowastG For in-stance in a scalar field theory C = M timesR has coordinates (xμ φ) wherexμ is a point in Minkowski space and φ a field value A surface α isdetermined by the two functions

α = [xμ(τ) ϕ(τ)] (592)

and determines a boundary 3-surface xμ(τ) in Minkowski space M andboundary values φ(x(τ)) = ϕ(τ) of the field on this surface

A quantization of the theory can be obtained precisely as in the finite-dimensional case in terms of a boundary state space K of functionalsΨ[α] on G Notice however that the difference between the kinematical

53 Quantum field theory 185

state space K and the boundary state space K is far less significant infield theory than for finite-dimensional systems In the finite-dimensionalcase the states ψ(qa) in K are functions on the extended configurationspace C while the states ψ(qa qaprime) in K are functions on the boundaryspace G = C times C In the field theoretical case both states have the formΨ[α] The difference is that the states in K are functions of an ldquoinitialrdquosurface α where xμ(τ) can be for instance the spacelike surface x0 = 0in this case α contains only one-half of the data needed to determine asolution of the field equations On the other hand the states Ψ[α] in Kare functions of a closed surface α In fact the only difference betweenK and K is in the global topology of α If we disregard this and considerlocal equations we can confuse K and K (see Section 535)

The relativistic hamiltonian is given in (3192) The complete solutionof the classical dynamics is known if we know the Hamilton function S[α]which is the value of the action

S[α] = S[R φ] =int

RL(φ(x) partμφ(x))d4x (593)

where R is the four-dimensional region bounded by x(τ) and φ(x) isthe solution of the equations of motion in this region determined bythe boundary data φ(x(τ)) = ϕ(τ) If there is more than one of thesesolutions we write them as φi(x) and the Hamilton function is multivalued

Si[α] = S[R φi] =int

RL(φi(x) partμφi(x))d4x (594)

The relativistic Hamiltonian gives rise to the WheelerndashDeWitt equation

H

[xμ φminusih

δ

δxμminusih

δ

δϕ

](τ) Ψ[α] = 0 (595)

precisely as in the finite-dimensional case The HamiltonndashJacobi equation(3193) can be interpreted as the eikonal approximation for this waveequation

The complete solution of the dynamics is known if we know the propa-gator W [α] which is a solution of this equation Formally the field prop-agator can be written as a functional integral

W [α] =int

φ(x(τ))=ϕ(τ)[Dφ] eminus

ihS[Rφ] (596)

Of course one should not confuse the field propagator W [α] with theFeynman propagator The first propagates field the second the particlesof a QFT The first is a functional of a surface and the value of the field


on this surface the second is a function of two spacetime points To thelowest order in h the saddle-point approximation gives

W [α] simsum

i

Ai[α] eminusihSi[α] (597)

There are two characteristic difficulties in the field theoretical contextthat are absent in finite dimensions the definition of the scalar productand the need to regularize operator products

First in finite dimensions a measure dqa on C is sufficient to definean associated L2 Hilbert space of wave functions In the field theoreticalcase we have to define the scalar product in some other way The scalarproduct must respect the invariances of the theory and must be such thatreal classical variables be represented by self-adjoint operators This isbecause self-adjoint operators have a real spectrum and the spectrumdetermines the values that a quantity can take in a measurement Givena set of linear operators on a linear space the requirement that they areself-adjoint puts stringent conditions on the scalar product As we shallsee in all cases of interest these requirements are sufficient to determinethe scalar product

Second local operators are in general distributions and their productsare ill defined Operator products arise in physical observable quantities aswell as in the dynamical equation namely in the WheelerndashDeWitt equa-tion In particular functional derivatives are distributions In the classicalHamiltonndashJacobi equation we have products of functional derivatives ofthe HamiltonndashJacobi functional which are well-defined products of func-tions In the corresponding quantum WheelerndashDeWitt equation thesebecome products of functional-derivative operators which are ill definedwithout an appropriate renormalization procedure The definition of gen-erally covariant regularization techniques will be a major concern in thesecond part of the book

531 Functional representation

Consider a simple free scalar theory where V = 0 I describe this well-known QFTin some detail in order to illustrate certain techniques that play a role in quantumgravity In particular I illustrate the functional representation of quantum field the-ory a simple form of the WheelerndashDeWitt equation the general form of W [α] and itsphysical interpretation The functional representation is the representation in whichthe field operator is diagonal The quantum states will be represented as functionalsΨ[φ] = 〈φ|Ψ〉 where |φ〉 is the (generalized) eigenstate of the field operator with eigen-value φ(x) The relation between this representation and the conventional one on the

Fock basis |k1 k1〉 is precisely the same as the relation between the Schrodingerrepresentation ψ(x) and the one on the energy basis |n〉 for a simple harmonic oscilla-tor I also illustrate the way in which the scalar product on the space of the solutions


of the WheelerndashDeWitt equation is determined by the reality properties of the fieldoperators

To start with and to connect the generally covariant formalism de-scribed above with conventional QFT letrsquos restrict the surface x(τ) in α toa spacelike surface xμ(τ) = (t τ) in Minkowski space Then α = [t φ(x)]and Ψ[α] = Ψ[t φ(x)] The HamiltonndashJacobi equation (3186) reduces to(3190) The corresponding quantum WheelerndashDeWitt equation becomes

ihpart

parttΨ = H0Ψ (598)

where the nonrelativistic hamiltonian operator H0 is

H0 =int

d3x H0

[φminusih

δ

δφ

](x) (599)

and H0[φ p](x) is given in (3195) The factor ordering of this operatorcan be chosen in order to avoid the divergence that would result from thenaive factor ordering

H0 naive =12

intd3x

[minush2 δ

δφ(x)δ

δφ(x)+|nablaφ|2(x) + m2φ2(x)

] (5100)

The Fourier modes

φ(k) = (2π)minus32

intd3x e+ikmiddotxφ(x) (5101)

decouple

H0 =12

intd3k

[p2(k) + ω2(k)φ2(k)

] (5102)

where ω =radic

|k|2 + m2 The dangerous divergence is produced by the

vacuum energy of the quantum oscillators associated with each mode kand can be avoided by normal ordering In terms of the positive andnegative frequency fields

a(k) =iradic2ω

p(k) +radic

ω

2φ(k) (5103)

adagger(k) = minus iradic2ω

p(minusk) +radic

ω

2φ(minusk) (5104)

the hamiltonian reads

H0 =int

d3k ω(k) adagger(k) a(k) (5105)


We define the quantum hamiltonian by this equation where

a(k) =hradic2ω

δ

δφ(k)+

radicω

2φ(k) (5106)

adagger(k) = minus hradic2ω

δ

δφ(minusk)+

radicω

2φ(minusk) (5107)

The lowest-energy eigenvector of the hamiltonian has vanishing eigen-value and is called the Minkowski vacuum state This state is usuallydenoted |0〉 I denote it here as |0M〉 where M stands for Minkowski inorder to distinguish it from other vacuum states that will be introducedlater on The Minkowski vacuum state is determined by a(k)|0M〉 = 0 Inthe functional representation this state reads

Ψ0M [φ] equiv 〈φ|0M〉 (5108)

and is determined by

a(k)Ψ0M [φ] =hradic2ω

δ

δφ(k)Ψ0M [φ] +

radicω

2φ(k)Ψ0M [φ] = 0 (5109)

The solution of this equation gives the functional form of the vacuumstate

Ψ0M [φ] = Neminus12h

intd3k ω(k)φ(k)φ(k) (5110)

The one-particle state with momentum k is created by adagger(k)

Ψk[φ] equiv 〈φ|k〉 = adagger(k)Ψ0M [φ] =

radic2ω φ(k) Ψ0M [φ] (5111)

It has energy hω(k) Therefore the time-dependent state

Ψk[t φ] equiv

radic2ω eminusiω(k)t φ(k) Ψ0M [φ] (5112)

is a solution of the WheelerndashDeWitt equation (598)A generic one-particle state with wave function f(k) is defined by

|f〉 equivint

d3kradic2ω

f(k) |k〉 (5113)

and its functional representation is therefore

Ψf [φ] equiv 〈φ|f〉 =int

d3k f(k) φ(k) Ψ0[φ] (5114)

or

Ψf [φ] = φ[f ] Ψ0[φ] (5115)


where

φ[f ] =int

d3k f(k) φ(k) (5116)

The corresponding solution of the WheelerndashDeWitt equation (598) is

Ψf [t φ] =int

d3k f(k) eminusiω(k)t φ(k) Ψ0[φ] (5117)

or in Fourier transform

Ψf [t φ] =int

d3x F (t x) φ(x) Ψ0[φ] (5118)

where

F (x) = F (t x) = (2π)minus32

intd3k ei(kmiddotxminusω(k)t) f(k) (5119)

is a positive-energy solution of the KleinndashGordon equationThe n-particle states |k1 kn〉 can be obtained using again the cre-

ation operator adagger(k) in the well-known way They have energy h(ω1 +middot middot middot + ωn) where ωi = ω(ki) The general solution of the WheelerndashDeWittequation is therefore

Ψ[t φ] =sum

n

intd3k1 d3knradic

2ω1 2ωnf(k1 kn) eminusi(ω1+middotmiddotmiddot+ωn)t

times adagger(k1) adagger(kn)Ψ0[φ] (5120)

The space F of these solutions labeled by the functions f(k1 kn) isthe physical state space H of the theory Since Ψ[t φ] is determined byΨ[φ] = Ψ[0 φ] we can also represent the quantum states by their valueon the t = 0 surface namely as functionals Ψ[φ]

Scalar product The scalar product can be determined on the space ofthe solutions of the WheelerndashDeWitt equation from the requirement thatreal quantities are represented by self-adjoint operators The scalar fieldφ(x) and its momentum p(x) are real Therefore we must demand thatthe corresponding operators are self-adjoint It follows that the operatoradagger(k) is the adjoint of the operator a(k) Using this we obtain easily

〈k|kprime〉 = 〈adagger(k)0|adagger(kprime)0〉 = 〈0|a(k)adagger(kprime)0〉 = hδ(k minus kprime) (5121)

It follows from (5113) that

〈f |f prime〉 = h

intd3k

2ωf(k) f prime(k) (5122)


(Recall that d3k2ω is the Lorentz-invariant measure) Therefore the one-particle state space is H1 = L2[R3 d3k2ω] Let us write

f(x) = (2π)minus32

intd3k

2ωeikx f(k) (5123)

The spacetime function

f(x) = f(t x) = (2π)minus32

intd3k

2ωei(kmiddotxminusω(k)t) f(k) (5124)

is a positive-frequency solution of the KleinndashGordon equation with initialvalue f(0 x) = f(x) (not to be confused with the one defined in (5119)which is F = iparttf) Then easily

〈f |f prime〉 = ihint

d3x[f(x)part0f

prime(x) minus f prime(x)part0f(x)]t=0

(5125)

This is the well-known KleinndashGordon scalar product which is positivedefinite on the positive-frequency solutions

Notice that the one-particle Hilbert space can be represented in various equivalentmanners It is

bull the space of the positive-frequency solutions f(x) of the KleinndashGordon equationwith the scalar product (5125)

bull the space H = L2[R3 d3k2ω] of the functions f(k)

bull the space H = L2[R4 δ(k2 + m2)θ(k0)d4k] of the functions f(k)

bull the space H = L2[R3 d3x] of the functions

f(x) =

intd3kradic2ω

eikmiddotx f(k) (5126)

(the position operator x in this representation is obviously self-adjoint it is thewell-known NewtonndashWigner operator which has a far more complicated form inother representations)

bull and so on

Using the same technique the entire space F can be equipped with ascalar product The resulting Hilbert space is of course the well-knownFock space over this one-particle Hilbert space

532 Field propagator between parallel boundary surfaces

Consider now a surface Σt formed by two parallel spacelike planes inMinkowski space say xμ1 (τ) = (t1 τ) and xμ2 (τ) = (t2 τ) Consider twoscalar fields ϕ1(τ) ϕ2(τ) on these planes Let α be the union of thesetwo surfaces with their fields that is α is formed by two disconnected


components α = α1 cup α2 = [xμ1 (τ) ϕ1(τ)] cup [xμ2 (τ) ϕ2(τ)] Consider thefield propagator (596) for this value of α Thus W [α] = W [t1 ϕ1 t2 ϕ2]In this case we can simply write

W [t1 ϕ1 t2 ϕ2] = 〈t1 ϕ1|t2 ϕ2〉 = 〈ϕ1|eminusihH0(t1minust2)|ϕ2〉 (5127)

The calculation of the propagator is simplified by the fact that the quan-tum field theory is essentially a collection of one harmonic oscillator foreach mode k Using the propagator of the harmonic oscillator given in(511) one obtains with some algebra

W [t1 ϕ1 t2 ϕ2]

= N exp

minus i2h

intd3k

(2π)3ω

[(|ϕ1|2 + |ϕ2|2

)cos[ω(t1 minus t2)] minus 2ϕ1ϕ2

sin[ω(t1 minus t2)]

]

(5128)

where N is the formal divergent normalization factor

N simprod

k

radicmω(k)

hexp

minusV

2

intd3k

(2π)3ln

[sin[ω(k)(t1 minus t2)]

] (5129)

This has the form (597) see the classical Hamilton function given in(3170)

Minkowski vacuum from the euclidean field propagator The state spaceat time zero Ht=0 is Fock space where the field operators ϕ(x) = φ(x t)and the hamiltonian H0 are defined Fock space is separable and thereforeadmits countable bases Choose a basis |n〉 of eigenstates of H0 witheigenvalues En and consider the operator

W (T ) =sum

n

eminusThEn |n〉〈n| (5130)

In the large-T limit this becomes the projection on the only eigenstatewith vanishing energy namely the Minkowski vacuum

limTrarrinfin

W (T ) = |0M〉〈0M| (5131)

In the functional Schrodinger representation the operator (5130) reads

W [ϕ1 ϕ2 T ] = 〈ϕ1|eminusihH0(minusiT )|ϕ2〉 = W [0 ϕ1 iT ϕ2] (5132)


Therefore it is the analytical continuation of the field propagator (5127)and satisfies the euclidean Schrodinger equation

minushpart

partTW [ϕ1 ϕ2 T ] = Hϕ1 W [ϕ1 ϕ2 T ] (5133)

We can obtain the vacuum (up to normalization) as

Ψ0M [ϕ] = 〈ϕ|0M〉 = limTrarrinfin

W [ϕ 0 T ] (5134)

We can derive all particle scattering amplitudes from the functionalW [ϕ1 ϕ2 T ] For instance the 2-point function can be obtained as theanalytic continuation of the Schwinger function

S(x1 x2) = limTrarrinfin

intDϕ1Dϕ2 W [0 ϕ1 T ]ϕ1(x1)

timesW [ϕ1 ϕ2 (t1 minus t2)]ϕ2(x2) W [ϕ2 0 T ] (5135)

This can be generalized to any n-point function where the times t1 tnare on the t = 0 and the t = T surfaces these in turn are sufficient tocompute all scattering amplitudes since time dependence of asymptoticstates is trivial

W [ϕ1 ϕ2 T ] admits the well-defined functional integral representation

W [ϕ1 ϕ2 T ] =int

φ|t=T =ϕ1

φ|t=0=ϕ2

Dφ eminus1hSET [φ] (5136)

Here the integral is over all fields φ on the strip R bounded by the twosurfaces t = 0 and t = T with fixed boundary value The action SE

T [φ]is the euclidean action Notice that using this functional integral repre-sentation the expression (5135) for the Schwinger function becomes thewell-known expression

S(x1 x2) =int

Dφ φ(x1) φ(x2) eminus1hSE [φ] (5137)

obtained by joining at the two boundaries the three functional integralsin the regions tltt2 t2lttltt1 and t1ltt The functional W [ϕ1 ϕ2 T ] canbe computed explicitly in the free field theory Its expression in terms ofthe Fourier transform ϕ of ϕ is the analytic continuation of (5128)

W [ϕ1 ϕ2 T ] = N expminus 1

2h

intd3k

(2π)3ω

( |ϕ1|2 + |ϕ2|2tanh (ωT )

minus 2ϕ1ϕ2

sinh (ωT )

)

(5138)


The dynamical vacuum |0ΣT〉 Consider the boundary state space KΣt

associated with the entire surface Σt as in Section 514 That is defineKΣt = Ht otimes Hlowast

0 Denote ϕ = (ϕ1 ϕ2) a field on Σt The field basis ofthe Fock space induces the basis |ϕ〉 = |ϕ1 ϕ2〉 equiv |ϕ1〉t otimes 〈ϕ2|0 in KΣt the vectors |Ψ〉 of KΣt are written in this basis as functionals Ψ[ϕ] =Ψ[ϕ1 ϕ2] equiv 〈ϕ1 ϕ2|Ψ〉 This is the field theoretical generalization of theboundary state space defined in (542)

The functional W defines a preferred state in this Hilbert space as in(543)ndash(544) Denote this state |0Σt〉 and call it the dynamical vacuum Itis defined by 〈ϕ|0Σt〉 equiv W [t ϕ1 0 ϕ2] This state expresses the dynamicsfrom t = 0 to t A state in the tensor product of two Hilbert spaces definesa linear mapping between the two spaces The linear mapping from Ht=0

to Ht=T defined by |0ΣT〉 is precisely the time evolution eminusiHt

The interpretation of this state is the same as in the finite-dimensionalcase The tensor product of two quantum state spaces describes the en-semble of the measurements described by the two factors Therefore KΣt

is the space of the possible results of all measurements performed at time0 and at time t Observations at two different times are correlated by thedynamics Hence KΣt is a ldquokinematicalrdquo state space in the sense that itdescribes more outcomes than the physically realizable ones Dynamics isthen a restriction on the possible outcomes of observations It expressesthe fact that measurement outcomes are correlated The linear functional〈0Σt | on KΣt assigns an amplitude to any outcome of observations Thisamplitude gives us the correlation between outcomes at time 0 and out-comes at time t

Therefore the theory can be represented as follows The Hilbert spaceKΣt describes all possible outcomes of measurements made on Σt The dy-namics is given by a single bra state 〈0Σt | Kt rarr C For a given collectionof measurement outcomes described by a state |Ψ〉 the quantity 〈0Σt |Ψ〉gives the correlation probability amplitude between these measurements

Using (5131) we have then the relation between the dynamical vacuumand the Minkowski vacuum (the braket mismatch is apparent only asthe three states are in different spaces)

limtrarrinfin

|0Σminusit〉 = |0M〉 otimes 〈0M| (5139)

533 Arbitrary boundary surfaces

So far I have considered only boundary surfaces formed by two parallelspacelike planes This restriction is sufficient and convenient in ordinaryQFT on Minkowski space but it has no meaning in a generally covariantcontext It is therefore necessary to consider arbitrary boundary surfacesso let us study the extension of the formalism to the case where the surface


Σ instead of being formed by two parallel planes is the boundary of a(sufficiently regular) arbitrary finite region of spacetime R

Let Σ be a closed connected 3d surface in Minkowski spacetime withthe topology (but in general not the geometry) of a 3-sphere and Σ =partR Let ϕ be a scalar field on Σ and consider the functional

W [ϕΣ] =int

φ|Σ=ϕDφ eminusSE

R[φ] (5140)

The integral is over all 4d fields on R that take the value ϕ on Σ and theaction in the exponent is the euclidean action where the 4d integral is overR In the free theory the integral is a well-defined gaussian integral andcan be evaluated The classical equations of motion with boundary valueϕ on Σ form an elliptic system which in general has a solution φcl[ϕ] thatcan be obtained by integration from the Green function for the shape R Achange of variable in the integral reduces it to a trivial gaussian integrationtimes eminusSE

R[ϕcl] Here SER[ϕ] is the field theoretical Hamilton function the

action of the bulk field determined by the boundary condition ϕ

W [ϕΣ] can be defined in the Minkowski regime as well If Σ is a rectangular boxin Minkowski space let ϕ = (ϕout ϕin ϕside) be the components of the field on thespacelike bases and timelike side Consider the field theory defined in the box withtime-dependent boundary conditions ϕside and let U [ϕside] be the evolution operatorfrom t = 0 to t = T generated by the (time-dependent) hamiltonian of the theoryThen we can write

W [ϕΣ] equiv 〈ϕout|U [ϕside]|ϕin〉 (5141)

In particular if ϕside is constant in time W can be obtained by analytic continuationfrom the euclidean functional More generally we can write the formal definition

W [ϕΣ] =

int

φ|Σ=ϕ

Dφ eiSR[φ] (5142)

Notice that W [ϕΣ] is a function on the space G defined in Section 333This space represents all possible ensembles of classical field measurementson a closed surface namely the minimal data for a local experimentFormally functions on G define the quantum state space K and W [ϕΣ]defines the preferred covariant vacuum state |0〉 in K

Local Schrodinger equation W [ϕΣ] satisfies a local functional equationthat governs its dependence on Σ Let τ be arbitrary coordinates on ΣRepresent the surface and the boundary fields as Σ τ rarr xμ(τ) andϕ τ rarr ϕ(τ) Let nμ(τ) be the unit length normal to Σ Then

nμ(τ)δ

δxμ(τ)W [ϕΣ] = H(τ) W [ϕΣ] (5143)


where H(x) is an operator obtained by replacing π(x) by minusiδδϕ(x) inthe hamiltonian density

H(x) = gminus12π2(x) + g

12 (|nablaϕ|2 + m2ϕ2) (5144)

Here g is the determinant of the induced metric on Σ and the norm istaken in this metric (see [145 146]) The local HamiltonndashJacobi equation(3186) can be viewed as the eikonal approximation of this equation SinceW is independent from the parametrization we have

partxμ(τ)partτ

δ

δxμ(τ)W [ϕΣ] = P (τ) W [ϕΣ] (5145)

where the linear momentum is P (τ) = nablaφ(τ) δδϕ(τ) If Σ is spacelike(5143) is the (euclidean) TomonagandashSchwinger equation

We expect a local equation like (5143) to hold in any field theory If thetheory is generally covariant the functional W will be independent fromΣ and therefore the left-hand side of the equation will vanish leavingonly the hamiltonian operator acting on the field variables namely aWheelerndashDeWitt equation

534 What is a particle

Choose Σ to be a cylinder ΣRT with radius R and height T with thetwo bases on the surfaces t = 0 and t = T Given two compact supportfunctions ϕ1 and ϕ2 defined on t = 0 and t = T respectively we canalways choose R large enough for the two compact supports to be includedin the bases of the cylinder Then

limRrarrinfin

W [ϕ1 ϕ2ΣRT ] = W [ϕ1 ϕ2 T ] (5146)

because the euclidean Green function decays rapidly and the effect ofhaving the side of the cylinder at finite distance goes rapidly to zero as Rincreases Equation (5135) illustrates how scattering amplitudes can becomputed from W [ϕ1 ϕ2 T ] In turn (5146) indicates how W [ϕ1 ϕ2 T ]can be obtained from W [ϕΣ] where Σ is the boundary of a finite regionTherefore knowledge of W [ϕΣ] allows us to compute particle scatteringamplitudes We expect this to remain true in the perturbative expansionof an interacting field theory as well where R includes the interactionregion

The limits TR rarr infin seem to indicate that arbitrarily large surfaces Σare needed to compute vacuum and particle scattering amplitudes Butnotice that the convergence of W [ϕ1 ϕ2 T ] to the vacuum projector isdictated by (5130) and is exponential in the mass gap or the Compton


frequency of the particle Thus T at laboratory scales is largely sufficientto guarantee arbitrarily accurate convergence In the euclidean regimerotational symmetry suggests the same to hold for the R rarr infin limitThus the limits can be replaced by choosing R and T at laboratory scales(At least for the vacuum which does not require analytic continuation)

The conventional notions of vacuum and particle states are global innature How is it possible that we can recover them from the local func-tional W [ϕΣ] This is an important question that plays a role in QFTon curved spacetime and in quantum gravity To answer this questionnotice that realistic particle detectors are finitely extended How can afinitely extended detector detect particles if particles are globally definedobjects

The answer is that there exist two distinct notions of particle Fock par-ticle states are ldquoglobalrdquo while the physical states detected by a localizeddetector (eigenstates of local operators describing detection) are ldquolocalrdquoparticle states Local particle states are close to (in a suitable topology)but distinct from the global particle states In conventional QFT we usea global particle state in order to conveniently approximate the local par-ticle state detected by a detector Global particle states indeed are fareasier to deal with

Therefore the global nature of the conventional definition of vacuumand particles is not dictated by the physical properties of particles it isan approximation adopted for convenience Replacing the limits Rrarrinfinand T rarr infin with finite macroscopic R and T we miss the exact globalvacuum or n-particle state but we can nevertheless describe local ex-periments The restriction of QFT to a finite region of spacetime mustdescribe completely experiments confined to this region

Global and local particles in a simple finite system The distinction between globalparticles and local particles can be illustrated in a very simple system Consider twoweakly coupled harmonic oscillators Let the total hamiltonian of the system be

H =1

2(p2

1 + q21) +

1

2(p2

2 + q22) minus 2λq1q2 = H1 + H2 minus λV (5147)

Consider a measuring apparatus that interacts only with the first oscillator and mea-sures the quantity H1 The Hilbert space of the system is H = L2[R

2 dq1dq2] On thisspace the quantity H1 is represented by the operator minush2part2partq2

1 + q21 The operator

has a discrete spectrum E = (n+ 12)h If the result of the measurement is the eigen-value (1 + 12)h let us say that ldquothere is one local particle in the first oscillatorrdquo Inparticular a one-local-particle state is the common eigenstate of H1 and H2

ψlocal(q1 q2) = q1eminus(q21+q22)2h (5148)

in which there is one local particle in the first oscillator and no local particles in thesecond

Next let us diagonalize the full hamiltonian H This can easily be done by findingthe normal modes of the system which are qplusmn = (q1 plusmn q2)

radic2 and have frequencies


ω2plusmn = 1 plusmn λ The eigenvalues of H are therefore E = h(n+ω+ + nminusωminus + 1) We call

|n+ nminus〉 the corresponding eigenstates and N = n+ + nminus the global-particle numberIn particular we call ldquoone-global-particle staterdquo all states with N = 1 namely anystate of the form |ψ〉 = α|1 0〉 + β|0 1〉 Notice that this is precisely the definition ofone-particle states in QFT a one-particle state is an arbitrary linear combination ofstates |k〉 where there is a single quantum in one of the modes In particular considerthe one-global-particle state |ψ〉 = (|1 0〉+ |0 1〉)

radic2 This is a global particle which is

maximally localized on the first oscillator A straightforward calculation gives to firstorder in λ

ψglobal(q1 q2) = (q1 +λ

4q2)e

minus(q21+q22minus2λq1q2)2h (5149)

The two states ψlocal and ψglobal are different and have different physical meaningThe state ψglobal is the kind of state that is called a one-particle state in QFT It is theone-particle state which is most localized on the first oscillator On the other hand ifour measuring apparatus interacts only with the first oscillator then what we measureis not ψglobal it is ψlocal which is an eigenstate of an operator that acts only on thevariable q1

In QFT we confuse the two kinds of states In the formalism we use global-particlestates such as ψglobal However particle detectors are localized in space (A local mea-suring apparatus can only interact with the components of the field in a finite regionlike the apparatus that interacts only with the variable q1 in the example) Thereforethey measure particle states such as ψlocal Strictly speaking therefore the interpreta-tion of the particle states measured by particle detectors as global-particle states is amistake because a global-particle state can never be an eigenstate of a local measuringapparatus and therefore cannot be detected by a local apparatus

The reason we can nevertheless use this interpretation successfully is that the statesψlocal and ψglobal are very similar In the example their distance in the Hilbert normvanishes to first order in lambda

(ψglobal ψlocal) = 0(λ) (5150)

The error we make in using ψglobal to describe the physical state ψlocal is small if λVis small In the field theoretical case λV represents the interaction energy betweenthe region inside the detector and the region outside the detector this energy is verysmall compared to the energy of the state itself for all the states of interest We caneffectively approximate the local-particle states that are detected by our measuringapparatus by means of the global-particle states which are easier to deal with

On the other hand the argument shows that global-particle states are not required fordealing with the realistic observed particles they are just a convenient approximationIf we can define local-particle states by means of a local formalism we are not makinga mistake rather we are simply not using an approximation that was convenient onflat space but may not be viable in a generally covariant context

535 Boundary state space K and covariant vacuum |0〉Finally consider the space G of the variable α = (Σ ϕ) where Σ is a closed3d surface in spacetime Call K the space of functions ψ[α] = ψ[Σ ϕ] Thisspace represents all possible outcomes of ensembles of measurements onthe boundary of a finite region of spacetime The measurements includespacetime localization measurements that determine the surface Σ as well


as field (or particle) measurements that determine ϕ (or a function of ϕ)K is the boundary quantum space

There is a preferred state |0〉 in K given by

〈Σ ϕ|0〉 = W [Σ ϕ] (5151)

If the functional integral can be defined this is given by (5142) The state|0〉 expresses the dynamics entirely As we shall see this formulation ofQFT makes sense in quantum gravity

In general K is a space of functions over G Recall that G is the spaceof data needed to determine a classical solution two events in the finite-dimensional case a set of events forming a 3d closed surface in the fieldcase

In the case of a finite-dimensional theory a classical solution in some in-terval is determined by two events in C In the quantum theory a completeexperiment consists of two events a preparation and a quantum measure-ment In this case K = L2[G] = L2[C times C] sim L2[C] otimes L2[C] = K otimes K isthe space representing two quantum events while K = L2[C] is the spacerepresenting a single quantum event

In the field theoretical case a classical solution in a region R is deter-mined by infinite events in C forming a closed 3d surface namely by a3d surface Σ = partR in spacetime and the field ϕ on it In the quantumtheory a complete experiment requires measurements (or assumptions)on the entire Σ In this case K sim L2[G] is the space describing the obser-vation of the entire boundary surface Σ and the measurements on it

The boundary of R can be formed by two (or even more) connectedcomponents Σ In this case we can decompose K into the tensor productof one factor K associated with each component The space K is then aspace of functionals of the connected surface Σ and the field on it Sincethe WheelerndashDeWitt equation is local it looks the same on K and on KTherefore the distinction between K and K is of much less importance inthe field theoretical context than in the finite-dimensional case The spaceK is associated with the idea of the full data characterizing an experimenton a closed surface Σ while the space K is associated with the idea of anldquoinitial datardquo surface Σ

536 Lattice scalar product intertwiners and spin network states

An interacting quantum field theory can be constructed as a perturbation expansion

around a free theory An alternative is to define a cut-off theory with a large but finite

number of degrees of freedom using a lattice One expects then to recover physical

predictions as suitable limits as the lattice spacing is taken to zero I illustrate here the

definition of the scalar product in a lattice gauge theory since the same technique is

used in quantum gravity


Consider a three-dimensional lattice Γ with L links l and N nodes nTo define a YangndashMills theory for a compact YangndashMills group G on thislattice we associate a group element Ul to each link l and we considerthe Hilbert space KΓ = L2[GLdUl] where GL is the product of L copiesof G and dUl equiv dUl dUl is the Haar measure on the group Quantumstates in KΓ are functions Ψ(Ul) of L group elements The scalar productof two states is given by

〈Ψ|Φ〉 equivint

dU1 dUL Ψ(U1 UL) Φ(U1 UL) (5152)

An orthonormal basis of states in KΓ can be obtained as follows Let jlabel unitary irreducible representations of G and let (Rj(U))αβ be thematrix elements of the representation The PeterndashWeyl theorem tells usthat the states |j β α〉 defined by 〈U |j β α〉 = (Rj(U))αβ form an or-thonormal basis in L2[GdU ] A basis in KΓ is therefore given by thestates

|jl βl αl〉 equiv |j1 jL β1 βL α1 αL〉 (5153)

defined by 〈Ul|jl βl αl〉 =prod

l(Rjl(Ul))

αlβl

The theory is invariant under local YangndashMills transformations on the

lattice These depend on a group element λn for each node n The variablesUl transform under a gauge transformation as Ul rarr λli

minus1Ulλlf where thelink l goes from the initial node li to the final node lf Hence the gauge-invariant states are the ones satisfying

Ψ(Ul) = Ψ(λliminus1Ulλlf ) (5154)

These states form a linear subspace K0Γ of KΓ the space K0

Γ is the(fixed-time) Hilbert space of the gauge-invariant states of the theory Anorthonormal basis of states in K0

Γ can be obtained using the notion of anintertwiner

Intertwiners Consider N irreducible representations j1 jN Considerthe tensor product of their Hilbert spaces

Hj1jN = Hj1 otimes otimesHjN (5155)

This space can be decomposed into a sum of irreducible components Inparticular let H0

j1jNbe the subspace formed by the invariant vectors

namely the subspace that transforms in the trivial representation Thisspace is k-dimensional where k is the multiplicity with which the trivialrepresentation appears in the decomposition It is of course a Hilbertspace and therefore we can choose an orthonormal basis in it We call


the elements i of this basis ldquointertwinersrdquo between the representationsj1 jN

More explicitly elements of Hj1jN are tensors vα1 αN with one index in eachrepresentation Elements of H0

j1jN are tensors vα1αN that are invariant under theaction of G on all their indices That is they satisfy

R(j1)α1β1(U) R(jN )αN

βN (U) vβ1 βN = vα1 αN (5156)

The intertwiners vα1 αNi are a set of k such invariant tensors which are orthonormal

in the scalar product of H0j1jN That is they satisfy

vα1αNi viprimeα1αN = δiiprime (5157)

If the space Hj carries the representation j its dual space Hlowastj carries the

dual representation jlowast An intertwiner i between n dual representationsjlowast1 j

lowastn and m representations j1 jm is an invariant tensor in the

space (otimesi=1nHlowastji) otimes (otimesk=1mHjk) that is a covariant map

i (otimesi=1nHji) minusrarr (otimesk=1mHjk) (5158)

or an invariant tensor with n lower indices and m upper indicesNow associate a representation jl to each link l and an intertwiner in

in each node n (in the tensor product of the representations associatedwith the links adjacent to the node) of the lattice The set s = (Γ jl in)is called a ldquospin networkrdquo Each spin network s defines a state |s〉 by

〈Ul|s〉 = ψs(Ul) =prod

l

Rjl(Ul) middotprod

n

in (5159)

where the raised dot indicates index contraction Notice that the indices(not indicated in the equation) match as on each side of the dot there isone index for each couple node-link The states |s〉 form a complete andorthonormal basis in K0

Γ〈s|sprime〉 = δssprime (5160)

This basis will play a major role in quantum gravity

54 Quantum gravity

Finally I sketch here the formal structure of quantum gravity The actual mathematicaldefinition of the quantities mentioned here is the task I undertake in the second partof the book

541 Transition amplitudes in quantum gravity

In the presence of a background QFT yields scattering amplitudes andcross sections for asymptotic particle states and these are compared with

54 Quantum gravity 201

data obtained in a lab The conventional theoretical definition of theseamplitudes involves infinitely extended spacetime regions and relies onsymmetry properties of the background In a background-independentcontext this procedure becomes problematic For instance backgroundindependence implies immediately that any 2-point function W (x y) isconstant for x = y as mentioned in Section 114 How can the formalismcontrol the localization of the measuring apparatus

We have seen above that in the context of a simple scalar field theory lo-cal physics can be expressed in terms of a functional W [ϕΣ] that dependson field boundary eigenstates ϕ and the geometry of the 3d surface Σthat bounds R Physical predictions concerning measurements performedin the finite region R including scattering amplitudes between particlesdetected in the lab can be expressed in terms of W [ϕΣ] The functionalsatisfies a local version of the Schrodinger equation The geometry of Σcodes the relative spacetime localization of the particle detectors W [ϕΣ]can be expressed as a functional integral over a finite spacetime regionR of spacetime In the euclidean regime the functional integral is welldefined and can be used to determine the Minkowski vacuum state

This technique can be extended to quantum gravity namely to adiffeomorphism-invariant context The effect of diffeomorphism invarianceis that the functional W turns out to be independent of the location of ΣAt first sight this seems to leave us in the characteristic interpretative ob-scurity of background-independent QFT the independence of W from Σ isequivalent to the independence of W (x y) from x and y mentioned above

But a closer look reveals it is not so The boundary field includes thegravitational field which is the metric and therefore the argument of Wdoes describe the metric of the boundary surface that is the relativespacetime location of the detectors as explained in Section 412 There-fore the relative location of the detectors lost with Σ because of generalcovariance comes back with ϕ as this now includes the boundary valueof the gravitational field The boundary value of the gravitational fieldplays the double role previously played by ϕ and Σ In fact this is pre-cisely the core of the conceptual novelty of general relativity there is no apriori distinction between localization measurements and measurementsof dynamical variables

More formally in a background-dependent theory the space G is a spaceof couples (Σ ϕ) but in a general-relativistic theory the space G is just aspace of fields on a closed differential surface In pure GR we can take Gas the space of the gravitational connections A on a closed surface Ac-cordingly the space K is a space of functionals of the field A on a closedsurface These functionals are invariant under 3d diffeomorphisms of thesurface In the second part of the book the space K will be built explic-itly As explained in the previous section the functional W determines a


preferred state |0〉 in K This is the covariant vacuum state which containsthe dynamical information of the theory

A key result of the theory developed in the second part of the book isthat the eigenstates of the gravitational field on a 3d surface are notsmooth fields They present a characteristic Planck-scale discretenessThese eigenstates determine a preferred basis |s〉 in K labeled by theldquospin networksrdquo s that will be described in detail in the second part ofthe book Each state |s〉 describes a ldquoquantum geometry of spacerdquo namelythe possible result of a complete measurement of the gravitational fieldon the 3d surface We shall express W in this preferred basis

W (s) = 〈0|s〉 (5161)

Therefore because of the Planck-scale discreteness of space in the gravi-tational context the analog of W [ϕΣ] is the functional W (s) A definitionof W (s) in the canonical quantum theory will be given below in (737)As we shall see the covariant vacuum state |0〉 will simply be related tothe spin network state with no nodes and no links A sum-over-historiesdefinition of W (s) will be given below in (921)

A case of particular interest is the one in which we can separate theboundary surface Σ into two components For instance these can be dis-connected Accordingly we can write s as (sout sin) and the associatedamplitude as

W (sout sin) = 〈0|sout sin〉 = 〈sout|P |sin〉 (5162)

where P is the projector on the solutions of the WheelerndashDeWitt equa-tion A sum-over-histories expression of W (sout sin) is given in terms ofhistories that go from sin to sout

542 Much ado about nothing the vacuum

The notion of ldquovacuum staterdquo plays a central role in QFT on a backgroundspacetime The vacuum is the basis over which Fock space is built In grav-ity on the other hand the notion of vacuum is very ambiguous This factcontributes to make quantum gravity sharply different from conventionalQFTs However this is not a difficulty a preferred notion of vacuum isnot needed for a quantum theory to be well defined The quantum theoryof a harmonic oscillator has a vacuum state but the quantum theory ofa free particle does not In this respect general relativity resembles morea free particle than a harmonic oscillator

Notice that even the terminology of classical GR is confusing with re-spect to the notion of vacuum in relativistic parlance all solutions of theEinstein equations without a source term are called ldquovacuum solutionsrdquo


We use three distinct notions of vacuum in quantum gravity

Covariant vacuum The first is the nonperturbative or covariant vac-uum state |0〉 defined in Sections 514 and 532 This is the statein the boundary state space that defines the dynamics Intuitivelyit is defined by the sum-over-histories on a region bounded by thegiven boundary data If the metric boundary data are chosen to bespacelike this is the HartlendashHawking state In the context we areconsidering instead the boundary surface bounds a finite 4d regionof spacetime and the state |0〉 is a background-independent way ofcoding quantum dynamics

Empty state The state |empty〉 is the kinematical quantum state of thegravitational field in which the volume of space is zero namely inwhich there is no physical space As we shall see it is related to thecovariant vacuum state |0〉

Minkowski vacuum A different notion of vacuum is the Minkowskivacuum state |0M〉 The quantum state |0M〉 that describes theMinkowski vacuum is not singled out by the dynamics alone In-stead it is singled out as the lowest eigenstate of an energy HT

which is the variable canonically conjugate to a nonlocal functionof the gravitational field defined as the proper time T along a givenworldline This is analogous to the identification of the energy witha momentum p0 under the choice of a specific Lorentz time x0 Tofind this state in quantum gravity we can use the procedure em-ployed in (549) and (5139) This will be briefly discussed at theend of Chapter 9 Alternatively in an asymptotically flat contextwe expect |0M 〉 to be the lowest eigenstate of the ADM energy

The notion of vacuum is strictly connected to the notion of energy Thevacuum can be defined as the state with lowest energy In GR the notionof energy is ambiguous and the ambiguity in the definition of energy isreflected in the ambiguity in defining the vacuum Indeed we can identifyseveral notions of energy in GR

Canonical energy The canonical energy namely the generator H oftranslations in coordinate time vanishes identically in any general-relativistic theory In this sense all physical states of quantum grav-ity are vacuum states

Matter energy The energy-momentum tensor T Iμ of the nongravita-

tional fields is well defined and therefore the energy Ematter = T 00

of the nongravitational fields is well defined In classical GR a vac-uum solution is a solution with Ematter = 0 In this sense vacuumstates are all the pure gravity physical states without matter


Gravitational energy The energy of the gravitational field Egravity isstrictly speaking (minus) the left-hand side of the timendashtime com-ponent of the Einstein equations so the timendashtime component ofthe Einstein equations reads Egravity+Ematter = 0 That is the totalenergy vanishes see for instance [147]

ADM energy We can associate an energy EADM to an isolated sys-tem surrounded by a region where the gravitational field is approxi-mately minkowskian Such a system can be described by asymptot-ically flat solutions of the Einstein equations For such a system wecan identify the energy with the generator EADM of time transla-tions in the asymptotic Minkowski space Given asymptotic flatnessEADM is minimized by the Minkowski solution In this sense theMinkowski solution is ldquothe vacuumrdquo of the asymptotic minkowskiantheory

The fact that the notions of energy and vacuum are so ambiguous in GRshould not be disconcerting There is nothing essential in these notions aquantum theory and its predictions are meaningful also in the absence ofthem The notions of energy and vacuum play an important role in non-general-relativistic physics just because of the accidental fact that we livein a region of the Universe which happens to have a peculiar symmetrytranslation invariance in newtonian or special-relativistic time

55 Complements

551 Thermal time hypothesis and Tomita flow

The thermal time hypothesis discussed in Section 34 extends nicely toQM and very nicely to QFT

QM In QM the time flow is given by

At = αt(A) = eitH0AeminusitH0 (5163)

A statistical state is described by a density matrix ρ It determines theexpectation values of any observable A via

ρ[A] = tr[Aρ] (5164)

This equation defines a positive functional ρ on the observablesrsquo algebraThe relation between a quantum Gibbs state ρ0 and H0 is the same as in(3202) That is

ρ0 = NeminusβH0 (5165)

55 Complements 205

Correlation probabilities can be written as

WAB(t) = ρ0[αt(A)B] = tr[eitH0A eminusitH0B eminusβH0 ] (5166)

Notice that it follows immediately from the definition that

ρ0[αt(A)B] = ρ0[α(minustminusiβ)(B)A] (5167)

namely

WAB(t) = WBA(minustminus iβ) (5168)

A state ρ0 over an algebra satisfying the relation (5167) is said to beKMS (KubondashMartinndashSchwinger) with respect to the flow αt

We can generalize easily the thermal time hypothesis Given a genericstate ρ the thermal hamiltonian is defined by

Hρ = ln ρ (5169)

and the thermal time flow is defined by

Atρ = αtρ(A) = eitρHρAeminusitρHρ (5170)

ρ is a KMS state with respect to the thermal time flow

QFT Tomita flow In QFT finite-temperature states do not live in thesame Hilbert space as the zero-temperature states H0 is a divergent oper-ator on these finite-temperature states This is to be expected since in athermal state there is a constant energy density and therefore a divergingtotal energy H0 Therefore (5165) makes no sense in QFT How thendo we characterize the Gibbs states The solution to this problem is wellknown equation (5167) can still be used to characterize a Gibbs stateρ0 in the algebraic framework and can be taken as the basic postulate ofstatistical QFT a Gibbs state ρ0 over an algebra of observables is a KMSstate with respect to the time flow α(t)

It follows that if we want to extend the thermal time hypothesis tofield theory we cannot use (5169) Can we get around this problem Isthere a flow αtρ which is KMS with respect to a generic thermal stateρ Remarkably the answer is yes A celebrated theorem by Tomita statesprecisely that given any2 state ρ over a von Neumann algebra3 there isalways a flow αt called the Tomita flow of ρ such that (5167) holds

2Any separating state ρ A separating density matrix has no zero eigenvalues This isthe QFT equivalent of the condition stated in footnote 11 of Chapter 3

3The observablesrsquo algebra is in general a Clowast algebra We obtain a von Neumann algebraby closing in the Hilbert norm of the quantum state space


This theorem allows us to extend (3205) to QFT the thermal timeflow αtρ is defined in general as the Tomita flow of the statistical state ρ

Thus the thermal time hypothesis can be readily extended to QFTwhat we call the ldquoflow of timerdquo is simply the Tomita flow of the statisticalstate ρ in which the world happens to be when it is described in termsof macroscopic parameters

The flow αtρ depends on the state ρ However a von Neumann algebra possessesalso a more abstract notion of time flow independent of ρ This is given by the one-parameter group of outer automorphisms formed by the equivalence classes of auto-morphisms under inner (unitary) automorphisms Alain Connes has shown that thisgroup is independent of ρ It only depends on the algebra itself Connes has stressedthe fact that this group provides an abstract notion of time flow that depends only onthe algebraic structure of the observables and nothing else

The thermal time hypothesis and the notion of thermal time have notyet been extensively investigated They might provide the key by whichto relate timeless fundamental mechanics with our experience of a worldevolving in time

552 The ldquochoicerdquo of the physical scalar product

The solutions of the WheelerndashDeWitt equation (564) form the linear space H Thisspace is naturally equipped with a scalar product that makes it a Hilbert space Thisscalar product is often denoted the ldquophysicalrdquo scalar product in order to distinguishit from the scalar product in K denoted the ldquokinematicalrdquo scalar product

The relation between kinematical and physical scalar product depends on the hamil-tonian H The space H is the eigenspace of H corresponding to the eigenvalue zeroIn order for solutions to exist the spectrum of H must therefore include zero If zerois part of the discrete spectrum of H then H is a proper subspace of K that is thesolutions of the WheelerndashDeWitt equation (564) are normalizable states in K In thiscase the physical scalar product is the same as the kinematical scalar product andthere is no complication But if zero is part of the continuum spectrum of H then H isformed by generalized eigenvectors which are in S prime and not in K That is the solutionsof the WheelerndashDeWitt equation (564) are nonnormalizable states in K In this casethe physical scalar product is different from the kinematical scalar product What isit

In the quantum gravity (and quantum cosmology) literature there is a certain con-fusion regarding the issue of the definition of the physical scalar product For instanceone often reads that this issue has to do with the notion of time This is a conceptualmistake that derives from the observation that in a nonrelativistic theory there is a pre-ferred time variable and the problem of defining H starting from K does not appearBut the fact that the issue of defining the product in H appears in timeless systemsdoesnrsquot imply that it cannot be resolved unless there is a time variable

In fact there are a large number of solutions to this issue all essentially equivalentPreferences vary here are some of the solutions proposed

(i) The scalar product can be defined on H using the matrix elements of the projectoras illustrated above

55 Complements 207

(ii) Here is a general theorem on the issue If H is a self-adjoint operator on a Hilbertspace K then we can write

K =

int

S

ds Hs (5171)

Here S is the spectrum of H ds is a measure on this spectrum and Hs is a family ofHilbert spaces labeled by the eigenvalues s The meaning of this integral over Hilbertspaces is the following any vector ψ isin K can be written as a family ψs where forevery s ψs isin Hs and

(ψ φ)K =

int

S

ds (ψs φs)Hs (5172)

where ( )H is the scalar product in the Hilbert space H and in this instance theintegral is a standard numerical integral The relevance of this theorem is that it statesthat there is a Hilbert space H0 That is a scalar product on the space of the solutionsof Hψ = 0

Here is a simple example of how the theorem works Consider the space K =L2[R

2 dxdy] and the self-adjoint operator H = minusiddy The solutions of Hψ = 0or

minusid

dyψ(x y) = 0 (5173)

are functions ψ(x y) constant in y and are nonnormalizable in K However the decom-position (5171) (5172) is immediate

K =

int

R

dy Hy (5174)

where H(y) = L2[Rdx] In fact

(ψ φ)K =

int

R2dx dy ψ(x y) φ(x y) =

int

R

dy (ψy φy)Hy (5175)

where ψy(x) = ψ(x y) and

(ψy φy)Hy =

int

R

dx ψy(x) φy(x) (5176)

The space of the solutions of (5173) is H(0) and has the natural Hilbert structureH(0) = L2[R dx]

(iii) Here is another solution Pick a set of self-adjoint operators Ai in K thatcommute with H These are well defined on the space H because if Hψ = 0 thenH(Aiψ) = AiHψ = 0 Now require that the operators Ai be self-adjoint in the phys-ical scalar product For a sufficient number of operators this requirement fixes thescalar product of H

In the example given in (ii) above the obvious self-adjoint operators that commutewith H = minusiddy are x and minusiddx These are well defined on the space of the functionsof x alone There is only one scalar product on this space of functions that makes xand minusiddx self-adjoint the one of L2[R dx]

(iv) A convenient way of addressing the problem especially in the case in which His not a single operator but has many components is given by the ldquogroup averagingrdquotechnique Assume the WheelerndashDeWitt equation has the form Hiψ = 0 where theself-adjoint operators Hiψ = 0 are the generators of a unitary action of a group U on


K Assume also that S is invariant under this action and that we can find an invariantmeasure on the group or at least on the orbit of the group in K Then we can generalizethe operator P S rarr H of (558)

P =

int

U

dτ U(τ) (5177)

and write the physical scalar product as

(Pψ Pφ)H equiv (Pψ)(φ) =

int

U

dτ (ψ|U(τ)|φ)K (5178)

There certainly are other techniques as well This is a field in which the same ideashave independently reappeared many times under different names (and with differentlevels of mathematical precision) All these techniques are generally equivalent If thereis a case in which they differ wersquoll have to resort to physical arguments to find thephysically correct choice

553 Reality conditions and scalar product

Section 327 illustrated the possibility of using mixed complex and real dynamicalvariables a strategy that will turn out to be useful in GR Here I illustrate whathappens with the same choice in quantum theory In particular I illustrate the key rolethat the reality conditions play in quantum theory Recall the simple example discussedin Section 327 a free particle described in the coordinates x and z = xminus ip We canwrite the quantum theory in terms of wave functionals ψ(z) of the complex variable zThe Schrodinger equation gives immediately (see (3134))

ihpartψ(z t)

partt= H0

(hpart

partz z

)ψ(z t) = minus 1

2m

(hpart

partzminus z

)2

ψ(z t) (5179)

A complete family of solutions is given by

ψk(z t) = eminusihS(ztk) (5180)

where S(z t k) is given in (3135) Observe now that in the quantum theory the realitycondition (3138) becomes a relation between operators

z + zdagger = 2hpart

partz (5181)

Notice that classical complex conjugation is translated into the adjoint operation thisis necessary in order for real quantities to be represented by self-adjoint operators Now(5181) makes sense only after we have specified the scalar product because the daggeroperation is defined in terms of and therefore depends on the scalar product Indeedrequiring the reality condition (5181) to hold amounts to posing a condition on thescalar product of the theory Let us search for a scalar product of the form

(ψ φ) =

intdzdz f(z z) ψ(z) φ(z) (5182)

where f is a function to specify Imposing (5181) gives the condition on f

(z + z)f(z z) = minus2hpart

partzf(z z) (5183)

This gives

f(z z) = eminus(z+z)24h (5184)

56 Relational interpretation of quantum theory 209

Let us check whether the states (5180) are well defined with respect to this productInserting (5180) (at t = 0 for simplicity) and (5184) in (5182) gives

(ψk ψkprime) =

intdzdz eminus(z+z)24h e

ih

(kzminus i2 z2) eminus

ih

(kprimez+ i2 z2) (5185)

A simple change of variables shows that the integral in the imaginary part of z is finiteand the integral in the real part of z is proportional to δ(k kprime) Therefore the statesψk form a standard continuous orthogonal basis of generalized states They are clearlyeigenstates of the momentum since

pψk = i(xminus z)ψk = i

(part

partzminus z

)ψk = kψk (5186)

In fact what we have developed is a simple rewriting of the standard Hilbert space ofa quantum particle

Notice that appearances can be misleading For instance for k = 0 the state ψk

readsψ0(z t) = e+z22h (5187)

This looks like a badly nonnormalizable state but it is not It is a well-defined general-ized state since the negative exponential in the measure compensates for the positiveexponential in the state

56 Relational interpretation of quantum theory

Quantum mechanics is one of the most successful scientific theories everHowever its interpretation is controversial What does the theory actuallytell us about the physical world This question sparked off a lively debatewhich was intense during the 1930s the early days of the theory and isgenerating new interest today

The possibility that the interpretation of an empirically successful the-ory could be debated should not surprise examples abound in the historyof science For instance the great scientific revolution was fueled by thedebate on whether the efficacy of the copernican system should be takenas an indication that the Earth really moves In more recent times Ein-steinrsquos celebrated contribution to special relativity consisted to a largeextent just in understanding the physical interpretation (simultaneityis relative) of an already existing effective mathematical formalism (theLorentz transformations) In these cases as in the case of quantum me-chanics an overly strictly empiricist position could have circumventedthe problem altogether by reducing the content of the theory to a list ofpredicted numbers But science would not then have progressed

Quantum theory was first constructed for describing microscopic ob-jects (atoms electrons photons) and the way these interact with macro-scopic apparatuses built to measure their properties Such interactionswere called ldquomeasurementsrdquo The theory is formed by a mathematicalformalism which allows probabilities of alternative outcomes of such mea-surements to be calculated If used just for this purpose the theory raises


no difficulty But we expect the macroscopic apparatuses themselves ndash infact any physical system in the world ndash to obey quantum theory andthis seems to raise contradictions within the theory Here I discuss theseapparent contradictions and a possible resolution This resolution offersa precise answer to the question of what the quantum theory actually tellsus about the physical world

561 The observer observed

Measurements A ldquomeasurementrdquo of the variable A of a system S is aninteraction between the system S and another system O whose effecton O depends on the value that the variable A has at the time of theinteraction We say that the variable A is ldquomeasuredrdquo and that its valuea is the ldquooutcome of the measurementrdquo For instance let S be a particlethat impacts on O let the effect of this impact depend on the positionof the particle and let q be the value of the position at the moment ofthe impact Then we say that the position Q is measured and that theoutcome of the measurement is q

The term ldquomeasurementrdquo and the common terminology used to de-scribe measurement situations (S for ldquoSystemrdquo and O for ldquoObserverrdquo)are very misleading because they evoke a human intentionally ldquoobservingrdquoS and using an apparatus to gather data about it There is nothingldquohumanrdquo or ldquointentionalrdquo in the definition of measurement given aboveThe system O does not need to be human nor to be a special ldquoapparatusrdquonor to be macroscopic The measured value need not be stored Any in-teraction between two physical systems is a measurement The measuredvariable of the system S is the variable that determines the effect that theinteraction has on O This is true in classical as well as in quantum theory

Classical states and quantum states In classical mechanics a system Sis described by a certain number of physical variables ABC For in-stance a particle is described by its position Q and velocity V Thesevariables change with time They represent the contingent properties ofthe system We say that the values of these variables determine at everymoment the ldquostaterdquo of the system If the value of the position Q of theparticle is q and the value of its velocity V is v we say that the state is(q v) In classical mechanics a state is therefore a list of values of physicalvariables

Quantum mechanics differs from classical mechanics because it assumesthat the variables of the system do not have a determined value at alltimes Werner Heisenberg introduced this key idea According to quantumtheory an electron does not have a well-determined position at every


time When it is not interacting with an external system sensitive to itsposition the electron can be ldquospread outrdquo over different positions It is ina ldquoquantum superpositionrdquo of different positions

It follows that in quantum mechanics the state of the system cannotbe captured by giving the value of its variables Instead quantum theoryintroduces a novel notion of the ldquostaterdquo of a system different from theclassical list of variable values The new notion of ldquoquantum staterdquo wasintroduced in the work of Erwin Schrodinger in the form of the ldquowavefunctionrdquo of the system Paul Adrien Maurice Dirac gave it its generalabstract formulation in terms of a vector Ψ moving in an abstract vectorspace From the knowledge of the state Ψ we can compute the probabilityof the different measurement outcomes a1 a2 of any variable A Thatis the probability of the different ways in which the system S can affecta system O in the course of an interaction

The theory prescribes that at every such measurement we must updatethe value of Ψ to take into account which of the different outcomes hasbeen realized This sudden change of the state Ψ depends on the outcomeof the measurement and is therefore probabilistic This is the ldquocollapse ofthe wave functionrdquo

The notion of ldquostate of the systemrdquo of classical mechanics is there-fore split into two distinct notions in quantum theory (i) the state Ψthat expresses the probability for the different ways the system S caninteract with its surroundings and (ii) the actual sequence of valuesq1 q2 q3 that the variables of S take in the course of the interac-tions These are the called ldquomeasurement outcomesrdquo I prefer calling themldquoquantum eventsrdquo

We can either think that Ψ is a ldquorealrdquo entity or that it is nothing morethan a theoretical bookkeeping for the quantum events which are theldquorealrdquo events The choice of the relative ontological weight we attribute tothe state Ψ or the quantum events q1 q2 q3 is a matter of convenienceempirical evidence alone does not uniquely determine what is ldquorealrdquoI think the second choice is cleaner but in the following I refer to both

The observer observed The key problem of the interpretation of quantummechanics is illustrated by the following situation Consider a real physicalsituation illustrated in Figure 51 in which at some time t a system Ointeracts with a system S and then at a later time tprime a third system Oprime

interacts with the coupled system [S + O] formed by S and O togetherLet the effect on O of the first interaction depend on the variable A ofthe system S and the effect on Oprime of the second interaction depend onthe variable B of the coupled system [S+O] (That is we can say that O


Fig 51 The observer observed

measures the variable A of S at time t and then Oprime measures the variableB of [S +O] at time tprime) Before the first interaction say S was in a statewhere A is a quantum superposition of two values a1 and a2 Say thatat the first interaction O measures the value a1 of the variable O Thepuzzling question can be formulated in various equivalent manners

bull What is the state of S and O between the two interactions

bull Has the quantum event a1 happened or not

bull Does the quantity A have a determined value after the first interac-tion or not

Say that before the first interaction the state of S was Ψ = c1Ψ1 +c2Ψ2

where Ψ1Ψ2 are states where A has values a1 a2 respectively Then attime t we have

c1Ψ1 + c2Ψ2 rarr Ψ1

A takes the value a1(5188)

However the system O obeys the laws of quantum theory as well There-fore we can also give a quantum description of the evolution of the coupledquantum system (S + O) formed by S and O together If we do so no


collapse happens Instead the effect of the interaction is the Schrodingerevolution

(c1Ψ1 + c2Ψ2

)otimes Φ rarr

(c1Ψ1 otimes Φ1 + c2Ψ2 otimes Φ2

)

A is still in the superposition of the two values a1 a2

(5189)

for suitable states ΦΦ1Φ2 of OWhat is real seems to depend on how we choose to describe the world

What is the real state of affairs of the world after the interaction betweenS and O (5188) or (5189) In either case we get a difficulty If we saythat after t the state has collapsed as in (5188) and A has the value a1we get the wrong predictions about the second measurement at time tprimeIn fact quantum theory allows us to predict the probability distributionof the possible outcomes of the second measurement but to computethis we have to use the state (5189) and not the state (5188) Indeedif B and A do not commute this probability distribution can be affectedby the interference between the two different ldquobranchesrdquo in (5189) Inother words we have to assume that the variable A was in a quantumsuperposition of the values a1 a2 and not determined

But if we do so and say that after the first measurement the state is(5189) then we must say that A has no determined value at time t Butthe situation is general any measurement can be thought of as the firstmeasurement of the example and therefore we must conclude that novariable can take a definite value ever

Thus we seem to get a contradiction in both cases whether we thinkthat the wave function has collapsed and a1 was realized or whether wethink it hasnrsquot This is the core of the difficulty of the interpretation ofquantum theory

Real wave functions or real quantum events Let us examine the abovedifficulty in a bit more detail from the two points of view of the twopossible ontologies of quantum theory

If we think that Ψ is real but it never truly collapses there is no sim-ple and compelling reason why the world should appear as described byvalues of physical quantities that take determined values at each interac-tion as it does We experience particles in given positions not particlewavefunctions The relation between a noncollapsing wavefunction ontol-ogy and our experience of the world is very indirect and involuted Weneed some complicated story to understand how specific observed valuesq1 q2 q3 can emerge from the sole Ψ If this story is given (which ispossible) we are then in a situation similar to the one of a quantum eventontology to which I now turn


I think it is preferable to take the quantum events as the actual elementsof reality and view Ψ just as a bookkeeping device coding the events thathappened in the past and their consequences For instance I prefer to saythat the ldquorealityrdquo of a subatomic particle is expressed by the sequence ofthe positions of the particle revealed by the bubbles in a bubble chambernot by the spherically symmetric wave function emerging from the inter-action area The reality of the electron is in the events where it revealsitself interacting with its surrounding not in the abstract probabilityamplitude for such events From this perspective the real events of theworld are the ldquorealizationsrdquo (the ldquocoming to realityrdquo the ldquoactualizationrdquo)of the values q1 q2 q3 in the course of the interactions between physicalsystems These quantum events have an intrinsically discrete ldquoquantizedrdquogranular structure

This perspective however does not solve the above puzzle either Thekey puzzle of quantum mechanics becomes the fact that the statementthat the quantum event a1 ldquohas happenedrdquo can be at the same time trueand not-true has the quantum event a1 happened or not If we answer nothen we are forced to say that no quantum event ever happens becausethe situation described above is completely general any quantum eventhappening in the interaction of two systems S and O is ldquonon-happeningrdquoas far as the effect of (S + O) on a further system Oprime is concerned If wesay yes then we contradict the predictions of quantum mechanics (aboutthe second interaction)

The ldquosecond observerrdquo puzzle captures the core conceptual difficultyof the interpretation of quantum mechanics reconciling the possibilityof quantum superposition with the fact that the world we observe anddescribe is characterized by determined values of physical quantities Moreprecisely the puzzle shows that we cannot disentangle the two accordingto the theory a quantum event (a1) can be at the same time realized andnot realized

A possible escape from the puzzle is to assume that there are ldquospecialrdquosystems that produce the collapse and cause quantum events to happenFor instance these could be ldquomacroscopicrdquo systems or ldquosufficiently com-plexrdquo systems or ldquosystems with memoryrdquo or the ldquogravitational fieldrdquo orhuman ldquoconsciousnessrdquo All these systems and others have been sug-gested as causing quantum collapse and generating quantum events Ifthis were correct at some point we shall be able to measure violations ofthe predictions of QM That is QM as we know it would break down forthose systems

So far this breaking down of QM has never been observed We canfancy a phenomenology that we have not yet observed that could bringback reality to the way we used to think it is It is certainly worthwhile to


investigate this possibility theoretically and experimentally But we shouldnot forget that reality might be truly different from what we thought andmight be simply demanding us to renounce some old prejudice I thinkthat the history of physics indicates that the productive attitude is not toresist the conceptual novelty of empirically successful theories but ratherto make an effort to understand it We should not force reality into ourprejudices but rather try to adapt our conceptual schemes to what welearn about the world

562 Facts are interactions

I think that the key to the solution of the difficulty can be found inthe observation that the two descriptions (5188) and (5189) refer todifferent systems the first to O the second to Oprime More precisely the firstis relevant for describing the effects of interactions on O the second fordescribing the effects of interactions on Oprime

The solution of the puzzle can be found in the idea that quantum eventsare the elements of reality but they are always relative to a physicalsystem the quantum event a1 happens with respect to O but it does nothappen with respect to Oprime

In other words the way out from the puzzle is that the values of thevariables of any physical system are relational They do not express aproperty of the system S alone but rather refer to the relation betweenthis system and another system The variable A has value a1 with respectto O but it has no determined values with respect to Oprime This pointof view is called the relational interpretation of quantum mechanics orsimply relational quantum mechanics

The central idea of relational quantum mechanics is that there is nomeaning in saying that a certain variable of the system S takes the valueq There is only meaning in saying that a variable has value q with respectto a system O In the example discussed above for instance the fact thatA takes the value a1 with respect to O does not imply that A has thevalue a1 also with respect to Oprime

If we avoid all statements that are not referred to a physical systemwe can get rid of all apparent contradictions of quantum theory Theapparent contradiction between the two statements that a variable hasor hasnrsquot a value is resolved by referring the statements to the differentsystems with which the system in question interacts If I observe anelectron at a certain position I cannot conclude that the electron isthere I can only conclude that the electron as seen by me is there

Indeed quantum theory must be understood as an account of the waydistinct physical systems affect one another when they interact and not


the way physical systems ldquoarerdquo This account exhausts all that can besaid about the physical world The physical world can be described asa network of interacting components where there is no meaning to ldquothestate of an isolated systemrdquo The state of a physical system is the networkof its relationships with the surrounding systems The physical structureof the world is identified with this network of relationships

The unique account of the state of the world of the classical theoryis thus shattered into a multiplicity of accounts one for each possibleldquoobservingrdquo physical system Quantum mechanics is a theory about thephysical description of physical systems relative to other systems and thisis a complete description of the world

Of course we can pick a system O once and for all as ldquothe observersystemrdquo and be concerned only with the effects of the rest of the worldon this system Each interaction between the rest of the world and O iscorrectly described by standard quantum mechanics In this descriptionthe quantum state Ψ collapses at each interaction with O This descriptionis completely self-consistent but it treats O as if it were a special systema classical nonquantum system If we want to describe O itself quantummechanically we can but we have to pick a different system Oprime as theobserver and describe the way O interacts with Oprime In this descriptionthe quantum properties of O are taken into account but not the ones ofOprime because this description describes the effects of the rest of the worldon Oprime

Consistency This relativisation of actuality is viable thanks to a remark-able property of the formalism of quantum mechanics

John von Neumann was the first to notice that the formalism of thetheory treats the measured system (S) and the measuring system (O) dif-ferently but the theory is surprisingly flexible on the choice of where to putthe boundary between the two Different choices give different accounts ofthe state of the world (for instance the collapse of the wave function hap-pens at different times) but this does not affect the predictions about thefinal observations This flexibility reflects a general structural propertyof quantum theory which guarantees consistency among all the distinctldquoaccounts of the worldrdquo of the different observing systems The mannerin which this consistency is realized however is subtle

As a simple illustration of this phenomenon consider the case in whicha system O with two states Φ1 and Φ2 (say a light-bulb which can be onor off ) interacts with a system S with two states Ψ1 and Ψ2 (say thespin of the electron which can be up or down) Assume the interactionis such that if the spin is up (down) the light goes on (off ) To start


with the electron can be in a superposition of its two states In theaccount of the state of the electron that we can associate with the lightthe wave function of the electron collapses to one of two states duringthe interaction as in (5188) and the light is then either on or off Butwe can also consider the lightelectron composite system as a quantumsystem and study the interactions of this composite system with anothersystem Oprime In the account associated to Oprime there is no collapse at the timeof the interaction and the composite system is still in the superpositionof the two states [spin uplight on] and [spin downlight off ] after theinteraction as in (5189) As remarked above it is necessary to assumethis superposition because it accounts for measurable interference effectsbetween the two states if quantum mechanics is correct these interferenceeffects are truly observable by Oprime

So we have two discordant accounts of the same events the one associ-ated to O where the spin has a determined value and the one associatedto Oprime where the spin is in a superposition Now can the two discordantaccounts be compared and does the comparison lead to a contradiction

They can be compared because the information on the first accountis stored in the state of the light and Oprime has access to this informationTherefore O and Oprime can compare their accounts of the state of the worldHowever the comparison does not lead to a contradiction because the com-parison is itself a physical process that must be understood in the contextof quantum mechanics

Indeed Oprime can physically interact with the electron and then with thelight (or equivalently with the light and then with the electron) If forinstance Oprime finds the spin of the electron up quantum mechanics predictsthat the observer will then consistently find the light on because in thefirst measurement the state of the composite system collapses on its [spinuplight on] component namely on the first term of the right-hand sideof (5189)

That is the multiplicity of accounts does not lead to a contradictionprecisely because the comparison between different accounts can only be aphysical quantum interaction Many common paradoxes of quantum me-chanics follow from assuming that the communication between differentobservers violates quantum mechanics4 This internal self-consistency ofthe quantum formalism is general and is perhaps its most remarkable

4The EPR (EinsteinndashPodolskindashRosen) apparent paradox might be among these Thetwo observers far from each other are physical systems The standard account neglectsthe fact that each of the two is in a quantum superposition with respect to the otheruntil the moment they physically communicate But this communication is a physicalinteraction and must be strictly consistent with causality


aspect5 This self-consistency is a strong indication of the relational na-ture of the world

563 Information

What appears with respect to O as a measurement of the variable A (witha specific outcome) appears with respect to Oprime simply as a dynamicalprocess that establishes a correlation between S and O As far as theobserver O is concerned the variable A of a system S has taken a certainvalue As far as the second observer Oprime is concerned the only relevantelement of reality is that a correlation is established between S and O

Concretely this correlation appears in all further observations that Oprime

would perform on the [S + O] system That is the way the two systemsS and O will interact with Oprime is characterized by the fact that thereis a correlation Oprime will find some properties of O correlated with someproperties of S

On the other hand until it physically interacts with [S+O] the systemOprime has no access to the actual outcomes of the measurements performedby O on S These actual outcomes are real only with respect to O

The existence of a correlation between the possible outcomes of a mea-surement performed by Oprime on S and the outcomes of a measurementperformed by Oprime on O can be interpreted in terms of information In factthis correlation corresponds precisely to Shannonrsquos definition of informa-tion According to this definition ldquoO has information about Srdquo meansthat we shall observe O and S in a subset of the set formed by the carte-sian product of the possible states of O and the possible states of S Thusa measurement of S by O has the effect that ldquoO has information aboutSrdquo This statement has a precise technical meaning which refers to thepossible outcomes of the observations by a third system Oprime

On the other hand if we interact a sufficient number of times with aphysical system S we can then predict (the distribution probability ofthe) future outcomes of our interactions with this system In this senseby interacting with S we can say we ldquohave informationrdquo about S (This

5In fact one may conjecture that this peculiar consistency between the observationsof different observers is the missing ingredient for a reconstruction theorem of theHilbert space formalism of quantum theory Such a reconstruction theorem is stillunavailable On the basis of reasonable physical assumptions one is able to derivethe structure of an orthomodular lattice containing subsets that form Boolean alge-bras which ldquoalmostrdquo but not quite implies the existence of a Hilbert space and itsprojectorsrsquo algebra Perhaps an appropriate algebraic formulation of the condition ofconsistency between subsystems could provide the missing hypothesis to complete thereconstruction theorem


information need not be stored or utilized but its existence is the neces-sary physical condition for being able to store it or utilize it for predic-tions)

Therefore we have two distinct senses in which the physical theory isabout information But a moment of reflection shows that the two simplyreflect the same physical reality as it affects two different systems On theone hand O has information about S because it has interacted with S andthe past interactions are sufficient to ldquogive informationrdquo namely to deter-mine (the probability distribution of) the result of future interactions Onthe other hand O has information about S in the sense that there are cor-relations in the outcomes of measurements that Oprime can make on the two

There is a crucial subtle difference that can be figuratively expressedas follows O ldquoknowsrdquo about S while Oprime only knows that O knows aboutS but does not know what O knows As far as Oprime is concerned a physicalinteraction between S and O establishes a correlation it does not selectan outcome

These observations are sufficient to conclude that what precisely quan-tum mechanics is about is the information that physical systems haveabout one another

The common unease with taking quantum mechanics as a fundamentaldescription of Nature referred to as the measurement problem can betraced to the use of an incorrect notion in the same way that uneasewith Lorentz transformations derived from the notion shown by Einsteinto be mistaken of an observer-independent time The incorrect notionthat generates the unease with quantum mechanics is the notion of anobserver-independent state of a system or observer-independent valuesof physical quantities or an observer-independent quantum event

We can assume that all systems are equivalent there is no a prioriobserverndashobserved distinction the theory describes the information thatsystems have about one another The theory is complete because thisdescription exhausts the physical world

In physics the move of deepening our insight into the physical worldby relativizing notions previously treated as absolute has been appliedrepeatedly and very successfully Here are a few examples

The notion of the velocity of an object has been understood as mean-ingless unless it is referred to a reference body with respect to whichthe object is moving With special relativity simultaneity of two distantevents has been understood as meaningless unless referred to a specificstate of motion of something (This something is usually denoted as ldquotheobserverrdquo without of course any implication that the observer is humanor has any other peculiar property besides having a state of motion Simi-larly the ldquoobserver systemrdquo O in quantum mechanics need not be humanor have any other property beside the possibility of interacting with the


ldquoobservedrdquo system S) With general relativity the position in space andtime of an object has been understood as meaningless unless it is referredto the gravitational field or to another dynamical physical entity

The step proposed by the relational interpretation of quantum mechan-ics has strong analogies with these In a sense it is a longer jump sinceall the contingent (variable) properties of all physical systems are takento be meaningful only as relative to a second physical system This is notan arbitrary step It is a conclusion which is difficult to escape followingfrom the observation ndash explained above in the example of the ldquosecond ob-serverrdquo ndash that a variable (of a system S) can have a well-determined valuea1 for one observer (O) and at the same time fail to have a determinedvalue for another observer (Oprime)

This way of thinking of the world has perhaps heavy philosophical im-plications But it is Nature that is forcing us to this way of thinking If wewant to understand Nature our task is not to frame Nature into our philo-sophical prejudices but rather to learn how to adjust our philosophicalprejudices to what we learn from Nature

564 Spacetime relationalism versus quantum relationalism

I close with a very speculative suggestion As discussed in Section 23 themain idea underlying GR is the relational interpretation of localizationobjects are not located in spacetime They are located with respect toone another In the present section I have observed that the lesson ofQM is that quantum events and states of systems are relational theymake sense only with respect to another system Thus both GR and QMare characterized by a form of relationalism Is there a connection betweenthese two forms of relationalism

Let us look closer at the two relations In GR the localization of anobject S in spacetime is relative to another object (or field) O to whichS is contiguous Contiguities or equivalently Einsteinrsquos ldquospacetime coin-cidencesrdquo are the basic relations that construct spacetime In QM thereare no absolute properties or facts properties of a system S are relativeto another system O with which S is interacting Facts are interactionsThus interactions form the basic relations between systems

But there is a strict connection between contiguity and interaction Onthe one hand S and O can interact only if they are contiguous if they arenearby in spacetime this is locality Interaction requires contiguity Onthe other hand what does it mean that S and O are contiguous Whatelse does it mean besides the fact that they can interact6 Therefore

6The very word ldquocontiguousrdquo derives from the Latin cum-tangere to touch each otherthat is to inter-act


contiguity is manifested by interacting In a sense the fact that inter-actions are local means that there is a sort of identity between beingcontiguous and interacting

Thus locality ties together very strictly the spacetime relationalism ofGR with the relationalism underlying QM It is tempting to try to developa general conceptual scheme based on this observation This could be aconceptual scheme in which contiguity is nothing else than a manifesta-tion or can be identified with the existence of a quantum interactionThe spatiotemporal structure of the world would then be directly deter-mined by who is interacting with whom This is of course very vagueand might lead nowhere but I find the idea intriguing

mdashmdash


Textbooks on quantum theory are numerous I think the best of all is thefirst of them Dirac [148] because of Diracrsquos crystal-clear thinking In theearlier editions Dirac uses a relativistic notion of state (that does notevolve in time) as is done here He calls these states ldquorelativisticrdquo as isdone here In later editions he switches to Schrodinger states that evolvein time explaining in a preface that it is easier to calculate with thesebut it is a pity to give up relativistic states which are more fundamental

I have discussed the idea that QM remains consistent also in the absenceof unitary time evolution in [98] and [149] The same idea is developed bymany authors see [26] [150] and references therein

In the past I have discussed relativistic systems only in terms of ldquoevolv-ing constantsrdquo The two-oscillators example used in the text was consid-ered in these terms in [151 152] The probabilistic interpretation of thecovariant formulation presented in this chapter is an evolution of thispoint of view and derives from [144]

I have taken the discussion on the boundary formulation of QFT from[145] The idea that quantum field theory must be formulated in termsof boundary data on a finite surface has been advocated by Robert Oeckl[153] The derivation of the local Schrodinger equation is in [146] and[154] The TomonagandashSchwinger equation was introduced in [155] On thedifficulty of a direct interpretation of the n-point functions in quantumgravity see for instance [156] The HartlendashHawking state was introducedin [157]

The possibility of defining the physical scalar product on the space ofthe solutions of the WheelerndashDeWitt equation even when these solutionsare nonnormalizable in the kinematical Hilbert space has been discussed


by many authors using a variety of techniques A nice mathematical con-struction has been given by Don Marolf see [158] and references therein

The thermal time hypothesis was extended to QM and QFT in [125]The relational interpretation presented here is discussed in [159 160]

see also [161 162] An overview of similar points of view is in the onlineStanford Encyclopedia of Philosophy [163] on a possibly related point ofview see also [164] The role of information in the foundations of quantumtheory has been stressed by John Wheeler in [165 166] For a recent dis-cussion on the role of information in the foundation of quantum theorysee for instance [167] and references therein An original and fascinat-ing point of view on the relational aspects of quantum and relativity isexplored by David Finkelstein in [168]

Part II

Loop quantum gravity

ndash Now itrsquos time to leave the capsuleif you dare

ndash This is Major Tom to Ground ControlIrsquom stepping through the doorAnd Irsquom floating in a most peculiar wayAnd the stars look very different today

David Bowie Space Oddity

6Quantum space

It is time to begin to put together the tools developed in the first part of the bookand build the quantum theory of spacetime The strategy is simple We ldquoquantizerdquothe canonical formulation of GR described at the beginning of Chapter 4 accordingto the relativistic QM formalism detailed in Chapter 5 This chapter deals with thekinematical part of the theory states partial observables and their eigenvalues Thenext chapter deals with dynamics namely with transition amplitudes

61 Structure of quantum gravity

In Chapter 4 we have seen that GR can be formulated as the dynamicalsystem defined by the HamiltonndashJacobi equation (49)

F ijab(τ)

δS[A]δAi

a(τ)δS[A]

δAjb(τ)

= 0 (61)

where the functional S[A] is defined on the space G of the 3d SU(2)connections Ai

a(τ) and is invariant under internal gauge transformationsand 3d diffeomorphisms That is

δfAia(τ)

δS[A]δAi

a(τ)= 0 δλA

ia(τ)

δS[A]δAi

a(τ)= 0 (62)

where the variations δfAia and δλA are given in (412) Equivalently the

theory is defined by the hamiltonian H[AE] = F ijab E

ai E

bj on T lowastG

Following the prescription of Chapter 5 a quantization of the theorycan be obtained in terms of complex-valued Schrodinger wave functionalsΨ[A] on G The quantum dynamics is inferred from the classical dynamicsby interpreting S[A] as times the phase of Ψ[A] Namely interpretingthe classical HamiltonndashJacobi theory as the eikonal approximation of aquantum wave equation as in [142] semiclassical ldquowave packetsrdquo will thenbehave according to the classical theory This can be obtained defining

225

226 Quantum space

the quantum dynamics by replacing derivatives of the HamiltonndashJacobifunctional S[A] with derivative operators The two equations (62) remainunchanged they simply force Ψ[A] to be invariant under SU(2) gaugetransformations and 3d diffeomorphisms Equation (61) gives

F ijab (τ)

δ

δAia(τ)

δ

δAjb(τ)

Ψ[A] = 0 (63)

This is the WheelerndashDeWitt equation or EinsteinndashSchrodinger equationIt governs the quantum dynamics of spacetime In other words the dy-namics is defined by the hamiltonian operator H = H[Aminusi δδA]

More precisely we want a rigged Hilbert space S sub K sub S prime where S is asuitable space of functionals Ψ[A] Partial observables are represented byself-adjoint operators on K Their eigenvalues describe the quantization ofphysical quantities The operator P formally given by the field theoreticalgeneralization of (558)

P simint

[DN ] eminusiint

d3τ N(τ)H(τ) (64)

sends S to the space of the solutions of (63) Its matrix elements be-tween eigenstates of partial observables define the transition amplitudesof quantum gravity These determine all (probabilistic) dynamical rela-tions between any measurement that we can perform

A preferred state in K is |empty〉 the eigenstate of the geometry with zerovolume and zero area The covariant vacuum is given by |0〉 = P |empty〉 If weassume that the surface Σ coordinatized by τ is the entire boundary of afinite spacetime region then we can identify K with the boundary spaceK The correlation probability amplitude associated with a measurementof partial observables on the boundary surface is A = W (s) = 〈0|s〉where |s〉 is the eigenstate of the partial observables corresponding to themeasured eigenvalues

We must now build this structure concretely As we shall see in a precisetechnical sense this structure is unique

62 The kinematical state space KI construct here the quantum state space defined by the real connection (see Section42) There are three reasons for this First the physical lorentzian theory can beformulated in terms of this connection the only difference is that the hamiltonianoperator is slightly more complicated than (63) as explained in Section 422 Secondthings are far easier with the real connection and it is better to do easy things first inthe construction of the quantum state space defined by the complex connection thereare still some open technical complications [169] Third the real connection with thehamiltonian operator (63) defines the quantum euclidean theory namely the quantumtheory which has the theory defined in Section 421 as its classical limit this is aninteresting model by itself and is likely to be related to the physical theory as wasdiscussed in Section 42

62 The kinematical state space K 227

Cylindrical functions Let G be the space of the smooth 3d real connec-tions A defined everywhere on a 3d surface Σ except possibly at isolatedpoints1 Fix the topology of Σ say to a 3-sphere I now define a space Sof functionals on G

We are now going to make use of the geometric interpretation of thefield A as a connection The so(3) Lie algebra is the same as the su(2)Lie algebra and it is convenient to view A as an su(2) connection Let τibe a fixed basis in the su(2) Lie algebra I choose τi = minus i

2 σi where σiare the Pauli matrices (A14) Write

A(τ) = Aia(τ) τi dxa (65)

Recall from Section 215 that an oriented path γ in Σ and a connectionA determine a group element U(A γ) = P exp

intγ A called the holonomy

of the connection along the path For a given γ the holonomy U(A γ)is a functional on G Consider an ordered collection Γ of smooth orientedpaths γl with l = 1 L and a smooth function f(U1 UL) of L groupelements A couple (Γ f) defines a functional of A

ΨΓf [A] = f(U(A γ1) U(A γL)) (66)

S is defined as the linear space of all functionals ΨΓf [A] for all Γ and f We call these functionals ldquocylindrical functionsrdquo In a suitable topologywhich is not important to detail here S is dense in the space of all con-tinuous functionals of A

I call Γ an ldquoordered oriented graphrdquo embedded in Σ I call simplyldquographrdquo an ordered oriented graph up to ordering and orientation anddenote it by the same letter Γ Clearly as far as cylindrical functionsare concerned changing the ordering or the orientation of a graph is justthe same as changing the order of the arguments of the function f orreplacing arguments with their inverse

Scalar product I now define a scalar product on the space S If twofunctionals ΨΓf [A] and ΨΓg[A] are defined with the same ordered ori-ented graph Γ define

〈ΨΓf |ΨΓg〉 equivint

dU1 dUL f(U1 UL) g(U1 UL) (67)

where dU is the Haar measure on SU(2) Notice the similarity with thelattice scalar product (5152) equation (67) is the scalar product of aYangndashMills theory on the lattice Γ

1The reason for this technical choice will become clear below

228 Quantum space

The extension of this scalar product to functionals defined on the samegraph but with different ordering or orientation is obvious But also theextension to functionals defined on different graphs Γ is simple In factobserve that different couples (Γ f) and (Γprime f prime) may define the same func-tional For instance say Γ is the union of the Lprime curves in Γprime and Lprimeprime othercurves and that f(U1 ULprime ULprime+1 ULprime+Lprimeprime) = f prime(U1 ULprime)then clearly ΨΓf = ΨΓprimef prime Using this fact it is clear that we can rewriteany two given functionals ΨΓprimef prime and ΨΓprimeprimegprimeprime as functionals ΨΓf and ΨΓg

having the same graph Γ where Γ is the union of Γprime and Γprimeprime Using thisfact (67) becomes a definition valid for any two functionals in S

〈ΨΓprimef prime |ΨΓprimeprimegprimeprime〉 equiv 〈ΨΓf |ΨΓg〉 (68)

Notice that even if (67) is similar to the scalar product of a lattice YangndashMills theory the difference is profound Here we are dealing with a gen-uinely continuous theory in which the states do not live on a single latticeΓ but on all possible lattices in Σ There is no cut-off on the degrees offreedom as in lattice YangndashMills theory

Loop states and the loop transform An important example of a finite norm state isprovided by the case (Γ f) = (α tr) That is Γ is formed by a single closed curve αor a ldquolooprdquo and f is the trace function on the group We can write this state as Ψαor simply in Dirac notation as |α〉 That is

Ψα[A] = Ψαtr[A] = 〈A|α〉 = tr U(Aα) = tr Pe∮α A (69)

The very peculiar properties that these states have in quantum gravity which will beillustrated later on have motivated the entire LQG approach (and its name) The normof Ψα is easily computed from (67)

|Ψα|2 =

intdU |tr U |2 = 1 (610)

A ldquomultilooprdquo is a collection [α] = (α1 αn) of a finite number n of (possiblyoverlapping) loops A ldquomultiloop staterdquo is defined as

Ψ[α][A] = Ψα1 [A] Ψαn [A] = tr U(Aα1) tr U(Aαn) (611)

The functional on loop space

Ψ[α] = 〈Ψα|Ψ〉 (612)

is called the ldquoloop transformrdquo of the state Ψ[A] The functional Ψ[α] represents thequantum state as a functional on a space of loops This formula called the ldquolooptransformrdquo is the formula through which LQG was originally constructed (see forinstance [170]) Using the measure dμ0[A] mentioned below this can be written as

Ψ[α] =

intdμ0[A] tr Pe

∮α AΨ[A] (613)

Intuitively this is a sort of infinite-dimensional Fourier transform from the A space tothe α space


Kinematical Hilbert space Define the kinematical Hilbert space K ofquantum gravity as the completion of S in the norm defined by the scalarproduct (67)2 and S prime as the completion of S in the weak topology definedby (67)3 This completes the definition of the kinematical rigged Hilbertspace S sub K sub S prime

Why this definition The main reason is that the scalar product (67) isinvariant under diffeomorphisms and local gauge transformations (Section622 below) and it is such that real classical observables become self-adjoint operators (Section 65 below) These very strict conditions arethe ones that the scalar product must satisfy in order to give a consistenttheory with the correct classical limit Furthermore the main feature ofthis definition is that the loop states Ψα are normalizable As we shallsee later on loop states are natural objects in quantum gravity Theydiagonalize geometric observables and they are solutions of the WheelerndashDeWitt equation Hence the kinematics as well as the dynamics selectthis space of states as the natural ones in gravity

There are two objections that can be raised against the definition ofK we have given First K is nonseparable This objection would be fatalin the context of flat-space quantum field theory but it turns out tobe harmless in a general-relativistic context because of diffeomorphisminvariance Indeed the ldquoexcessive sizerdquo of the nonseparable Hilbert spacewill turn out to be just gauge the physical Hilbert space H is separable Aswe shall see in Section 64 it is sufficient to factor away the diffeomorphismgauge to obtain a separable Hilbert space Kdiff

Second loop states are normalizable in lattice YangndashMills theory butthey are nonnormalizable in continuous YangndashMills theory By analogyone might object that they should not be normalizable states in continu-ous quantum gravity either As we shall see below however this analogyis misleading again precisely because of the great structural differencebetween a diffeomorphism-invariant QFT and a QFT on a backgroundAs we shall see in continuous YangndashMills theory a loop state describesan unphysical excitation that has infinitesimal transversal physical sizeIn gravity on the other hand a loop state describes a physical excitationthat has finite (planckian) transversal physical size This will become clearbelow in Section 662

Boundary Hilbert space There are two natural ways of defining theboundary space K We can either define K = Klowast otimes K and describe thequantum geometry of a spacetime region bounded by an initial and a

2The space of the Cauchy sequences Ψn where ||Ψm minus Ψn|| converges to zero3The space of the sequences Ψn such that 〈Ψn|Ψ〉 converges for all Ψ in S

230 Quantum space

final surface or simply define K = K interpreting the closed connectedsurface Σ as the boundary of a finite 4d spacetime region

621 Structures in KThe space K has a rich and beautiful structure I mention here only a fewaspects of this structure which are important for what follows referringto the more mathematically oriented literature on this subject (see [20]and references therein) for more details

Graph subspaces The cylindrical functions with support on a given graphΓ form a subspace KΓ of K By definition KΓ = L2[SU(2)L] where L isthe number of paths in Γ The space KΓ is the (unconstrained) Hilbertspace of a lattice gauge theory with spatial lattice Γ as described inSection 536 If the graph Γ is contained in the graph Γprime the Hilbert spaceKΓ is a proper subspace of the Hilbert space KΓprime This nested structureof Hilbert spaces is called a projective family of Hilbert spaces K is ndash andcan be defined as ndash the projective limit of this family

An orthonormal basis The tool for finding a basis in K is the PeterndashWeyltheorem which states that a basis on the Hilbert space of L2 functions onSU(2) is given by the matrix elements of the irreducible representations ofthe group Irreducible representations of SU(2) are labeled by half-integerspin j Call Hj the Hilbert space on which the representation j is definedand vα its vectors Write the matrix elements of the representation j as

R(j)αβ(U) = 〈U |j α β〉 (614)

For each graph Γ choose an ordering and an orientation Then a basis

|Γ jl αl βl〉 equiv |Γ j1 jL α1 αL β1 βL〉 (615)

in KΓ is simply obtained by tensoring the basis (614) That is

〈A|Γ jl αl βl〉 = R(j1)α1β1(U(A γ1)) R(jL)αL

βL(U(A γL)) (616)

This set of vectors in K is not a basis because the same vector appearsin KΓ and KΓprime if Γ is contained in the graph Γprime However it is very easyto get rid of the redundancy because all KΓ vectors belong to the trivialrepresentation of the paths that are in Γprime but not in Γ Therefore anorthonormal basis of K is simply given by the states |Γ jl αl βl〉 definedin (616) where the spins jl = 1

2 132 2 never take the value zero This

fact justifies the following definition


Proper graph subspaces For each graph Γ the proper graph subspace KΓ

is the subset of KΓ spanned by the basis states with jj gt 0 It is easy tosee that all proper subspaces KΓ are orthogonal to each other and theyspan K we can write this as

K simoplus

Γ

KΓ (617)

The ldquonullrdquo graph Γ = empty is included in the sum the corresponding Hilbertspace is the one-dimensional space spanned by the state Ψ[A] = 1 Thisstate is denoted |empty〉 thus 〈A|empty〉 = 1

K as an L2 space I have defined K as the completion of S in the scalarproduct defined by the bilinear form (67) Can this space be viewed asa space of square integrable functionals in some measure The answer isyes and it involves a beautiful mathematical construction that I will notdescribe here since it is not needed for what follows and for which I referthe reader to [20] Very briefly K sim L2[A dμ0] where A is an extensionof the space of the smooth connection The extension includes distribu-tional connections The measure dμ0 is defined on this space and is calledthe AshtekarndashLewandowski measure The construction is analogous to thedefinition of the gaussian measure dμG[φ] sim ldquoeminus

intdxdy φ(x)G(xy)φ(y)[dφ]rdquo

which as is well known needs to be defined on a space of distributionsφ(x) The space A has the beautiful property of being the Gelfand spec-trum of the abelian Clowast algebra formed by the smooth holonomies of theconnection A

622 Invariances of the scalar product

The kinematical state space S sub K sub S prime carries a natural representationof local SU(2) and Diff(Σ) simply realized by the transformations ofthe argument A The scalar product defined above is invariant underthese transformations Therefore K carries a unitary representation oflocal SU(2) and Diff(Σ) Let us look at this in some detail

Local gauge transformations Under (smooth) local SU(2) gauge trans-formations λ Σ rarr SU(2) the connection A transforms inhomogeneouslylike a gauge potential ie

A rarr Aλ = λAλminus1 + λdλminus1 (618)

This transformation of A induces a natural representation of local gaugetransformations Ψ(A) rarr Ψ(Aλminus1) on K Despite the inhomogeneous

232 Quantum space

transformation rule (618) of the connection the holonomy transformshomogeneously (see Sec 215)

U [A γ] rarr U [Aλ γ] = λ(xγf )U [A γ]λminus1(xγi ) (619)

where xγi xγf isin Σ are the initial and final points of the path γ For a given

(Γ f) define

fλ(U1 UL) = f(λ(xγ1

f )U1λminus1(xγ1

i ) λ(xγLf )ULλminus1(xγLi )) (620)

It is then easy to see that the transformation of the quantum states is

ΨΓf (A) rarr [UλΨΓf ](A) = ΨΓf (Aλminus1) = ΨΓfλminus1 (A) (621)

Since the Haar measure is invariant under right and left group transforma-tions it follows immediately that (67) is invariant From (621) and fromtheir definition it is easy to see that basis states |Γ jl αl βl〉 transform as

Uλ|Γ jl αl βl〉 = R(j1)α1αprime

1(λminus1(xf1)) R(j1)βprime

1β1(λ(xi1)) middot middot middotR(jL)αL

αprimeL(λminus1(xfL)) R(jL)βprime

LβL(λ(xiL))

|Γ jl αprimel β

primel〉 (622)

where il and fl are the points where the link l begins and ends

(Extended) diffeomorphisms Consider maps φ Σ rarr Σ that are con-tinuous invertible and such that the map and its inverse are smootheverywhere except possibly at a finite number of isolated points Callthese maps ldquoextended diffeomorphismsrdquo (or sometimes loosely justldquodiffeomorphismsrdquo) Call the group formed by these maps Diff lowast

An example of an extended diffeomorphism which is not a proper diffeomorphismin two dimensions is the following In polar coordinates the map

rprime = r φprime = φ +1

2sinφ (623)

is continuous everywhere while it is differentiable everywhere except at r = 0 whereits jacobian is ill defined

Under an extended diffeomorphism the transformation of the connec-tion is well defined (recall A isin G is defined everywhere on Σ except on afinite number of isolated points) A transforms as a one-form

A rarr φlowastA (624)

Hence S carries the representation Uφ of Diff lowast defined by UφΨ(A) =Ψ((φlowast)minus1 A) The holonomy transforms as

U [A γ] rarr U [φlowastA γ] = U [A φminus1γ] (625)


where (φγ)(s) equiv (φ(γ(s)) That is dragging A by a diffeomorphism φis equivalent to dragging the curve γ (Notice that if φ is not a properdiffeomorphism the curve φγ may fail to be smooth at a finite num-ber of points at most) In turn a cylindrical function ΨΓf [A] is sentto a cylindrical function ΨφΓf [A] namely one which is based on theshifted graph Since the right-hand side of (67) does not depend explic-itly on the graph the diffeomorphism invariance of the inner product isimmediate

623 Gauge-invariant and diffeomorphism-invariant states

The kinematical state space K is a space of arbitrary wave functionals ofthe connection Ψ[A] but recall that we need a space of states that arewave functionals invariant under local gauge transformations and diffeo-morphisms More formally the two classical equations (480) and (481)must be implemented in the quantum theory giving

Daδ

δAia(τ)

Ψ[A] = 0 (626)

F iab(τ)

δ

δAia(τ)

Ψ[A] = 0 (627)

The same argument as used in Section 434 shows that these equationsdemand the invariance of Ψ under local SU(2) transformations and diffeo-morphisms More precisely the smearing functions fa(τ) of Section 434must be chosen in an appropriate class here the relevant group is Diff lowast

and therefore fa(τ) can be any infinitesimal generator of Diff lowast Underthis choice (627) is equivalent to the requirement of invariance of thestate under Diff lowast

Call K0 the space of the states invariant under local SU(2) and Kdiff

the space of the states invariant under local SU(2) and Diff lowast Recallingthat we call H the space of solutions of the WheelerndashDeWitt equationswe have therefore the sequence of Hilbert spaces

K SU(2)minusrarr K0Diff lowastminusrarr Kdiff

Hminusrarr H (628)

where the three steps correspond to the implementation of the threeequations that the wave functional must satisfy namely (626) (627) and(63) respectively I construct explicitly K0 in the next section and Kdiff

in the following one (Except for the first one the domain of the mapsin (628) will turn out to be given by the first term of the correspondingrigged Hilbert space)

234 Quantum space

63 Internal gauge invariance The space K0

The space K0 is the space of the states in K invariant under local SU(2)gauge transformations I call S0 the gauge-invariant subspace of S and S prime

0

its dual It is not difficult to see that K0 is a proper subspace of K Exam-ples of finite norm SU(2) invariant states are provided by the loop statesdefined in (69) In fact a moment of reflection shows that the multiloopstates are sufficient to span K0 In the first years of the development ofLQG multiloop states were used as a basis for K0 however this basis isovercomplete and this fact complicates the formalism Nowadays we havea much better control of K0 thanks to the introduction of the spin net-work states which can be seen as finite linear combinations of multiloopstates forming a genuine orthonormal basis

The spin network basis of quantum gravity is a simple extension ofthe spin network basis defined in Section 536 in the context of latticegauge theory As we shall see in the next section however diffeomorphisminvariance will soon make the two cases very different and connect thequantum gravity spin network basis to Penrosersquos old ldquospin networkrdquo ideathat quantum states of the geometry can be described as abstract graphscarrying spins

Spin networks Denote ldquonodesrdquo the end points of the oriented curves in ΓWithout loss of generality assume that each set of curves Γ is formed bycurves γ that if they overlap at all overlap only at nodes Viewed in thisway Γ is in fact a graph immersed in the manifold that is a collectionof nodes n which are points of Σ joined by links l which are curves inΣ The ldquooutgoing multiplicityrdquo mout of a node is the number of links thatbegin at the node The ldquoingoing multiplicityrdquo min of a node is the numberof links that end at the node The multiplicity or valence m = min +mout

of a node is the sum of the twoGiven the graph Γ for which an ordering and an orientation have been

chosen let jl be an assignment of an irreducible representation differ-ent from the trivial one to each link l Let in be an assignment of anintertwiner in to each node n The notion of intertwiner was defined inSection 536 The intertwiner in associated with a node is between therepresentations associated with the links adjacent to the node The tripletS = (Γ jl in) is called a ldquospin network embedded in Σrdquo A choice of jland in is called a ldquocoloringrdquo of the links and the nodes respectively

631 Spin network states

Consider a spin network S = (Γ jl in) with L links and N nodes Thestate |Γ jl αl βl〉 defined above in (616) has L indices αl and L indices

63 Internal gauge invariance The space K0 235

j

j

2= 12

S =n1

n2

1 = 1

j3

= 12

Fig 61 A simple spin network with two trivalent nodes

βl The N intertwiners in have altogether precisely a set of indices dualto these The contraction of the two

|S〉 equivsum

αlβl

vβ1βn1i1 α1αn1

vβn1+1βn2i2 αn1+1αn2

vβ(nNminus1+1)βL

iN α(nNminus1+1)αL

|Γ jl αl βl〉 (629)

defines the spin network state |S〉 The pattern of the contraction of theindices is dictated by the topology of the graph itself the index αl (respβl) of the link l is contracted with the corresponding index of the inter-twiner vin of the node n where the link l starts (resp ends) The gaugeinvariance of this state follows immediately from the transformation prop-erties (622) of the basis states and the invariance of the intertwiners Asa functional of the connection this state is

ΨS[A] = 〈A|S〉 equiv(otimes

l

R(jl)(H[A γl])

)

middot(otimes

n

in

)

(630)

The raised dot notation indicates the contraction between dual spaces onthe left the tensor product of the matrices lives in the space otimesl (Hlowast

jlotimesHjl)

On the right the tensor product of all intertwiners lives precisely in thedual of this space

Example Letrsquos say Γ has two nodes n1 and n2 and three links l1 l2 l3 each link begin-ning at n1 and ending at n2 Let the coloring of the links be j1 = 1 j2 = 12 j3 = 12see Figure 61 At each of the two nodes we must therefore consider the tensor productof two fundamental and one adjoint representation of SU(2) As is well known the ten-sor product of these representations contains a single copy of the trivial representationtherefore there is only one possible intertwiner A moment of reflection shows that thisis given by the triple of Pauli matrices viAB = 1radic

3σiAB since these have precisely the

invariance property

(R(U))ij UAC UB

D σjCD = σiAB (631)

236 Quantum space

Here (R(U))ij is the adjoint representation with i j = 1 2 3 vector indices and UAC

is the fundamental representation with AB = 0 1 spinor indices The 1radic3

is a nor-malization factor to satisfy (5157) Therefore there is only one possible coloring of thenodes in this case The spin network state is then

ΨS[A] =1

3σiAB (R(H[A γ1]))

ij (H[A γ2])

AC (H[A γ3])

BD σjCD (632)

Let me now enunciate the main fact concerning the spin network statesthe ensemble of the spin network states |S〉 forms an orthonormal basisin K0 Orthonormality can be checked by a direct calculation The basisis labeled by spin networks namely graphs Γ and colorings (jl in)

Some comments First I have assumed the spins jl to be all differentfrom zero (a spin network containing a link l with jl = 0 is identified withthe spin network obtained by removing the link l) Second this result isa simple consequence of the PeterndashWeyl theorem namely of the fact thatthe states |Γ jl αl βl〉 form a basis in K and the very definition of theintertwiners Third the spin network basis is not unique as it dependson the (arbitrary) choice of a basis in each space of intertwiners at eachnode Notice also that in the basis |S〉 = |Γ jl in〉 the label Γ runs overall nonoriented and nonordered graphs However for the definition of thecoloring an orientation and an ordering has to be chosen for each Γ

The space S0 is the space of the finite linear combinations of spinnetwork states which is dense in K0 and S prime

0 is its dual

632 Details about spin networks

Orientation There is an isomorphism ε between a representation j and its dual jlowast Ifj is the fundamental

ε C2 rarr (C2)lowast

ψA rarr ψA = εABψB (633)

where εAB is the antisymmetric tensor This extends to all other representations sincethey can be obtained from tensor products of the fundamental Using this we can raiseand lower the indices of the intertwiners and identify intertwiners with the same totalnumber of indices We can then ask what happens if we change the orientation of one ofthe links Using (A6) a straightforward calculation that I leave to the reader showsthat the only change is to the overall sign if the representation has half-integer spinThus the only relevant orientation of the spin network is an overall global orientation

Spin networks versus loop states A spin network state can be decomposed into a finitelinear combination of (multi-)loop states The representation j can be written as thesymmetrized tensor product of 2j fundamental representations Therefore we can writethe elements of Hj as completely symmetric complex tensors ψA1A2j with 2j spinorindices Ai = 0 1 In this basis the representation matrices have the simple form

R(j)A1A2jB1B2j (U) = U (A1

(B1 UA2j)

B2j) (634)


=

g2

g

g

1

12

1

12

+3

Fig 62 Decomposition of a spin network state into loop states

where the parentheses indicate complete symmetrization (see Appendix A1) In thisbasis the intertwiners are simply combinations of the only two SU(2) invariant ten-sors namely εAB and δBA For instance a (nonnormalized) trivalent intertwiner betweenincoming representations j jprime and an outgoing representation jprimeprime is

vA1A2j B1B2jprimeC1C2jprimeprime = εA1B1 εAaBa δC1

Ba+1 δ

CbB2jprime

δCb+1Aa+1

δC2jprimeprimeA2j

(635)

where j = a + c jprime = a + b and jprimeprime = b + c Now when the two holonomy matrices oftwo contiguous links γ1 and γ2 are joined by δBA they give the holonomy of the curveobtained joining γ1 and γ2 which I denote γ1γ2

H[A γ1]ABδ

BCH[A γ2]

CD = H[A γ1γ2]

AD (636)

On the other hand recall that εABUA

CεCD = (Uminus1)DA Therefore

εDBH[A γ1]ABεACH[A γ2]

CE = H[A γminus1

1 γ2]DE (637)

Therefore the tensors εAB and δBA in the intertwiners simply join the segments in thearguments of the holonomies Since the graph of the spin network is finite a line ofjoining must close to a loop A moment of reflection will convince the reader thata spin network state (629) is therefore equal to a linear combination of products ofholonomies of closed lines that wrap along the graph That is it is a linear combinationof multiloop states

The decomposition of a spin network state in loop states can be obtained graphi-cally as follows Replace each link of the graph colored with spin j with 2j parallelstrands Symmetrize these strands along each link The intertwiners at the nodes canbe represented as collections of segments joining the strands of different links By join-ing these segments with the strands one obtains a linear combination of multiloopsThe spin network states can then be expanded in the corresponding loop states Noticethat this is analogous to the construction at the basis of the KaufmanndashLins recouplingtheory illustrated in Appendix A1 (care should be taken with signs) For details on thisconstruction see [171]

Applying this rule to the state (632) illustrated in Figure 61 it is easy to see that

ΨS(A) =1

2

[Ψ

((γ1γminus12 )(γ1γ

minus13 ))

+ Ψγ1γ

minus12 γ1γ

minus13

] (638)

See Figure 62 for a graphical illustration of this decomposition

Details on intertwiners Given a graph and a coloring of its links it may happen thatthere is no nonvanishing intertwiner at all associated with a node For instance a nodewith valence unity cannot exist in a spin network because a single nontrivial irreducible

238 Quantum space

representation does not contain any invariant subspace If the node is bivalent there isan intertwiner only if the incoming and outgoing representations are the same and theintertwiner is the identity The spin network with two links joined by a bivalent nodeis identified with one obtained replacing the two links with a single link A trivalentnode may have adjacent links with colorings j1 j2 j3 only if these satisfy the ClebschndashGordan conditions (A10)ndash(A11) The intertwiners are directly given by the Wigner3j-coefficients (A16) up to normalization Nontrivial intertwiner spaces begin onlywith nodes of valence four or higher Intertwiners of an n-valent node can be labeledby nminus 2 spins as detailed in Appendix A1

64 Diffeomorphism invariance The space Kdiff

Let me now come to the second and far more crucial invariance 3d dif-feomorphism invariance We have to find the diffeomorphism-invariantstates

Transformation properties of spin network states under diffeomorphisms The spin net-work states |S〉 are not invariant under diffeomorphisms A diffeomorphism moves thegraph around on the manifold and therefore changes the state Notice however that adiffeomorphism may change more than the graph of a spin network that is the equa-tion Uφ|Γ jl in〉 = |(φΓ jl in)〉 is not always correct In particular a diffeomorphismthat leaves the graph Γ invariant may still affect a spin network state |Γ jl in〉 This isbecause for each graph the definition of the spin network state requires the choice ofan orientation and ordering of the links and these can be changed by a diffeomorphism

Here is an example Let Γ be an ldquoeyeglassesrdquo graph formed by two loops α and βin the j = 12 representation connected by a path γ in the j = 1 representation Thespace of the intertwiners at each node is one-dimensional but this does not imply thatthere is no choice to be made for the basis since if i is a normalized intertwiner so isminusi With one choice the state is

ΨS[A] = (U(Aα))AB σiB

A (R(1)(U(A γ)))ij σjDC (U(A β))CD (639)

Using elementary SU(2) representation theory this can be rewritten (up to a normal-ization factor) as

ΨS[A] sim trH[Aαγβγminus1] minus trH[Aαγβminus1γminus1] (640)

Now consider a diffeomorphism φ that turns the loop β around namely it reverses itsorientation while leaving α and γ as they are Clearly this diffeomorphism will sendthe two terms of the last equation into each other giving

UφΨS[A] = minusΨS[A] (641)

while φΓ = ΓGiven an oriented and ordered graph Γ there is a finite discrete group GΓ of maps

gk such as the one of the example that change its order or orientation and that canbe obtained as a diffeomorphism The elements gk of this group act on KΓ

A moment of reflection will convince the reader that the diff-invariantstates are not in K0 they are in S prime

0 We are therefore in the commonsituation in which the solutions of a quantum equation must be searched

64 Diffeomorphism invariance The space Kdiff 239

in the extension of the Hilbert space and the scalar product must be ap-propriately extended to the space of the solutions as explained in Section552

The elements of S prime0 are linear functionals Φ on the functionals Ψ isin S0

The requirement of diff invariance makes sense in S prime0 because the action

of the diffeomorphism group is well defined in S prime0 by duality

(UφΦ)(Ψ) equiv Φ(Uφminus1Ψ) (642)

Therefore a diff-invariant element Φ of S prime0 is a linear functional such that

Φ(UφΨ) equiv Φ(Ψ) (643)

The space Kdiff is the space of these diff-invariant elements of S prime0 Remark-

ably we have a quite good understanding of this space whose elementscan be viewed as the quantum states of physical space

The space Kdiff I now define a map Pdiff S0 rarr S prime0 and show that the

(closure in norm of the) image of this map is precisely Kdiff Let the statePdiffΨ be the element of S prime

0 defined by

(Pdiff Ψ)(Ψprime) =sum

Ψprimeprime=UφΨ

〈ΨprimeprimeΨprime〉 (644)

The sum is over all states Ψprimeprime in S0 for which there exist a φ isin Diff lowast

such that Ψprimeprime = UφΨ The key point is that this sum is always finite andtherefore well defined To see this notice that since Ψ and Ψprime are in S0they can be expanded in a finite linear combination of spin network statesIf a diffeomorphism changes the graph of a spin network state ΨS thenit takes it to a state orthogonal to itself If it doesnrsquot change the graphthen either it leaves the state invariant so that no multiplicity appears in(644) or it changes the ordering or the orientation of the links but theseare discrete operations giving at most a discrete multiplicity in the sum in(644) Therefore the sum in (644) is always well defined Clearly Pdiff Ψis diff invariant namely it satisfies (643) Furthermore it is not difficultto convince oneself that the functionals of the form (644) span the spaceof the diff-invariant states Therefore the (closure in norm of the) imageof Pdiff is Kdiff States related by a diffeomorphism are projected by Pdiff

to the same element of Kdiff

Pdiff ΨS = Pdiff (UφΨS) (645)

Finally the scalar product on Kdiff is naturally defined by

〈Pdiff ΨS Pdiff ΨSprime〉Kdiffequiv (Pdiff ΨS)(ΨSprime) (646)

240 Quantum space

(see section 552) This completely defines Kdiff Equivalently Kdiff isdefined by the bilinear form

〈ΨΨprime〉Kdiffequiv 〈Ψ|Pdiff |Ψprime〉 equiv

sum

Ψprimeprime=φΨ


in S0

To understand intuitively why the above definition works consider the followingformal argument Imagine that we were able to define a measure dφ on Diff lowast Wecould then write diffeomorphism-invariant states by simply integrating an arbitrarystate on the orbit of the diffeomorphism group

PdiffΨ equivint

Diff lowast[dφ] UφΨ (648)

Therefore

(PdiffΨprime)(Ψ) =

int

Diff lowast[dφ] (ΨprimeUφΨ) (649)

Let us assume for simplicity that Ψ isin KΓ and Ψprime isin KΓprime (the general case following bylinearity) Then the right-hand side of (649) vanishes unless there is a φ that sends Γin Γprime If this is the case the integral has support just on the subgroup of Diff lowast thatleaves Γ invariant Elements of this subgroup can either change the state Ψ or leave itinvariant Thus we can rewrite (649) as


sum

Ψprimeprime=UφprimeΨ

int

DΨprimeprime[dφ] (ΨprimeUφΨprimeprime) (650)

where the integral is over the subgroup DΨprimeprime of Diff lowast that leaves Ψprimeprime invariant Butthen we can take the scalar product out of the integral and write


sum

Ψprimeprime=UφΨ

(ΨprimeUφΨprimeprime)

(int

DΨprime[dφ]

)

(651)

If we now assume that the measure dφ is such that the volume of DΨprime is unity werecover the definition given above Therefore the definition (644) can be seen as arigorous implementation of the intuitive ldquointegration of the diffeomorphism grouprdquo of(648)

641 Knots and s-knot states

To understand the structure of Kdiff consider the action of Pdiff on thestates of the spin network basis To this aim observe that a diffeomor-phism sends a spin network state |S〉 to an orthogonal state or to a stateobtained by a change in the order of the orientation of the links Denotegk|S〉 the states that are obtained from |S〉 by changes of orientation orordering and that can be obtained via a diffeomorphism as in the exam-ple above The maps gk form the finite discrete group GΓ therefore therange of the discrete index k is finite Then it is easy to see that

〈S|Pdiff |Sprime〉 =

0 if Γ = φΓprimesum

k 〈S|gk|Sprime〉 if Γ = φΓprime (652)


An equivalence class K of unoriented graphs Γ under diffeomorphisms iscalled a ldquoknotrdquo Knots without nodes have been widely studied by thebranch of mathematics called knot theory with the aim of classifyingthem Knots with nodes have also been studied in knot theory but to alesser extent From the first line of (652) we see that two spin networksS and Sprime define orthogonal states in Kdiff unless they are knotted in thesame way That is unless they are defined on graphs Γ and Γprime belongingto the same knot class K Therefore the basis states in Kdiff are first ofall labeled by knots K We call KK the subspace of Kdiff spanned by thebasis states labeled by the knot K That is

KK = Pdiff KΓ (653)

for any Γ isin KThe states in KK are then distinguished only by the coloring of links and

nodes As observed before the colorings are not necessarily orthonormaldue to the nontrivial action of the discrete symmetry group GΓ To findan orthonormal basis in KK we have therefore to further diagonalize thequadratic form defined by the second line of (652) Denote |s〉 = |K c〉 theresulting states The discrete label c is called the coloring of the knot KUp to the complications due to the discrete symmetry GΓ it correspondsto the coloring of the links and the nodes of Γ The states |s〉 = |K c〉 arecalled spin-knot states or s-knot states

642 The Hilbert space Kdiff is separable

The key property of knots is that they form a discrete set Therefore thelabel K is discrete It follows that Kdiff admits a discrete orthonormalbasis |s〉 = |K c〉 Thus Kdiff is a separable Hilbert space The ldquoexcessivesizerdquo of the kinematical Hilbert space K reflected in its nonseparabilityturns out to be just a gauge artifact

The fact that knots without nodes form a discrete set is a classic result of knottheory It is easy to understand it intuitively first if two loops without nodes can becontinuously deformed into each other without crossing then there is a diffeomorphismthat sends one into the other second to change the node class we have to deform a linkacross another link and this is a discrete operation On the other hand the fact thatknots with nodes form a discrete set is nontrivial Indeed it depends on the fact thatwe have chosen the extension Diff lowast of the diffeomorphism group Diff Had we chosenDiff as the invariance group the space of the knot classes would have been continuous

To understand this recall that the action of Diff on the tangent space is linearConsider a graph Γ that can be deformed continuously into a graph Γprime Let p be thelocation of an n-valent node of Γ and pprime the corresponding node on Γprime Is there adiffeomorphism φ in Diff sending Γ to Γprime The answer in general is negative for thefollowing reason The diffeomorphism must send p to pprime Hence φ(p) = pprime The tangentspace to p is sent into the tangent space to pprime by the jacobian Jp of φ at p which is a

242 Quantum space

linear transformation in 3d Let vi i = 1 n be the tangents of the n links at p andvprimei i = 1 n be the tangents of the n links at pprime For φ to send the two nodes intoeach other we must have

Jpvi = vprimei (654)

But in general there is no linear transformation sending n given directions into n othergiven directions In other words (654) gives n linear conditions on the nine degreesof freedom of the jacobian matrix (Jp)

ab = partφa(x)partxb|p Therefore in general two

graphs that can be transformed into each other continuously cannot be transformed intoeach other by a diffeomorphism The equivalence classes are characterized by continuousparameters at the nodes

On the other hand maps in Diff lowast can freely transform these parameters The reasonis that thanks to the relaxation of the differentiability condition an extended diffeo-morphism φ isin Diff lowast can act nonlinearly on the tangents In Section 67 after thediscussion of the physical interpretation of the knot states I will discuss the physicalreasons why Diff lowast is more appropriate than Diff as gauge group

This concludes the construction of the kinematical quantum state spaceof LQG The physical meaning of the s-knot states in Kdiff will becomeclear later on It is now time to define the operators

65 Operators

There are two basic field variables in the canonical theory from which allmeasurable quantities can be constructed the connection Ai

a(τ) and itsmomentum Ea

i (τ) I now define quantum operators corresponding to sim-ple functions of these Quantum states are functionals Ψ[A] of the connec-tion A The momentum conjugate to the real connection A is (18πG)E(see (440)) We can therefore define the two field operators

Aia(τ)Ψ[A] = Ai

a(τ) Ψ[A] (655)1

8πGEa

i (τ)Ψ[A] = minusiδ

δAia(τ)

Ψ[A] (656)

on functionals of Ψ[A] In the following I choose units in which 8πG = 1I will then restore physical units when needed The first is a multiplica-tive operator the second a functional derivative However both theseoperators send Ψ[A] out of the state spaces that we have constructed Inparticular they are not well defined in K This can be easily cured bytaking instead of A and E some simple function of these

651 The connection A

The holonomy U(A γ) is well defined on S More precisely let UAB(A γ)

be the matrix elements of the group element U(A γ) Then

(UAB(A γ)Ψ)[A] = UA

B(A γ)[A] Ψ[A] (657)

65 Operators 243

The right-hand side is clearly in S if Ψ[A] is In fact any cylindricalfunction of the connection is immediately well defined as a multiplicativeoperator in K

For instance consider a closed loop α and let Tα[A] = trU(Aα) Consider the actionof this operator on a spin network state |S〉 with a graph that does not intersect withα Then clearly

Tα|S〉 = |S cup α〉 (658)

where S cup α is the spin network formed by S plus the loop α in the j = 12 represen-tation

Notice that all this is quite different from quantum field theory on abackground spacetime where field operators are operator-valued distri-butions and therefore are well defined only when smeared in three dimen-sions In (657) a well-defined operator is obtained by simply smearing(the path-ordered exponential of) the field in just one dimension alongthe loop γ This is a characteristic feature of diff-invariant quantum fieldtheories

652 The conjugate momentum E

To understand the action of E we have to compute the functional deriva-tive of the holonomy the building block of the cylindrical functions It isnot hard to show that

δ

δAia(x)

U(A γ) =int

ds γa(s) δ3(γ(s) x) [U(A γ1) τi U(A γ2)] (659)

Here s is an arbitrary parametrization of the curve γ γa(s) are the coor-dinates of the curve γa(s) equiv dγa(s)ds is the tangent to the curve in thepoint s γ1 and γ2 are the two segments in which γ is separated by thepoint s This is a crucial formula that plays a major role in what followsThe diligent reader is therefore invited to derive it and understand it indetail There are several possible derivations The naive one is just to usea formal functional derivation of the expression (280) The rigorous one isto consider variations of the defining equation (278) See [172] for details

Notice that the right-hand side of (659) is a distribution but only atwo-dimensional one since one of the three deltas in δ3 is in fact integratedover ds It is therefore natural to search for an operator well defined onK by smearing E in two-dimensions To this purpose consider a two-dimensional surface S embedded in the 3d manifold

244 Quantum space

g

S

P

g

2

1

Fig 63 A curve that intersects the surface at an individual point P

Let σ = (σ1 σ2) be coordinates on the surface S The surface is definedby S (σ1 σ2) rarr xa(σ1 σ2) Consider the operator

Ei(S) equiv minusiint

Sdσ1dσ2 na(σ)

δ

δAia

(x(σ)) (660)

where

na(σ) = εabcpartxb(σ)partσ1

partxc(σ)partσ2

(661)

is the normal one-form on S and εabc is the completely antisymmetricobject (for the relativists the LevindashCivita tensor of density weight (minus1))

The ldquograsprdquo Let us now compute the action of the operator Ei(S) onthe holonomy U(A γ) Assume for the moment that the end points of γdo not lie on the surface S For simplicity let us also begin by assumingthat the curve γ crosses the surface S at most once and denote P theintersection point (if any) see Figure 63

The curve is separated into two parts γ = γ1 cup γ2 by P By using(659) and (660) we obtain

Ei(S)U(A γ)

= minusiint

Sdσ1dσ2 εabc

partxa(σ)partσ1

partxb(σ)partσ2

δ

δAic

(x(σ)) U(A γ)

= minusiint

S

int

γdσ1dσ2ds εabc

partxa

partσ1

partxb

partσ2

partxc

partsδ3(x(σ) x(s)

)

times U(A γ1) τi U(A γ2) (662)

A closer look at this result reveals a great simplification of the last integralThe integral vanishes unless the surface and the curve intersect Assume

65 Operators 245

that there is a single intersection point and further assume that it hascoordinates xa = 0 In the neighborhood of this point consider the map(σ1 σ2 s) rarr (x1 x2 x3) from the integration domain to coordinate spacedefined by

xa(σ1 σ2 s) = xa(σ1 σ2) + xa(s) (663)

The jacobian of this map

J equiv part (x1 x2 x3)part (σ1 σ2 s3)

= εabcpartxa

partσ1

partxb

partσ2

partxc

parts(664)

appears in the integral We can therefore make the change of variables(σ1 σ2 s) rarr (x1 x2 x3) in the integral

The jacobian is nonvanishing since I have required that there is only a single non-degenerate point of intersection The jacobian (664) and the integral (662) wouldvanish if the tangent vectors given by the partial derivatives in (664) were coplanarie if a tangent partxab(σ)partσ12 to the surface were parallel to the tangent partxc(s)partsof the curve This happens for instance if the curve lies entirely in S Then therewouldnrsquot be just a single intersection point I consider these limiting cases later on

With a change of variables in the integral we can easily perform theintegration and get rid of the delta function We obtain remarkablyint

S

int

γdσ1dσ2ds εabc

partxa(σ)partσ1

partxb(σ)partσ2

partxc(s)parts

δ3(x(σ) x(s)

)= plusmn1 (665)

In fact this integral is a well-known analytic coordinate-independent ex-pression for the intersection number between the surface S and the curveγ It vanishes if there is no intersection The sign is dictated by the relativeorientation of the surface and the curve Hence we obtain the simple result

Ei(S)U(A γ) = plusmni U(A γ1) τi U(A γ2) (666)

The action of the operator Ei(S) on holonomies consists of just insertingthe matrix (plusmni τi) at the point of intersection We say that the operatorEi(S) ldquograspsrdquo γ

The generalization to multiple intersections is immediate Using P tolabel different intersection points we have

Ei(S)U(A γ) =sum

Pisin(S cap γ)

plusmni U(A γP1 ) τi U(A γP2 ) (667)

For later use I give here also the action of the operator Ei(S) on the holonomy inan arbitrary representation j

Ei(S) Rj(U(A γ))

= plusmni Rj(U(A γ1)) (j)τi R

j(U(A γ2)) (668)

where (j)τi is the SU(2) generator in the spin-j representation

246 Quantum space

Thus Ei(S) is a well-defined operator on K The fact that it is a surfaceintegral of Ea

i (τ) which is well defined can be understood as follows Geo-metrically Ea

i (τ) is not a vector field but rather a vector density The nat-ural associated geometric quantity is the two-form Ei = εabcE

ai dxb and dxc

But a two-form can be naturally integrated over a surface giving an ob-ject which is well behaved under diffeomorphisms In fact the operatorwe have defined corresponds precisely to the classical quantity

Ei(S) =int

SEi (669)

The same is true in the quantum theory The functional derivative is avector density therefore a two-form and this is naturally integrated overa surface Geometry and operator properties begin here to go nicely handin hand

The operators Tα and Ei(S) defined on the Hilbert space K form a rep-resentation of the corresponding classical Poisson algebra A major resultof the mathematically rigorous approach to loop quantum gravity is theproof of a unicity theorem for this representation The theorem is usu-ally called the ldquoLOSTrdquo theorem from the initials of the people that havediscovered it (more precisely one of its versions see the Bibliographicalnotes below) The theorem is analogous to the Stone-vonNeuman theo-rem in nonrelativistic quantum mechanics which shows the unicity of theSchrodinger representation It relies heavily on the hypothesis of diffeo-morphism invariance The theorem shows under certain general hypothe-ses that the loop representation which was built largely ldquoby handrdquo isthe only possible way of quantizing a diffeomorphism invariant theoryNo such theorem hold in conventional quantum field theory In particu-lar this shows that in the diffeomorphism invariant context the theory israther tightly determined

66 Operators on K0

To be well defined on K0 an operator must be invariant under internalgauge transformations As far as the connection is concerned this is veryeasy to obtain We noticed above that any cylindrical function gives awell-defined operator The cylindrical function needs simply to be gaugeinvariant to be well defined in K0 The operator Tα for instance definedin (658) is well defined on K0

661 The operator A(S)

The situation with E is slightly more complicated The operator Ei(S)clearly cannot be gauge invariant as the index i transforms under internal

66 Operators on K0 247

gauges On the other hand we cannot obtain a gauge-invariant quantityby simply contracting this index as

E2(S) equivsum

i

Ei(S)Ei(S) (670)

because the transformation property of Ei(S) is complicated by the in-tegral over S Let us nevertheless compute its action on a spin networkstate S since this is a crucial step for what follows Let us assume thatthere is a single intersection P between the surface S and (the graph Γ)of the spin network S Let j be the spin of the link at the intersectionUsing (668) we see that the first operator Ei(S) inserts a matrix (j)τi atthe intersection So does the second but minus(j)τi

(j)τi = j(j + 1) times 1 is theCasimir operator of SU(2) Therefore

E2(S)|S〉 = 2 j(j + 1) |S〉 (671)

This beautiful result however is completely spoiled if Γ intersects S morethan once because in this case the τi matrices at different points getcontracted and we do not get a gauge-invariant state

To circumvent this difficulty let us define a gauge-invariant operatorA(S) associated to the surface S as follows For any N partition thesurface S into N small surfaces Sn that become smaller and smaller asN rarr infin and such that for each N

⋃n Sn = S Then define

A(S) equiv limNrarrinfin

sum

n

radicE2(Sn) (672)

Do not confuse the A chosen to denote this operator with the A thatdenotes the connection there is no relation between these two quantitiesThe reason for choosing the letter A to denote the operator (672) willbecome clear shortly

In the classical case by Riemannrsquos very definition of the integral wehave

A(S) =int

S

radicnaEa

i nbEbi d2σ (673)

which is a well-defined gauge-invariant quantity In the quantum casethe action of the operator (672) is easy to compute Let us evaluate iton a spin network state under the simplifying assumption that no spinnetwork node lies on S For sufficiently high N no Sn will contain morethan one intersection with Γ see Figure 64 Therefore the sum over nreduces to a sum over the intersection points P between S and Γ and isindependent from N for N sufficiently high Using (671) we have then

248 Quantum space

Γ

Sn

S

Fig 64 A partition of S

P2

SP3P1

Fig 65 A simple spin network S intersecting the surface S

immediately (see Figure 65)

A(S)|S〉 =

sum

Pisin(ScupΓ)

radicjP (jP + 1) |S〉 (674)

where jP is the color of the link that crosses S at P This is a key resultFirst of all the operator A(S) is well defined in K This is the operatorcorresponding to the classical quantity (673) Second spin network statesare eigenfunctions of this operator

To summarize we have obtained for each surface S isin M a well-definedSU(2) gauge-invariant and self-adjoint operator A(S) which is diagonalon the spin networks that do not have a node on S The correspondingspectrum (with the restrictions mentioned) is labeled by multiplets j =(j1 jn) i = 1 n and n arbitrary of positive half-integers ji This


is called the main sequence of the spectrum and is given by

Aj =

sum

i


We will compute the rest of the spectrum which is also real and discretein Section 664

Since the operator is diagonal on spin network states (with appro-priate choices of intertwiners) and all its eigenvalues are real it is alsoself-adjoint In fact this operator is well defined at the level of rigor-ous mathematical physics For a completely rigorous detailed construc-tion see for instance [20] and [173] The fact that this operator can berigorously constructed at all it is finite and its spectrum can even becomputed is a rather striking result considering that its definition in-volves an operator product and a square root This is the first remarkablepay-off of a well-defined diffeomorphism-invariant formalism for quantumfield theory As we will see in a moment this result has major physicalsignificance

In principle the operators Tα and A(S) are sufficient to define thequantum theory In practice it will be convenient later on to define otheroperators as well Before doing so however let us discuss the physicalmeaning of the mathematical result achieved so far

662 Quanta of area

In the previous section I have constructed and diagonalized the SU(2)gauge-invariant and self-adjoint operator A(S) What is the physical in-terpretation of this operator A direct comparison of (673) with (428)shows that A(S) is precisely the physical area of the surface S

Therefore we obtain immediately an important physical result Thepartial observable given by the area of a fixed two-dimensional surfaceis represented in the quantum theory by a self-adjoint operator with adiscrete spectrum But this yields immediately a physical prediction anymeasurement of the area of any physical surface can only give an outcomewhich is in the spectrum of this operator Since the spectrum is discretethis means that the physical area is a quantized partial observable Ameasurement of the area can only give a result contained in the spectrum(675)ndash(6125) of A(S)

Restoring physical units for 8πG and c the area operator is 8πGcminus3

times (672) and its eigenvalues are 8πGcminus3 times the ones in (675)ndash(6125) The main sequence for instance gives

Aj = 8πGcminus3sum

i


250 Quantum space

Had we used the more general Barbero connection described in Section423 instead of the real connection then from (446) we would have theoperator E given by

c3

8πγGEa


δAia(τ)

Ψ[A] (677)

instead of (656) In this case the spectrum is modified by an overallconstant factor

Aj = 8πγGcminus3sum

i


Up to the single Immirzi parameter γ this is a precise and quantitativeprediction of LQG It can in principle be verified or falsified Alter-natively indirect consequences of this prediction could have observableeffects In fact this quantization of the area is the basis of many resultsof the theory such as the derivation of black-hole entropy

The smallest (nonvanishing) eigenvalue in (676) taking the Immirziparameter equal to 1 is

A0 = 4radic

3πGcminus3 sim 10minus66 cm2 (679)

This is a sort of elementary quantum of area of the order of the Planckarea It is the quantum of area carried by a link in the fundamental j =12 representation The fact that there is no area that can be measuredbelow a minimum amount indicates that there is a sort of minimal sizeof physical space at the Planck scale

An intrinsic discreteness of physical space at the Planck length haslong been expected in quantum gravity Notice that in the context ofLQG this discreteness is not imposed or postulated Rather it is a directconsequence of a straightforward quantization of GR Space geometryis quantized in the same manner in which the energy of an harmonicoscillator is quantized

663 n-hand operators and recoupling theory

The two-hand loop operator The area operator can be defined also in a different man-ner which is of interest because it employs a technique that we will use below For eachsmall surface S define E2(S) in an SU(2) gauge-invariant manner as follows Given apath γ with end points r and s define the ldquotwo-handed loop operatorrdquo

T abγ = Ea

i (r)R(1)(U(A γ))ijEbj (s) (680)

where R(1)(U)ij is the adjoint j = 1 representation Given two points r and s in asmall surface S let γrs be a straight path (in the coordinate chosen) from r to s and

T ab(r s) = T abγrs

(681)


Then define

E2(S) =

int

Sd2σ

int

Sd2σprimena(σ)nb(σ

prime)T ab(σ σprime) (682)

In the limit in which the surface is small only the first term of the holonomy whichis the identity survives and therefore this definition converges to the one in (670) forsmall surfaces

The advantage of using this kind of regularization in the quantum theory is that itsimplifies the SU(2) representation calculations This is because the regularized oper-ator is itself SU(2) invariant

The action of the quantum operator (681) on a spin network state is easy to computeThere is a contribution for each intersection of a spin network link with the surface S foreach E Each of the two intersections is called a grasp For each of these contributionsthe spin network is modified by the creation of two nodes at the points r and s onefor each grasp and the addition of the loop γrs to the spin network We say that theldquohandsrdquo of the operator ldquograsprdquo the spin network Each node is trivalent with two linksbeing the ones of the grasped spin network say in a representation j and the otherbeing γrs in the representation j = 1 The intertwiner between these representationsis the SU(2) generator (j)(τi)

αβ in the representation j This is not normalized If we

call the normalized intertwiner iiαβ we have

(j)(τi)αβ = nj ii

αβ (683)

where nj can be computed easily by taking the norm of this equation This gives

n2j = tr((j)τ i (j)τi) = j(j + 1) tr(1) = j(j + 1) (2j + 1) (684)

Recoupling theory In the limit in which the surface is small the two grasps are onthe same link and at the same point and the line between them is infinitesimal It isnevertheless useful to write the two grasps as separated and the lines between them asfinite lines just remembering that the connection on these lines is trivial namely thatthey are associated to identities In this representation the result of the grasp on a linkof spin j of the spin network can therefore be represented as follows

E2(S) j sim 2j(j + 1) (2j + 1)

j

j

j 1

(685)

This picture can be directly interpreted in terms of recoupling theory which is a sim-ple graphical way of making calculations with SU(2) representation theory In thisrepresentation lines represent contraction of representation indices and nodes repre-sent normalized intertwiners

j

α

β

equiv δβα

γ δ ε

j jprime jprimeprime equiv vγδε (686)

Here α β and γ are indices in an orthonormal basis in the representation j and δand ε in the representations jprime and jprimeprime respectively Index contraction is represented byjoining open ends Since the picture in (685) represents an overall intertwiner between

252 Quantum space

the representation j and the representation j it must be proportional to the identityin the representation j Namely

j

j

j 1 = c j

(687)

The coefficient c can be computed by closing both sides namely tracing the matricesThis gives

c =

j

j

1

j13

(688)

The theta-shaped diagram in the numerator has value unity because it is the norm ofan intertwiner while the denominator is the trace of the identity namely the dimensionof the representation Therefore

c =1

2j + 1 (689)

Putting everything together the action of E2 gives

E2(S) j sim 2 j(j + 1) j (690)

In the present case this result was obtained earlier in a simpler way But the idea canbe used to define a general method of computing in LQG which is very convenientThe general idea is that an operator such as (680) can be represented by the picture

αr s

(691)

where the dots represent the operator E that can grasp a link The result of a graspis the formation of a node and multiplication by the factor

radicj(j + 1) (2j + 1) More

precisely if we include also the numerical part the action of the grasp of a hand locatedat a point x over a link γ with spin j is

x

j

γ

= njΔa[γ x] j

γ

(692)

where

Δa[γ x] equivint

ds γa(s) δ3(γ(s) x) (693)

Calculations have mostly been done with a slightly different notation which derivesfrom [174] It is the KauffmanndashLins (KL) notation and is explained in Appendix A2


Tables of formulas exist in this notation Let us therefore now change to the KLnotation in this section In it one uses the ldquocolorrdquo p = 2j of a link which is twice itsspin4 and is integer and the trivalent nodes are not normalized to unity The relationis given in (A68) In the case of a vertex with spins j j 1 namely colors p p 2 as theone above the normalization factor of the node easily derived from the formulas ofAppendix A2 is

⎛

⎝

j

j

1

⎞

⎠

spin network

=

⎛

⎜⎜⎜⎜⎝

radicj

(j+1)(2j+1)

p = 2j

p = 2j

2

⎞

⎟⎟⎟⎟⎠

KL

(694)

and using (692) the action of the grasp operator in this notation is therefore

x

p

γ

= p Δa[γ x] γ

p (695)

In the next section I give an example of a full calculation using this grasp operator forcomputing the complete spectrum of the area operator

Paths with many hands The definition (680) can be generalized to paths with anarbitrary number of ldquohandsrdquo For instance let

T abc(x r s t) =1

3εijkR

(1)(U(A γxr))il Ea

l (r)

timesR(1)(U(A γxs))jm Eb

m(s)R(1)(U(A γxt))kn Ec

n(t) (696)

Given a closed surface S define the three-hand generalization of the operator (690) as

E3(S) =

int

Sd2σ

int

Sd2σprime

int

Sd2σprimeprime |na(σ) nb(σ

prime) nc(σprimeprime) T abc(x σ σprime σprimeprime)| (697)

where x is a point in the interior of S (whose exact position is irrelevant as we willalways consider the limit of small S) The absolute value in the definition is for laterconvenience This can be represented by the picture

(698)

As we will see in a moment this operator also plays an important physical role

4The expression ldquocolorrdquo is routinely used with two distinct meanings It indicates twicethe spin as here Or it may designate any label of links or nodes (or later edges orfaces of a spinfoam) as in ldquothe links of the spin network are colored with represen-tations and the nodes with intertwinersrdquo

254 Quantum space

664 Degenerate sector

The simplification that we took above in order to compute the spectrum of the areawas to assume that no node is on the surface Here we drop this assumption in orderto find the full spectrum If we drop this assumption the regularization of the areaoperator considered above is not sufficient because we obtain ill-defined expressions ofthe kind int 1

0

dx δ(x) = (699)

We need a better regularization of the operator To this end it is sufficient to smearthe operator transverse to the surface Introduce a smooth coordinate τ over a finiteneighborhood of S in such a way that S is given by τ = 0 Consider then the three-dimensional region around S defined by minusδ2 le τ le δ2 Partition this region into anumber of blocks D of coordinate height δ and square horizontal section of coordinateside ε For each fixed choice of ε and δ we label the blocks by an index I Later we willsend both δ and ε to zero In order to have a one-parameter sequence we now choose δas a fixed function of ε For technical reasons the height of the block D must decreasemore rapidly than ε in the limit thus we put δ = εk with any k greater than 1 andsmaller than 2

Consider one of the blocks The intersection of the block and a τ = constant surfaceis a square surface let AI(τ) be the area of such a surface Let AIε be the average overτ of the areas of the surfaces in the block namely

AIε equiv 1

δ

int δ2

minusδ2

AI(τ)dτ =1

δ

int

DI

d3xradic

EaiEbinanb (6100)

Summing over the blocks yields the average of the areas of the τ = constant surfacesand as ε (and therefore δ) approaches zero the sum converges to the area of the surfaceS Therefore we have

A(S) = limεrarr0

sum

I

AIε equiv limεrarr0

Aε(S) (6101)

The quantity AIε associated with each block can be expressed as follows Write

AIε =radic

A2Iε (6102)

and notice that

A2Iε =

1

δ2

int

DotimesDd3xd3y na(x)nb(y)T

ab(x y) + O(ε5) (6103)

Equation (6103) holds because of the following We have

T ab(x y) = Eai(xI)Ebi (xI) + O(ε) (6104)

for any three points x y and xI in D It follows that

ε4 na(xI)nb(xI)Eai(xI)E

bi (xI) =

1

2δ2

int


ab(x y) + O(ε5)

(6105)

Equation (6103) follows from

A2I =

(1

δ

int δ2

minusδ2

AI(τ)dτ

)2

=

(1

δ

int

Dd3xradic

EaiEbinanb

)2

= ε4 na(xI)nb(xI)Eai(xI)E

bi (xI) + O(ε5) (6106)


jd

ju

j tP

Fig 66 The three classes of links that meet at a node on the surface

Equations (6101) (6102) and (6103) define the regularization of the area The quan-tum operator A(S) is defined by (6101) where

A2Iε equiv 1

2δ2

int


ab(x y) (6107)

The action of A(S) on the quantum states is found from the action of the T ab

operators The operator T ab(x y) annihilates the state |S〉 unless its hands x and y fallon some links of the graph of S If this happens the action of the operator on the stategives the union of S and α with two additional nodes at the points x and y Moreprecisely if x and y fall over two edges of β with color p and q respectively using thegrasp operator (695) we have

T ab(x y)p

q = 2 p qΔa[β x] Δb[β y] 2

p

q x

y

(6108)

Since the loop α runs back and forth between the intersection points x and y (the twograsps) it has spin one or color 2

Consider now the action of the operator A(S) on a generic spin network state |S〉Due to the limiting procedure involved in its definition the operator A(S) does notaffect the graph of |S〉 Furthermore since the action of T ab inside a specific coordinateblock D vanishes unless the graph of the state intersects D the action of A(S) ulti-mately consists of a countable sum of terms one for each intersection P of the graphwith the surface

Consider an intersection P between the spin network and the surface For the purposeof this discussion we can consider a generic point on a link as a ldquobivalent noderdquo andthus say without loss of generality that P is a node In general there will be n linksemerging from P Some of these will emerge upward(u) some downward(d) and sometangential(t) to the surface S see Figure 66 Since we are taking the limit in which theblocks shrink to zero we may assume without loss of generality that the surface andthe links are linear around P (see below for subtleties concerning higher derivatives)Due to the two integrals in (6107) the positions of the two hands of the area operatorare integrated over each block As the action of T ab is nonvanishing only when bothhands fall on the spin network we obtain n2 terms one for every couple of graspedlinks Consider one of these terms in which the grasped links have color p and q Let uswrite the result of the action of T ab with a finite ε on the links p and q of an n-valent

256 Quantum space

intersection P (up to the prefactor) as

p

q

2P

ε (6109)

The irrelevant links are not shown The links labeled p and q are generic in the sensethat their angles with the surface do not need to be specified at this point (the two linksmay also be identical) From the definition (6101) and (6107) of the area operator andthe definition of the T ab operator each term in which the grasps run over two links ofcolor p and q is of the form

T =1

2δ2

int

DotimesDd3xd3y na(x)Δa[β x]nb(y)Δ

b[β y] p q p

q

2P

ε (6110)

giving

T =1

2δ2

int

DotimesD

(

na(x)

int

β

ds βa(s)δ3[β(s) x] (6111)

times nb(y)

int

β

dt βb(t)δ3[β(t) y] p q p

q

2P

ε

)

d3xd3y

=1

2δ2

int

β

ds na(s) βa(s)

int

β

dt nb(t) βb(t)p q p

q

2P

ε

=p q

2δ2

(int

β

ds na(s) βa(s)

int

β

dt nb(t) βb(t)

)

p

q

2P

+ O(ε)

In the last step I have pulled the state out of the integral This is possible because the

ε-dependent states p

q

2P

ε all have the same limit state as ε rarr 0 I write this limit

simply as p

q

2P

without ε that is

p

q

2P

ε = p

q

2P

+ O(ε) (6112)

Hence the substitution of the ε-dependent states with their limit in the integral ispossible up to terms of order O(ε) Note that

int

β

dt nb(t) βb(t) =

0 if β is tangent to Sδ2 otherwise

(6113)

This result is independent of the angle the link makes with the surface because δcan always be chosen sufficiently small so that β crosses the top and bottom of thecoordinate block D (This is the reason for requiring that δ goes to zero faster than ε)Also since we have chosen k smaller than 2 it follows that any link tangential to the


PeP

eP

eP

d

u

t

Fig 67 Trivalent expansion of an n-valent node The dashed lines indicate thelines tangent to the surface

surface exits the box from the side irrespective of its second (and higher) derivativesfor sufficiently small ε and gives a vanishing contribution as ε goes to zero Thereforein the limit the links tangent to the surface do not contribute to the action of the areawhereas every nonvanishing term takes the form

2 p q

8 p

q

2P

(6114)

Generically there will be several links above below and tangential to the surface SExpand the node P into a virtual trivalent spin network We choose to perform theexpansion in such a way that all links above the surface converge to a single ldquoprincipalrdquovirtual link eu all links below the surface converge to a single principal virtual linked and all links tangential to the surface converge to a single principal virtual link etThe three principal links join in the principal trivalent node This trivalent expansionis shown in Figure 67

This choice simplifies the calculation of the action of the area since the sum of thegrasps of one hand on all real links above the surface is equivalent to a single grasp oneu (and similarly for the links below the surface and ed) This follows from the identity

p p q

r

2 +q

p q

r

2 = r

p q

r

2 (6115)

which can be proven as follows Using the recoupling theorem (A65) the left-hand sideof (6115) can be written as

sum

j

(

p

2 p jq r p

minus q λ2r

j

r p jq 2 q

)

2

p

r

q

j (6116)

where j can take the values r minus 2 r and r + 2 A straightforward calculation using

258 Quantum space

(A61) gives

p

2 p jq r p

minus q λ2r

j

r p jq 2 q

= r δjr (6117)

and (6115) follows A repeated application of the identity (6115) allows us toslide all grasps from the real links down to the two virtual links eu and ed Thuseach intersection contributes as a single principal trivalent node regardless of itsvalence

We are now in a position to calculate the action of the area on a generic intersectionFrom the discussion above the only relevant terms are as follows

A2P

q

p

r =

2

8

(

p2

q

p

r

p

p 2

+ q2

q

p

r

q

q2

+ 2 p q

q

p

rp

q

2

)

(6118)

where the first term comes from grasps on the links above the surface the second fromgrasps on two links below the surface and the third from the terms in which one handgrasps a link above and the other grasps a link below the surface Each term in thesum is proportional to the original state (see (A63) (A64)) Therefore we have

A2P

q

p

r = minus l408

( p2 λu + q2 λd + 2 p q λt )

q

p

r (6119)

The quantities λu λd and λt are easily obtained from the recoupling theory Using theformulas in Appendix A2 we obtain

λu =θ(p p 2)

Δ(p)= minus (p + 2)

2p (6120)

λd is obtained by replacing p with q in (6120) λt has the value

λt =

Tet

[p p rq q 2

]

θ(p q r)=

minus2p(p + 2) minus 2q(q + 2) + 2r(r + 2))

8pq (6121)

(Tet is defined in (A60))Substituting in (6118) we have

A2P

q

p

r =

2

16( 2 p (p + 2) + 2 q (q + 2) minus r (r + 2) )

q

p

r (6122)


Since A2P is diagonal the square root can be easily taken

AP

q

p

r =radic

A2P

q

p

r

=

radicradicradicradic2

4

(

2p

2

(p2

+ 1)

+ 2q

2

( q2

+ 1)minus r

2

( r2

+ 1))

q

p

r (6123)

Adding over the intersections and getting back to the spin notation p2 = ju q2 = jd

and r2 = jt the final result is

A(S)|S〉 =

⎛

⎝

2

sum

PisinScapS

radic2ju

P (juP +1) + 2jd

P (jdP +1) minus jt

P (jtP +1)

⎞

⎠ |S〉 (6124)

This expression provides the complete spectrum of the area It reduces to the earlierresult (675) for the case jt

P = 0 and jdP = ju

P (for every P )The complete spectrum of A(S) is therefore labeled by n-tuplets of triplets of positive

half-integers ji namely ji = (jui j

di j

ti ) i = 1 n and n arbitrary It is given

restoring natural units and the Immirzi parameter as

Aji(S) =

4πGγ

c3

sum

i

radic2ju

i (jui + 1) + 2jd

i (jdi + 1) minus jt

i (jti + 1) (6125)

It contains the previous case (675) which corresponds to the choice jui = jd

i andjti = 0 The eigenvalues which are contained in (6125) but not in (675) are called the

degenerate sector

665 Quanta of volume

A second operator that plays a key role in the physical interpretationof the quantum states of the gravitational field is the operator V(R)corresponding to the volume of a region R As for the area operatorconstructed above this quantity requires a bit of work to be defined inthe quantum theory because of the care to be taken in the definition ofthe operator products involved in detE and the square root Consider athree-dimensional region R The volume of R is

V (R) =int

Rd3x

radic13

∣∣∣∣εabcεijkEaiEbjEck

∣∣∣∣ (6126)

To construct a regularized form of this expression consider the classicalquantity (696) In the limit in which r s and t converge to x we have

T abc(x s t r) rarr 2εijk Eai(x)Ebj(x)Eck(x) = 2 εabc detE(x)

(6127)

We can therefore use the 3-hand loop operator to regularize the volume

260 Quantum space

Fix an arbitrary chart of the 3-manifold and consider a small cubicregion RI of coordinate volume ε3 Let xI be an arbitrary but fixed pointin RI Since classical fields are smooth we have E(s) = E(xI) + O(ε) forevery s isin RI and Hα(s t) B

A = 1 BA + O(ε) for any s t isin RI and straight

segment α joining s and t Consider the quantity

WI =1

16ε6 3E3(partRI) (6128)

where E3 is defined in (697) Because of (6127) we have to lowest orderin ε

WI =1

8ε6 3

∣∣det(E(xI)

∣∣int

partRI

d2σ

int

partRI

d2τ

int

partRI

d2ρ∣∣na(σ)nb(τ)nc(ρ)εabc

∣∣

=∣∣detE(xI)

∣∣ (6129)

Thus WI is a nonlocal quantity that approximates the volume elementfor small ε Using the Riemann theorem as in the case of the area we canthen write the volume V(R) of the region R as follows For every ε wepartition R into cubes RIε of coordinate volume ε3 Then

V(R) = limεrarr0

Vε(R) (6130)

Vε(R) =sum

Iε

ε3W12Iε

(6131)

Volume operator Returning now to the quantum theory we have thenimmediately a definition of the volume operator as

V(R) = limεrarr0

Vε(R) (6132)

Vε(R) =sum

Iε

ε3W12Iε

(6133)

WIε =1

16ε6 3E3(partRI) (6134)

where these quantities now are operators Notice the crucial cancellationof the ε6 factor when inserting (6134) into (6133)

The meaning of the limit in (6132) needs to be specified The specification of thetopology in which the limit is taken is an integral part of the definition of the oper-ator As is usual for limits involved in the regularization of quantum field theoreticaloperators the limit cannot be taken in the Hilbert space topology where in general itdoes not exist The limit must be taken in a topology that ldquoremembersrdquo the topologyin which the corresponding classical limit (6130) is taken This is easy to do in thepresent context We say that a sequence of quantum states Ψn converges to the stateΨ if Ψn[A] converges to Ψ[A] for all smooth connections A We use the correspondingoperator topology On rarr O if OnΨ rarr OΨ for all Ψ in the domain

An important consequence of the use of this topology is that a sequence of cylindricalfunctions converges to a cylindrical function defined on the limit graph The graphs


Γn converge to Γ in the topology of the 3-manifold This fact allows us to separate thestudy of a limit into two steps First we study the graph of the limit state Secondwe can study what happens to the coloring of states in order to express the limitrepresentation in terms of the spin network basis

Let us now begin to compute the action of this operator on a spinnetwork state The three surface integrals on the surface of the cube andthe line integrals along the loops combine as in the case of the areato give three intersection numbers which select three intersection pointsbetween the spin network and the boundary of the cube At these threepoints which we denote as r s and t the small graph γστρ of the operatorgrasps the spin network

Notice that the integration domain of the (three) surface integrals is a six-dimensional space ndash the space of the possible positions of three points on the surfaceof a cube Let us denote this integration domain as D6 The absolute value in (6134)plays a crucial role here contributions from different points of D6 have to be taken astheir absolute value while contributions from the same point of D6 have to be summedalgebraically before taking the absolute value The position of each hand of the operatoris integrated over the surface and therefore each hand grasps each of the three pointsr s and t producing 33 distinct terms However because of the absolute value a termin which two hands grasp the same point say r vanishes This happens because theresult of the grasp is symmetric but the operator is antisymmetric in the two hands ndashas follows from the antisymmetry of the trace of three sigma matrices Thus only termsin which each hand grasps a distinct point give nonvanishing contributions For eachtriplet of points of intersection r s and t between spin network and cube surface thereare 3 ways in which the three hands can grasp the three points These 3 terms havealternating signs because of the antisymmetry of the operator but the absolute valueprevents the sum from vanishing and yields the same contribution for each of the 3terms

If there are only two intersection points between the boundary of thecube and the spin network then there are always two hands graspingat the same point contributions have to be summed before taking theabsolute value and thus they cancel Thus the sum in (6133) reducesto a sum over the cubes Iiε whose boundaries have at least three distinctintersections with the spin network and the surface integration reducesto a sum over the triple-grasps at distinct points For ε small enough theonly cubes whose surfaces have at least three intersections with the spinnetwork are cubes containing a node i of the spin network Therefore thesum over cubes reduces to a sum over the nodes n isin S capR of the spinnetwork contained inside R Let us denote by Inε the cube containing thenode n We then have

V(R)|S〉 = limεrarr0

sum

nisinScapVε3radic

|WInε | |S〉

WInε |S〉 =1

16 ε6 3E3(partRI)|S〉 (6135)

262 Quantum space

The action of the operator E3(partRI) is the sum over the triplets (r s t) ofdistinct intersections between the spin network and the boundary of thecube For each such triplet let T (r s t)|S〉 be the result of this actionThen


sum

nisinScapVε3radic

|WInε | |S〉

WInε |S〉 =1

16 ε6 3

sum

rst

T (r s t) |S〉 (6136)

Next the key point now is that in the limit ε rarr 0 the operator does notchange the graph of the spin network state nor the coloring of the linksThe only possible action of the operator is therefore on the intertwinersTherefore

V(R) |Γ jl i1 iN 〉 = (16πG)32sum

nisinScapVVin

iprimen |Γ jl i1 iprimen iN 〉

(6137)The computation of the numerical matrices Vin

iprimen is an exercise in recou-pling theory For instance for a trivalent node we have to compute W in

j1 j2j3

1

1

113

13 = W j1

j2

j3

(6138)

and more complicated diagrams for higher-valence nodes The completecalculation is presented in great detail in [175] where a list of eigenvaluesis also given One of the interesting outcomes of the detailed calculationis that the node must be at least quadrivalent in order to have a non-vanishing volume

The operator can be shown to be a well-defined self-adjoint nonnegativeoperator with discrete spectrum For each given graph and labeling weshall choose from now on a basis in of intertwiner that diagonalizesthe matrices Vin and therefore the volume operator We denote Vin thecorresponding eigenvalues

67 Quantum geometry

Physical interpretation of the spin network states The essential propertyof the volume operator is that it has contribution only from the nodes of aspin network state |S〉 That is the volume of a region R is a sum of termsone for each node of S inside R Therefore each node of a spin networkrepresents a quantum of volume That is we can interpret a spin network

67 Quantum geometry 263

with N nodes as an ensemble of N quanta of volume or N ldquochunksrdquo ofspace located in the manifold ldquoaroundrdquo the node each with a quantizedvolume Vin

The elementary chunks of quantized volume are then separated fromeach other by surfaces The area of these surfaces is governed by the areaoperator The area operator A(S) has contribution from each link of Sthat crosses S Therefore the following interpretation follows Two chunksof space are contiguous if the corresponding nodes are connected by a linkl In this case there is an elementary surface separating them and thearea of this surface is determined by the color jl of the link l to be

Al = 8πcminus3Gradic

jl(jl + 1) (6139)

Therefore the intertwiners associated with the nodes are the quantumnumbers of the volume and the spins associated with the links are quan-tum numbers of the area Volume is on the nodes and area is on thelinks separating them The graph Γ of the spin network determines theadjacency relation between the chunks of space

In other words the graph Γ can be interpreted as the graph dual to acellular decomposition of physical space in which each cell is a quantumof volume

Thus a spin network state |S〉 determines a discrete quantized 3d metricThis physical picture is beautiful and compelling However its full beautyreveals itself only in going to the space of the diffeomorphism-invariantstates Kdiff

Physical interpretation of the s-knot states Consider an s-knot state |s〉For simplicity consider the generic case in which its symmetry group istrivial so that we can disregard the technicalities due to the diffeomor-phisms that change orientation and ordering Then we can view |s〉 as theprojection under Pdiff of a spin network state |S〉 In going from the spinnetwork state |S〉 to the s-knot state |s〉 we preserve the entire informationin |S〉 except for its localization on the 3d space manifold This is preciselyas the implementation of diffeomorphism invariance in the classical theorywhere a physical geometry is an equivalence class of metrics under diffeo-morphisms In the quantum case |s〉 retains the information about thevolume and the adjacency of the chunks of volumes and about the areaof the surfaces that separate these volumes But any information of thelocalization of the chunks of volume on the 3d manifold is lost under Pdiff

The physical interpretation of the resulting state |s〉 is therefore ex-tremely compelling it represents a discrete quantized geometry This isformed by abstract chunks of space which do not live on the 3d manifold

264 Quantum space


they are only localized with respect to one another Their spatial relationis only determined by the adjacency defined by the links see Figure 68These are not quantum excitations in space they are quantum excitationsof space itself Volume of the chunks and area of the surfaces are givenby the coloring of the s-knot The spins jl are the quantum numbers ofthe area and the intertwiners in are the quantum numbers of the volume

These are quantum states defined in a completely 3d diffeomorphism-invariant manner and with a simple physical interpretation These are thequantum states of space

Surfaces and regions on s-knots Recall that in classical GR we distin-guish between a metric g and a geometry [g] A geometry is an equiv-alence class of metrics under diffeomorphism For instance in three di-mensions the euclidean metric gab(x) = δab and a flat metric gprimeab(x) = δabare different metrics but define the same geometry [g] = [gprime] The no-tion of geometry is diffeomorphism invariant while the notion of metricis not On a given manifold with coordinates x we can define a surfaceby S = (σ1 σ2) rarr xa(σi) Then it makes sense to ask what is the areaof S in a given metric gab(x) but it makes no sense to ask what is thearea of S in a given geometry because the relative location of S and thegeometry is not defined

However given a geometry it is meaningful to define surfaces on thegeometry itself For instance (in 2d) given the geometry of the surface ofthe Earth (an ellipsoid) the equator is a well-defined (1d) surface and


Fig 69 Regions and surfaces defined on an s-knot the set of the thick blacknodes define a ldquoregionrdquo of space the set of the thick black links define theldquosurfacerdquo surrounding this region

so is the parallel 1 km north of the equator their location is determinedwith respect to the geometry itself Concretely a surface on a geometrycan be defined in various manners For instance it can be defined by thecouple (S g) with g isin [g] The couple (φS φlowastg) defines the same surfaceAlternatively the surface can be defined intrinsically (the equator is thelongest geodesic)

Now in quantum gravity we find precisely the same situation Abovewe have defined coordinate surfaces S and regions R and their areasand volumes Such coordinate surfaces and regions are not defined atthe diffeomorphism-invariant level However we can nevertheless definesurfaces and regions on the abstract quantum state |s〉 itself and associateareas and volumes with them A region R is simply a collection of nodesIts boundary is an ensemble of links and defines a surface we can say thatthis surface ldquocutsrdquo these links A moment of reflection will convince thereader that this is precisely the same situation as in the classical theory

For instance consider an s-knot state with two four-valent nodes andfour links with spins 12 12 1 1 On this quantum geometry we canidentify a closed surface separating the two quanta of volume This surfacecuts the four links and has area A = (8

radic3 + 16

radic2)πGcminus3 A more

complex situation is illustrated in Figure 69

Eigenvalues and measurements Suppose we had the technological capa-bility to measure the area of a surface or the volume of a region withPlanck-scale precision An example of an area measurement for instanceis the measurement of the cross section of an interaction Shall we obtainone of the eigenvalues computed in this section If the theory developed

266 Quantum space

so far is physically correct the answer is yes In fact area and volume arepartial observables Partial observables can be measured and the theorypredicts that the possible outcomes of a measurement are the numbers inthe spectrum of the corresponding operator

Therefore these spectra are precise quantitative physical predictions ofLQG

This prediction has raised a certain discussion The objection has been made thatin the classical theory area and volume of coordinate surfaces and regions are notdiffeomorphism-invariant quantities and therefore we cannot interpret them as trueobservables The objection is not correct it is generated by the obscurity of the con-strained treatment of diffeomorphism-invariant systems To clarify this point considerthe following simple example Consider a particle moving on a circle subject to a forceLet φ be the angular coordinate giving the position of the particle and pφ its conju-gate momentum As we know well pφ turns out to be quantized Now if we write thecovariant formulation of this system we have the WheelerndashDeWitt equation

Hψ(t φ) =

(i

part

parttminus

2 part2

partφ2+ V (φ)

)ψ(t φ) = 0 (6140)

which in the language of constrained systems theory is the hamiltonian constraintequation Notice that the momentum pφ is not a gauge-invariant quantity it doesnot commute with the operator H that is [pφ H] = 0 This happens precisely forthe same reason for which area and volume are not gauge-invariant quantities in GRBut this does not affect the simple fact that we can measure pφ and we do predictthat it is quantized The confusion originates from the distinction between completeobservables and partial observables which was explained in detail in Chapter 3 pφ isnot a complete observable but it is nevertheless a partial observable We cannot predictthe physical value of pφ from a physical state namely from a solution of the WheelerndashDeWitt equation because we do not know at which time this is to be measured Butwe can nevertheless compute and predict its eigenvalues

Alternatively we can define the evolving constant of the motion pφ(T ) as in Section515 This is a gauge-invariant quantity The spectral properties of pφ(T ) howeverare the same as those of pφ More importantly they are not affected by the potentialV (φ) namely they are not affected by the dynamics Similarly we could in principle usegauge-invariant definitions of areas of surfaces that would be genuinely diffeomorphisminvariant but this complicated exercise is useless because the spectral properties canbe directly determined by the partial observable operators

Why Diff It is now time to address the question of the choice of the precisefunctional space of coordinate transformations or active gauge maps φ M rarr M and to justify the fact that I have chosen Diff lowast instead of Diff Notice that in theclassical theory the precise functional space in which we choose the fields is dictated bymathematical convenience not by the physics In fact we always make measurementssmeared in spacetime which cannot be directly sensitive to what happens at pointsIndeed in classical field theory we choose to change freely the class of functions whenconvenient For instance twice differentiable fields allow us to write the equations ofmotion more easily but then we prefer to work with distributional fields in certainapplications Analytic fields are usually considered too rigid because we do not liketoo much the idea that a field in a small finite neighborhood could uniquely determinethe field everywhere The smooth category (Cinfin) is often easy and convenient and it


has been generally taken as the natural point of departure in quantum gravity but itis not God-given If we use smooth fields it is natural to consider smooth coordinatetransformations and Diff as the gauge group In fact this was the traditional choicein quantum gravity However at the end of Section 64 I pointed out that if we chooseDiff as the gauge group then the knot classes are labeled by continuous parameters(moduli) and the space Kdiff turns out to be nonseparable At first we may thinkthat these moduli represent physical degrees of freedom If they did there would beobservable quantities that are affected by them However none of the operators that wehave constructed in this chapter is sensitive to these moduli In particular a momentof reflection shows that all the geometric operators are only sensitive to features ofgraphs (and surfaces and volumes) that are invariant under continuous deformationsof the graphs (surfaces and regions) Therefore it is possible that these moduli are anartifact of the mathematics they have nothing to do with the physics They just reflectthe fact that we have not chosen the functional space of the maps φ appropriately Theφ in Diff are too ldquorigidrdquo in the sense that they leave invariant the linear structure ofthe tangent space at a node while this linear structure has no physical significance Thechoice of Diff lowast as gauge group is a simple extension of the gauge group that gets rid ofthe redundant parameters Accordingly we have to work with a space of fields slightlylarger that Cinfin Nothing changes in the classical theory while the quantum theory iscured of the double problem of having a nonseparable Hilbert space and redundantphysically meaningless moduli Of course choices other than Diff lowast are possible

Noncommutativity of the geometry I close this section with an observa-tion Consider a spin network state containing a four-valent node n Letl1 l2 l3 l4 be the four links adjacent to the node n Let i be the intertwineron this node Consider a surface S(12)(34) such that n is on S(12)(34) thelinks l1 and l2 are on one side of the surface while the links l3 and l4 are onthe other side of it We can choose a basis in the space of the intertwinersby splitting the node n into two trivalent nodes joined by a virtual link l(see Appendix A1) Let us do so by pairing the links as (l1 l2) and (l3 l4)That is the two trivalent nodes are between (l1 l2 l) and (l l3 l4) A basisin the space of intertwiners is then given by

vα1 α2 α3 α4j = vα1 α2 αjvαj α3 α4 (6141)

where the indices αi are in the representations of the links and the index αj

is in the representation j It is not hard to show that in order for the stateto be an eigenstate of the area of Sa the intertwiner must be one of thesebasis elements In other words the basis (6141) diagonalizes the areaof Sa Now consider a surface S(13)(24) such that n is on it and the linksl1 and l3 are on one side of the surface while the links l2 and l4 are onthe other side of it Clearly in this case it will be a different basis in thespace of the intertwiners that diagonalizes the area It will be the basis

wα1 α2 α3 α4k = vα1 α3 αkvαk α2 α4 (6142)

The two bases are related by a 6j symbol In general they are different Itfollows that the operator A(S(12)(34)) and the operator A(S(13)(24)) do

268 Quantum space

not commute (if they did they would be diagonalized by the same basis)Therefore the 3d geometry is in a sense noncommutative area operatorsof intersecting surfaces do not commute with each other

671 The texture of space weaves

What is the connection between the discrete and quantized geometrydescribed above and the smooth structure of physical geometry that weperceive around us The answer requires a few steps

Weaves Ordinary measurements of geometric quantities ndash that is mea-surements of the gravitational field ndash are macroscopic we observe thegeometry of space at a scale l much larger than the Planck length lP Atthis large scale planckian discreteness is smoothed out

Consider the fabric of a T-shirt as an analogy At a distance it is asmooth curved two-dimensional geometric surface At a closer look it iscomposed of thousands of one-dimensional linked threads The image ofspace given by LQG is similar Consider a very large spin network formedby a very large number of nodes and links each of Planck scale Micro-scopically it is a planckian-size lattice But probed at a macroscopic scaleit appears as a three-dimensional continuous metric geometry Physicalspace around us can therefore be described as a very fine weave Thehidden texture of reality is a weave of spins

This intuitive picture can be made precise Fix a classical macro-scopic 3d gravitational field e which determines a macroscopic 3d metricgab(x) = eia(x) eib(x) It is possible to construct a spin network state |S〉that approximates this metric at a scale l lP The precise relationbetween |S〉 and e is the following Consider a region R (or a surface S)with a size larger than l (in the metric g) and slowly varying at this scaleRequire that |S〉 is an eigenstate of the volume operator V(R) (and ofthe area operator A(S)) with eigenvalues equal to the volume of R (andof the area of S) determined by e up to small corrections in lPl That is

V(R)|S〉 =(V[eR] + O(lPl)

)|S〉

A(S)|S〉 =(A[eS] + O(lPl)

)|S〉 (6143)

where V[eR] (resp A[eS]) given in (273) (and (270)) is the volume(the area) of the region (the surface) determined by the gravitationalfield e

A spin network state |S〉 that satisfies these equations for any largeregion and surface is called a ldquoweaverdquo state of the metric g At largescale the state |S〉 determines precisely the same volumes and areas as g


This definition is given at the nondiff-invariant level but it can be easilycarried over to the diff-invariant level the s-knot state |s〉 = Pdiff |S〉 iscalled the weave state of the 3-geometry [g] the equivalence class of 3-metrics to which the metric g belongs

Several weave states were constructed and studied in the early daysof LQG for various 3d metrics including the ones of flat spaceSchwarzschild and gravitational waves They satisfied (6143) or equa-tions similar to these (at the time the area and volume operators werenot known and other operator functions of the gravitational field wereused to play the same role) Most of these weave states were constructedbefore the discovery of the spin network basis working with the morecumbersome loop basis Equations (6143) do not determine a weave stateuniquely from a given 3-metric There is a large freedom in constructinga weave state for a given metric because only the averaged properties areconstrained by (6143) The weave states constructed should not be takenas realistic proposals for the microstates of a given macroscopic geome-try They are only a proof of existence of microstates that have specifiedmacroscopic properties

On the other hand the weave states have played a very important rolein the historical development of the LQG I recall this role below becauseit contains an important physical lesson on the physics of Planck-scalediscreteness

The failure of the a rarr 0 limit and the emergence of Planck-scale dis-creteness There is a gap of several years between the construction of theloop representation of quantum GR (c 1988) and the calculation of theeigenvalues of area and volume (c 1995) which revealed that the theorypredicts a discrete structure of space During these years the fact that theloops have ldquoPlanck sizerdquo was not known and at first not even suspectedThe intuition was that a macroscopic geometry could be constructed bytaking a limit of an infinitely dense lattice of loops ndash roughly as a con-ventional QFT can be defined by taking the limit of a lattice theory asthe lattice size a goes to zero To construct a weave state approximatinga classical metric therefore the aim was at first to satisfy equations like(6143) by quantum states defined as limits where the spatial density ofthe loops was taken to infinity But something unexpected and very re-markable happened With increasing density of loops the accuracy of theapproximation did not increase Instead the eigenvalue of the operatorincreased

Let me be more precise Suppose we start with a 3d manifold withcoordinates x We want to define a weave on this manifold that approxi-mates the flat 3d metric g(0)

ab(x) = δab that is the field e(0)ia(x) = δia

270 Quantum space

We construct a spatially uniform weave state |Sa0〉 formed by a tangle ofloops of coordinate density ρ = aminus2

0 (The coordinate density ρ can bedefined as the ratio between the total coordinate length L of the loopsand the total coordinate volume V ) The loops are then at an average dis-tance a0 from each other Therefore one expects that the approximation(6143) would break down at the scale l sim a0 The idea was therefore toimprove the approximation by decreasing the ldquolattice spacingrdquo a0 namelyby increasing the coordinate density of the loops But decreasing a0 tobecome a lt a0 the calculations yielded instead

A(S) |Sa〉 sima2

0

a2

(A[e(0)S] + O(lPl)

)|S〉 (6144)

instead of a decrease in the error the area increases In other wordsby adding loops we do not obtain a better approximation Rather weapproximate a different field Since

a20

a2A[e0S] = A[(a0a) e(0)S] = A[e(a)S] (6145)

where e(a)ia(x) = a2

0a2 δai the weave with increased loop density approxi-

mates the metric

g(a)ab (x) =

a20

a2δab (6146)

But notice that the physical density of the loops ρa does not changewith decreasing a The physical density ρa is the ratio between the totallength of the loops and the total volume determined by the metric g(a)namely by the metric that the state |Sa〉 itself determines via equation(6143) This is

ρa =La

Va=

(a0a)L(a0a)3V

=a2

a20

ρ =a2

a20

aminus2 = aminus20 (6147)

The physical density remains aminus20 irrespective of the density of the loops

a chosen But then if a0 is not determined by the density of the loopsit must be given by a dimensional constant of the theory itself and sincethe only scale in the theory is the Planck scale we have necessarily thatup to numerical factors

a0 sim lP (6148)

At first this result was disconcerting The theory refused to approximatea smooth geometry at a physical scale lower than lP Then the reasonbecame clear there is no physical scale lower than lP Each loop carries


a quantum of geometry of Planck size more loops give more size not abetter approximation to a given geometry This was the first unexpectedhint that the loops themselves have an intrinsic geometric size and thatin the theory there is no spatial structure at physical scales smaller thanthe Planck scale

Quantum and classical discreteness superposition of weaves The weavepicture of space resembles the space of a lattice theory But there is astrong difference between the two and the analogy should be taken withgreat care

Planck-scale discreteness is predicted by loop quantum gravity on thebasis of a standard quantization procedure in the same manner in whichthe quantization of the energy levels of an atom is predicted by nonrela-tivistic quantum mechanics while the discretization of space in a latticetheory is assumed

But the difference is far more substantial than this In a lattice theorythe lattice is a fixed structure on which the theory is defined A weaveon the other hand is one of many quantum states that have a certainmacroscopic property and a very peculiar one since it is a single elementof the spin network basis There is no reason for the physical state of spacenot to be in a generic state and the generic quantum state that has thismacroscopic property is not a weave state it is a quantum superpositionof weave states Therefore it is reasonable to expect that at small scalespace is a quantum superposition of weave states

Therefore the picture of physical space suggested by LQG is not trulythat of a small-scale lattice Rather it is a quantum probabilistic cloudof such lattices

The Minkowski vacuum To a certain approximation macroscopic spacearound us is described by the Minkowski metric We should thereforeexpect that in the quantum theory there is a state |0M〉 (M for Minkowski)that reproduces the Minkowski metric at large scales

At fixed time Minkowski 3d space is described by a flat 3-metric gnamely by the gravitational field eia(x) = δia Does this imply that weshould expect |0M〉 to be a weave state of e It does not In quantumMaxwell theory the state that corresponds to the classical solution withvanishing electric and magnetic fields E = B = 0 is the vacuum state|0〉 But |0〉 is not an eigenstate of the electric field E Since E and Bdo not commute E and B have no common eigenstates Eigenstates ofE are maximally spread in B Instead E and B have vanishing meanvalue and minimal spread on |0〉 The situation is precisely the same asfor the vacuum state of an harmonic oscillator The classical solutionx(t) = p(t) = 0 corresponds to the vacuum state a state which is neither

272 Quantum space

an eigenstate of the position x nor an eigenstate of the momentum p Aneigenstate of x has Δp very large and would spread instantaneously

Similarly |0M〉 cannot be an eigenstate of the gravitational field EAn eigenstate of E has maximum spread in the gravitational magneticfield and would spread instantaneously Accordingly |0M〉 is not a weavestate of the flat geometry nor a superposition of weave states of the flatgeometry

|0M〉 must be a state with vanishing mean value and minimal spreadof the gravitational electric and magnetic fields On the other hand itshould be concentrated around weave states in the same sense in whichthe vacuum of the harmonic oscillator is concentrated around x = 0

We do not yet know the form of the state |0M〉 explicitly In fact theexploration of this functional is one of the main open problems in thetheory I will come back to this problem in Chapter 9

We can obtain some hints about the state |0M〉 from the classical fieldtheory obtained linearizing GR around the Minkowski solution In theclassical theory let us write

eIμ(x) = δIμ + hIμ(x) (6149)

and restrict our considerations to solutions of the Einstein equationswhere hIμ(x) 1 To first order in h these solutions satisfy the lin-earized Einstein equations The linearized Einstein equations are free waveequations on Minkowski space for a spin-2 field The solutions are super-positions of plane waves They can be gauge-fixed fixing h0

μ = hI0 = 0and restricting hia(x) to the sole transverse traceless components that ispartah

ia = partih

ia = haa = 0 For each momentum k there are then two inde-

pendent polarizations εplusmn In this gauge the linearized Einstein equationsdescribe simply a collection of uncoupled harmonic oscillators of frequencyω(k) = |k| =

radickaka one for each polarization and for each Fourier mode

hia(k) = (2π)minus32

intdx eikx hia(x) (6150)

The quantum field theory of this free system is a fully conventionalfree QFT In its vacuum state |0lin〉 (lin for linearized) all oscillators arein their ground state The Hilbert state of the theory is the Fock spacespanned by the states

|k1 ε1 kn εn〉 (6151)

containing n quanta with momenta and polarizations (ki εi) Thesequanta are called gravitons hia(x) is a conventional field operator on thisFock space (formed by creation and annihilation parts) For each (gauge-fixed) field configuration h we can write the (generalized) eigenstates


|h〉 These allow us to write a generic state |ψ〉 of the Fock space in theSchrodinger representation

Ψ[h] = 〈h|ψ〉 (6152)

In particular the vacuum state is given in this representation by

Ψ0[h] = 〈h|0lin〉 = Neminus12

intdk |k| ha

i (minusk) hia(k) (6153)

This is a gaussian functional concentrated around the field configurationh = 0 We can rewrite this state as a functional of the gravitational fieldas

Ψ0[e] = 〈eminus δ|0lin〉 = Neminus12

intdk |k| (eai (minusk)minusδai ) (eia(k)minusδia) (6154)

which is a functional concentrated around g = δIt is then reasonable to suspect that |0M〉 should satisfy

ΨM[Se] equiv 〈Se|0M〉 sim Ψ0[e] = Neminus12


for all spin network states Se that are weaves for the field e This is astate concentrated around the flat weave S0

The ldquoemptyrdquo state The Minkowski vacuum state |0M〉 should not be con-fused with the covariant vacuum state |0〉 and with the empty state |empty〉(see Section 542) The state

Ψempty[A] = 〈A|empty〉 = 1 (6156)

is an eigenstate of A(S) and V(R) with vanishing eigenvalue Thereforeit describes a space with no volume and no area Spin network statescan be constructed acting on |empty〉 with the holonomy operator In thisparticular sense |empty〉 is analogous to the Fock vacuum The state |empty〉 isgauge invariant and diff invariant hence this state is also in K0 and inKdiff In fact it represents the quantum state of the gravitational field inwhich there is no physical space at all As we shall see in the next chapter|empty〉 is a solution of the WheelerndashDeWitt equation and therefore it is in Has well

mdashmdash


The ldquoloop representation of quantum general relativityrdquo was introducedin [176 177] These papers present the first surprising results of the ap-proach solutions of the WheelerndashDeWitt equation and general solutions

274 Quantum space

of the diffeomorphism constraint The loop transform mapping betweenfunctionals of the connection and loop functionals was illustrated in [170]The approach was motivated by Ted Jacobsonrsquos and Lee Smolinrsquos discov-ery [178] of loop solutions of the WheelerndashDeWitt equation written inAshtekar variables Rodolfo Gambini and his collaborators have indepen-dently developed a formal loop quantization for YangndashMills theory [179]The importance for the theory of the nodes ndash where loops intersect ndash wasstressed by Jorge Pullin nodes were studied in [180] An account of thisfirst stage of LQG can be found in the review [2]

The importance for the theory of the graphs was understood by JerzyLewandowski The spin network basis was introduced in quantum grav-ity in [171] solving the longstanding difficulty of the overcompleteness ofthe loop basis The inspiration came from Penrosersquos speculations on thecombinatorial structure of space [181] For motivations and the history ofthe idea of spin networks see [182] The mathematical systematization ofthis idea is due to John Baez [183] The fact that the equivalence classesunder Diff of graphs with intersections are not discrete and the associatedproblem of the nonseparability of the state space was pointed out in [171]and studied in [184] where the structure of the corresponding modulispaces is analyzed The solution of the problem in terms of Diff lowast is dis-cussed in [185] A different approach to obtain a separable Hilbert space isin [186] The uniqueness theorem for the loop representation in quantumgravity (the ldquoLOSTrdquo theorem) has been proven by Jerzy LewandowskiAndrzej Okolow Hanno Sahlmann and Thomas Thiemann [187] and ina slightly different version by Christian Fleischhack [188]

The idea that LQG could predict a discrete Planck-scale geometryemerged in studying the weave states [189] The first explicit claim thatthe eigenvalues of the area represent a physical prediction of the theoryobservable in principle is in [190] The definition of the area and vol-ume operators and the first calculations of their discrete eigenvalues arein [191] The main sequence of the spectrum of the area was calculatedin this work The degenerate sector was computed in [173] and in [192]which I have followed here A calculation mistake in the spectrum of thevolume in the first version of [191] was soon found by Renate Loll [193]in developing the lattice version of the volume operator Loll noticed thatthe node must be at least quadrivalent in order to have a nonvanishingvolume There exist a number of equivalent constructions of the volumeoperator in the literature they all define the same operator except forone possible variant A systematic study of the variants in the definition ofthe volume operator has been completed by Jerzy Lewandowski in [194]Systematic techniques to compute geometry eigenvalues were developedin [175] using the mathematics of [174] For more details (and another


application) see also [195] A length operator (which is surprisingly farmore complicated than area and volume to deal with) is studied byThiemann in [196] The noncommutativity of the areas of intersectingsurfaces has been studied in detail in [197] On angle operators see [198]An applealing and well written introduction to spin networks and theirgeometry is Seth Majorrsquos [199]

The mathematical-physics version of LQG started from the seminalwork of Abhay Ashtekar Chris Isham and Jerzy Lewandowski [200 201]See Ashtekarrsquos 1992 Les Houches lectures [202] and John Baez [203] Thisdirection led to the paper [204] where the loop representation of [176 177]was mathematically systematized the relation between the different vari-ants of the formalisms is still very confused in [204] and was elucidatedby Roberto De Pietri [205] and Thomas Thiemann For a detailed intro-duction and full references see [9] The relation between the full theoryand the linearized theory has been explored in [206] The notion of weavestate was introduced in [189] for a recent discussion and references see[207]

7Dynamics and matter

In the previous chapter I constructed the Hilbert space and the basic operators ofLQG In this chapter I discuss the dynamical aspects of the theory For this we mustwrite a well-defined version of the WheelerndashDeWitt equation (63) in order to constructthe hamiltonian operator H

The construction of the hamiltonian operator of LQG has taken a long time pro-ceeding through a number of steps The first was to realize that the operator can bedefined via a simple regularization and that any loop state Ψα solves HΨα = 0 pro-vided that α is a loop without self-intersections This observation opened the door toLQG The result remains true in all subsequent definitions of H With this first simpleregularization however H diverges on intersections

The second step was to realize that a finite operator H could be obtained providedthat (i) its action is defined on diffeomorphism-invariant states and (ii) its densitycharacter is appropriately dealt with By a sort of magic the action of the operatorwith the correct density weight on a diff-invariant state converges trivially in the limitin which the cut-off is removed This result is the major pay-off of the background-independent approach to QFT It is a manifestation of the relation between backgroundindependence and the absence of UV divergences

The third step was the idea of writing H as a commutator a technique that allowsus to write the operator avoiding square roots and inversion of matrices This techniquesolved at once the remaining roadblocks on the one hand it made it possible to usethe real Barbero connection thus avoiding the difficulties of implementing nontrivialreality conditions in the quantum theory on the other hand it made it possible tocircumvent the difficulties due to the nonpolynomiality of H such as the square rootpreviously used to get a density-weight-one hamiltonian

In this chapter I do not follow the contorted historical path but rather define theoperator directly I discuss only the operator corresponding to the first term of thehamiltonian (443) that defines lorentzian GR with a real connection This first termalone defines euclidean quantum GR The technique described here extends directly tothe second term for which I refer the reader to [20]

Following common GR parlance I call ldquomatterrdquo anything which is not the gravita-tional field At best as we know the content of the Universe is the one described byGR and the standard model fermions YangndashMills fields gravitational field and pre-sumably Higgs scalars As explained in Chapter 1 LQG has no ambition of providinga naturally ldquounifiedrdquo theory or explaining the reasons of the content of the Universe

276

71 Hamiltonian operator 277

In this chapter I assume that the four entities noted above make up the Universe andI describe how the background-independent quantum theory of the gravitational fielddescribed thus far can be very naturally extended to a background-independent theoryfor all these fields

The remarkable aspect of this extension is that the finiteness of the gravitationaldynamics extends to matter In the theory there is ldquono spacerdquo for UV divergencesneither for gravity nor for matter

71 Hamiltonian operator

Regularization The form of the hamiltonian H which is most convenientfor the quantum theory is the one given in (416) namely

H =int

N tr(F and V A) (71)

The reason is that we have already defined the quantum operator V andthe operators F and A can be defined as limits of holonomy operators ofsmall paths while the classical Poisson bracket can readily be realized inthe quantum theory as a quantum commutator

Fix a point x and a tangent vector u at x consider a path γxu ofcoordinate length ε that starts at x tangent to u Then the holonomy canbe expanded as

U(A γxu) = 1 + ε ua Aa(x) + O(ε2) (72)

Similarly fix a point x and two tangent vectors u and v at x and considera small triangular loop αxuv with one vertex at x and two sides tangentto u and v at x each of length ε Then

U(Aαxuv) = 1 +12ε2 uavb Fab(x) + O(ε3) (73)

Using this and writing hγ = U(A γ) we can regularize the expression ofthe hamiltonian by writing it as

H = limεrarr0

1ε3

intNεijktr

(hγminus1

xukhαxuiuj

V hγxuk)

d3x (74)

Here (u1 u2 u3) are any three tangent vectors at x whose triple productis equal to unity Following the same strategy we used for the area andvolume operators let us partition the 3d coordinate space in small regionsRm of coordinate volume ε3 We can then write the integral as a Riemannsum and write

H = limεrarr0

1ε3

sum

m

ε3Nmεijktr(hγminus1

xmukhαxmuiuj

V(Rm) hγxmuk) (75)

278 Dynamics and matter

where xm is now an arbitrary point in Rm and Nm = N(xm) The factthat the limit is independent of the choice of this point is assured by theRiemann theorem Notice that the ε3 factors cancel and can therefore bedropped

Definition of the operator Since V(Rm) and hγ are well-defined operatorsin K we can then consider the corresponding quantum operator

H = minus i

limεrarr0

sum

m

Nm εijk tr(hγminus1

xmukhαxmuiuj

[V(Rm) hγxmuk]) (76)

To complete the definition of the operator we have yet to choose the pointxm the three vectors (u1 u2 u3) and the paths γxuk

and αxuiuj in eachregion Rm This must be done in such a way that the resulting quantumoperator is well defined covariant under diffeomorphism invariant underinternal gauges and nontrivial These requirements are highly nontrivialRemarkably there is a choice that satisfies all of them

The key observation to find it is the following When acting on a spinnetwork state this operator acts only on the nodes of the spin networkbecause of the presence of the volume (This is not changed by the pres-ence of the term hγxu in the commutator for the following reason Thevolume operator vanishes on trivalent nodes The operator hγxu can atmost increase the valence of a node by one Therefore there must be atleast a trivalent node in the state for H not to vanish)

Therefore in the sum (76) only the regions Rm in which there is anode n give a nonvanishing contribution Call Rn the region in which thenode n of a spin network S is located Then

H|S〉 = limεrarr0

Hε|S〉 (77)

where

Hε|S〉 = minus i

sum

nisinSNnε

ijk tr(hγminus1

xnukhαxnuiuj

[V(Rn) hγxnuk])|S〉 (78)

The sum is now on the nodes Now the only possibility to have a nontrivialcommutator is if the path γxnuk

itself touches the node We thereforedemand this This can be obtained by requiring that xn is precisely thelocation of the node Recall that the precise location of xn is irrelevant inthe classical theory because of the Riemann theorem but not so in thequantum theory This fixes xn

Finally there is a natural choice of the three vectors (u1 u2 u3) and forthe paths γxuk

and αxuiuj take (u1 u2 u3) tangent to three links l lprime lprimeprime

emerging from the node n (The condition that their triple product is


l

lprime

lprimeprime

γαxlprimelprimeprime

xl

x

Fig 71 The path γxl and the loop αxlprimelprimeprime at a trivalent node in the point x

unity can be satisfied by adjusting the length) Take γxukto be a path

γxl of coordinate length ε along the link l Take αxuiuj to be the triangleαxlprimelprimeprime formed by two sides of coordinate length ε along the other two linkslprime and lprimeprime and take the third side as a straight line (in the coordinates x)connecting the two end points This straight line is called an ldquoarcrdquo Thesum over i j k is a sum over all permutations of the three links see Figure71 If the node has valence higher than three that is if there are morethan three links at the node n we preserve covariance summing over allordered triplets of distinct links Thus we pose

Hε|S〉 = minus i

sum

nisinSNn

sum

llprimelprimeprimeεllprimelprimeprime tr

(hγminus1

xnlhαxnlprimelprimeprime [V(Rn) hγxnl

])|S〉

(79)

where εllprimelprimeprime is the parity of the permutation (determined by the sign of

the triple product) This completes the definition of the hamiltonian con-straint (up to one additional detail that we add below) Two major ques-tions are left open First whether the limit is finite Second whetherit is well behaved under diffeomorphisms and gauge transformations Inparticular whether it is independent of the coordinates chosen The twoquestions are intimately connected

A side remark There is a simple intuitive way of understanding why the hamiltonianacts only on nodes Consider the naive (divergent) form (63) of the WheelerndashDeWittoperator and act with this on a spin network state at a point where there are no nodesEquation (659) shows that the result of this action (disregarding the divergence) isproportional to

F ijab (τ)

δ

δAia(τ)

δ

δAjb(τ)

ΨS [A] sim F ijab(τ) γaγb (710)

where γa is the tangent to the spin network links at the point τ where the operatoracts But this expression vanishes because Fab is antisymmetric in ab while γaγb is


symmetric On the other hand acting on a node the two functional derivatives givemixed terms of the kind Fabγ

a1 γ

b2 where γa

1 and γa2 are the tangents of two different

links emerging from the node These terms may be nonvanishing Hence the operatorhas a nontrivial action only on nodes

711 Finiteness

In general the limit (77) does not exist This is no surprise as operatorproducts are generally ill defined in quantum field theory they can bedefined in regularized form but then a divergence develops when removingthe regulator ε The big surprise however is that the limit (77) does existon a subclass of states diffeomorphism-invariant states As these are thephysical states this is precisely what we need and is sufficient to definethe theory Here is where the intimate interplay between diffeomorphisminvariance and quantum field theoretical short-scale behavior begins toshine we are here at the core of diffeomorphism-invariant QFT

To compute H on diff-invariant states recall these are in the dual spaceS prime So far we have only considered H on spin network states or bylinearity on S The action of H on S prime is immediately defined by duality

(HΦ)(Ψ) equiv (Φ)(HΨ) (711)

(To be precise I should call Hdagger the operator in the left-hand side but forsimplicity I do not) Equivalently for every spin network state

(HΦ)(|S〉) = Φ(H|S〉) (712)

The key point is that we want to consider the regularized operator on S prime

and take the limit there Thus instead of simply inserting (77) in thelast equation and writing

(HΦ)(|S〉) = Φ(limεrarr0

Hε|S〉) (713)

I define the hamiltonian operator on S prime by

(HΦ)(|S〉) = limεrarr0

Φ(Hε|S〉) (714)

Notice that the limit is now a limit of a sequence of numbers (not a limitof a sequence of Hilbert space vectors) I now show that the limit exists(namely is finite) if Φ isin Kdiff namely if Φ is a diffeomorphism-invariantstate

The key to see this is the following crucial observation Given a spinnetwork S the operator in the parentheses modifies the state |S〉 in twoways by changing its graph Γ as well as its coloring The volume operatordoes not change the graph The graph is modified by the two operators


hγxnland hαxnllprime The first superimposes a path of length ε to the link l

of Γ The second superimposes a triangle with two sides of length propor-tional to ε along the links lprime and lprimeprime of Γ and a third side that is not onΓ as in Figure 71 The fundamental observation is that for ε sufficientlysmall changing ε in the operator changes the resulting state but not itsdiffeomorphism equivalence class (The maximal εm for this to happenis the value of ε such that the added paths cross or link other nodes orlinks of S) This is rather obvious adding a smaller triangle is the sameas adding a larger triangle and then reducing it with a diffeomorphismTherefore for ε lt εm the term in the parentheses remains in the samediffeomorphism equivalence class as ε is further reduced But Φ is invari-ant under diffeomorphisms and therefore the dependence on ε of theargument of the limit becomes constant for ε lt εm

Therefore the value of the limit (714) is simply given by


Φ(Hε|S〉) = Φ(H|S〉) (715)

where

H|S〉 = minus i

sum

nisinS

Nn

sum


(hγminus1


])|S〉

(716)

and the size ε of the regularizing paths is simply taken to be small enoughso that the added arc does not run over other nodes or link other links ofS The finiteness of the limit is then immediate

Discussion relation between regularization and background independenceThis result is very important and deserves a comment The first key pointis that the coordinate space x has no physical significance at all The phys-ical location of things is only location relative to one another not thelocation with respect to the coordinates x The diffeomorphism-invariantlevel of the theory implements this essential general-relativistic require-ment The second point is that the excitations of the theory are quan-tized This is reflected in the short-scale discreteness or in the discretecombinatorial structure of the states This is the result of the quantummechanical properties of the gravitational field When these two featuresare combined there is literally no longer room for diverging short-distancelimits The limit ε rarr 0 is a limit of small coordinate distance it becomesfinite simply because making the regulator smaller cannot change any-thing below the Planck scale as there is nothing below the Planck scaleOnce the regulating small loop αxlprimelprimeprime is smaller than the size needed tolink or cross other parts of the spin network any further decrease of its


size is gauge not physics This is how diffeomorphism invariance cures indepth the ultraviolet pathologies of quantum field theory

One last technicality In the definition given there is a residual (discrete) dependenceon the coordinates x In two different coordinate systems the arc may link the originallinks of the graph differently For instance a fourth link lprimeprimeprime may pass ldquooverrdquo or ldquounderrdquothe arc If the node is n-valent the possible alternatives are labeled by the homotopyclasses of lines (without intersection) going from the north to the south pole on asphere with n minus 2 punctures To have a fully diffeomorphism-invariant definition wemust therefore sum over these Nn alternatives On the other hand we can consistentlyexclude coordinate systems in which three links are coplanar and the arc intersects alink for all values of ε We can do this because we are using extended diffeomorphismsThus calling αr

xlprimelprimeprime r = 1 Nn a representative of the rth homotopy class wearrive at

(HΦ)(|S〉) = Φ(H|S〉) (717)

where

H|S〉 = minus i

sum

nisinSllprimelprimeprimer

Nnεllprimelprimeprime tr(hγminus1xnl

hαrxnlprimelprimeprime

[V(Rn) hγxnl ])

|S〉 (718)

This is the final form of the operator

712 Matrix elements

The resulting action of H on s-knot states is simple to derive and toillustrate (i) The action gives a sum of terms one for each node n of thestate (ii) For each node H gives a further sum of terms one term foreach triplet of links arriving at the node and for each triplet one termfor every permutation of the three links l lprime lprimeprime Each of these terms acts asfollows on the s-knot state (see Figure 72) (iii) It creates two new nodesnprime and nprimeprime at a finite distance from n along the links lprime and lprimeprime The exactlocation of these nodes is of course irrelevant for the s-knot state (iv) Itcreates a new link of spin-12 connecting nprime and nprimeprime (without linking anyother node) This new link is called ldquoarcrdquo (v) It changes the coloring j prime

of the link connecting n and nprime and the coloring j primeprime of the link connectingn and nprimeprime These turn out to be the colors of the links lprime and lprimeprime increasedor decreased by 12 (vi) It changes the intertwiner at the node n thenew intertwiner is between the representations corresponding to the newcolorings of the adjacent links

Remark Again it is easy to understand the origin of this action of the hamiltonianoperator on the basis of the simple form (63) of the hamiltonian constraint Thetwo functional derivatives ldquograsprdquo a spin network and as explained before the graspvanishes except in the vicinity of a node The curvature term Fab is essentially aninfinitesimal holonomy Therefore it creates a small loop next to the node This loop


D+ minus

j prime

j primeprime+

j primeprime j primeprime

j prime

j primen

nprime

nprimeprime

j j

1minus2

1minus2

1minus2

minus=^

Fig 72 Action of Dnlprimelprimeprimerεεprime

must be in the plane of two grasped links and can be identified with the triangle definedby the added arc

Notice that the ClebschndashGordan conditions always hold at the modified node Thisfollows immediately from the fact that the modified node is obtained from recouplingtheory the matrix element associated with nodes not satisfying the ClebschndashGordanconditions turns out to vanish

Call Dnlprimelprimeprimerplusmnplusmn an operator that acts around the node n by acting asdescribed in (iii) (iv) (v) This is illustrated in Figure 72 Then

H|S〉 =sum

nisinS

Nn

sum

llprimelprimeprimer

sum

εprimeεprimeprime=plusmnHnlprimelprimeprimeεprimeεprimeprime Dnlprimelprimeprimerεprimeεprimeprime |S〉 (719)

The operator Hnlprimelprimeprimeεprimeεprimeprime acts as a finite matrix on the space of the inter-twiners at the node n The explicit computation of its matrix elements isa straightforward problem in SU(2) representation theory It is discussedin detail in [208] where its matrix elements are explicitly given for simplenodes

The operator H is defined on S prime and as we have seen is finite when restricted toKdiff Notice however that the operator does not leave Kdiff invariant In general thestate H|s〉 is not a diffeomorphism-invariant state This is because of its dependenceon N To see this compute its action on a generic spin network state

〈s|H|S〉 =sum

nisinS

Nn

sum

llprimelprimeprimer

sum

εprimeεprimeprime=plusmn〈s|Hnlprimelprimeprimeεprimeεprimeprime Dnlprimelprimeprimerεprimeεprimeprime |S〉 (720)

On the right-hand side the quantities Nn are the values of N(x) at the points xn

where the nodes n of the spin network S are located The rest of the expression isdiff invariant but these values obviously change if we perform a diffeomorphism on|S〉 This has of course to be expected because the classical quantity H is itself notdiffeomorphism invariant Therefore there is no reason for the corresponding quantumoperator to be diff invariant and preserve Kdiff The theory this operator defines isnevertheless diffeomorphism invariant because the operator enters the theory via theWheelerndashDeWitt equation HΨ = 0 This equation is well defined on Kdiff Its solutionsare the (possibly generalized) states in Kdiff that are in the kernel of H Since H isfinite on the entire Kdiff this is well defined This is completely analogous to the classical


theory where H is not diff invariant but the equation H = 0 is perfectly sensible asan equation for diff-invariant equivalence classes of solutions (An alternative strategyyielding a version of the hamiltonian operator sending Kdiff into itself has been recentlyexplored by Thiemann in [209])

Recalling the definition of |s〉 we can write the matrix elements of Hamong spin network states

〈Sprime|H|S〉 =sum

nisinS

Nn

sum

|Ψ〉=Uφ|Sprime〉

sum

llprimelprimeprime

sum

εprimeεprimeprime=plusmn〈Ψ|HnlprimelprimeprimeεprimeεprimeprimeDnlprimelprimeprimeεprimeεprimeprime |S〉

(721)

The operator H is not symmetric This is evident for instance from thefact that it adds arcs but does not remove them Its adjoint Hdagger can bedefined simply by the complex conjugate of the transpose of its matrixelements

〈Sprime|Hdagger|S〉 = 〈S|H|Sprime〉 (722)

and a symmetric operator is defined by

Hs =12(H + Hdagger) (723)

This is an operator that adds as well as removes arcs It is reasonableto expect that this operator be better behaved for the classical limitTherefore we take this operator as the basic operator defining the theory

713 Variants

The striking fact about the hamiltonian operator is that it can be definedat all But how unique is it There are a number of possible variants ofthe operator that one may consider These can be seen as quantizationambiguities that is they define different dynamics in the quantum theoryall of which at least at first sight have the same classical limit So far itis not clear if these are truly all viable or whether there are physical ormathematical constraints that select among them

Higher j If we expand the matrix representing the holonomy of a con-nection in a representation j we obtain an expression analogous tothat in (72) Therefore we can regularize the terms F and A in thehamiltonian by using the holonomy in any arbitrary representationj That is we can write up to an irrelevant numerical factor

H = limεrarr0

1ε3

intN tr

(h

(j)

γminus1xw

h(j)αxuv

V h(j)γxw

)d3x (724)


where h(j) = R(j)(h) This has no effect on the classical limit How-ever the corresponding quantum operator

H(j)=minus i

limεrarr0

sum

m

Nm tr(h

(j)

γminus1xmw

h(j)αxmuv

[V(Rm) h(j)γxmw

])

(725)

is different from the operator (76) This can be easily seen by notic-ing that the added arc does not have spin 12 but rather spin j Ingeneral any arbitrary linear combination

H =sum

j

cj H(j) (726)

defines an hamiltonian operator with a correct classical limit Thereare indications that the coefficients cj should be different from zeroin a consistent theory discussed for instance in [210] In the samepaper matrix elements of the operator H(j) are computed and somearguments on criteria to fix the coefficients cj are discussed As weshall see in Chapter 8 this quantization ambiguity plays a role inloop quantum cosmology

Hs or H Both the symmetric operator Hs and the nonsymmetric oper-ator H define a quantum dynamics While there are arguments fortaking the symmetric one there are also arguments for taking thenonsymmetric one [20]

Other regularizing loops Loops different from αxlprimelprimeprime and γxl could bechosen for the regularization The freedom is strongly limited bydiffeomorphism invariance and by the condition that the result-ing operator is finite and nonvanishing But other choices might bepossible

Ordering A different ordering can be chosen between the volume and theholonomy operators

Others More generally no uniqueness theorem exists so far

This freedom in the definition of the hamiltonian operator is not aproblem it is an asset No complete and completely consistent theory ofquantum gravity with a well-understood low-energy limit exists so farHaving more than one is for the moment the very least of our worriesThe remarkable result is the existence of a finite and interesting hamil-tonian operator The fact that we have a certain residual latitude in itsdefinition might very well turn out to be helpful The ldquocorrectrdquo variantcould be selected by some internal consistency requirement that has notyet been considered or by requiring the correct classical limit If theseconditions turn out to be insufficient we shall simply have nonequivalentquantum theories with the same classical limit The physically correct onewill have to be determined by experiments I wish we were already there


72 Matter kinematics

The evolution of the mathematical description of the matter fields dur-ing the twentieth century has slowly converged with the evolution of themathematical description of the gravitational field In both cases differen-tial geometry notions such as fiber bundles sections and automorphismsof the bundle play a role in the description of the classical fields If wetake for instance a coupled EinsteinndashYangndashMills system and describethe gravitational field by means of an SO(3 1) connection the structuresof the two fields the gauge and the gravitational field are barely differentfrom each other

Indeed as I have argued in Section 232 the distinction between matterand spacetime (gravity) is not profound it is largely conventional Thegravitational field is not substantially different from the other matterfields It is the full coupled gravity + matter theory which is profoundlydifferent from a theory on a fixed background When the dynamical grav-itational field is not approximated by a fixed background the full theoryis generally covariant and the physical fields live only ldquoon one anotherrdquoas the animals and the whale of the metaphor

Accordingly the methods developed above for the gravitational fieldextend naturally to other fields The theory of pure quantum gravity andthe theory of quantum gravity and matter do not differ much from eachother The second has just some additional degrees of freedom The sim-ilarity and compatibility of the classical mathematical structures makesthe extension of the quantum theory to these additional degrees of free-dom very natural

721 YangndashMills

The easiest extension of the theory described in the previous chapter isto YangndashMills fields Let GYM be a compact YangndashMills group such asin particular the group SU(3)times SU(2)timesU(1) that defines the standardmodel Let AYM be a 3d YangndashMills connection for this group and A bethe gravitational connection The two 3d connections A (gravitational)and AYM (YangndashMills) can be considered together as a single connectionA=(AAYM) for the group G = SU(2)timesGYM The construction of K K0

and Kdiff of Chapter 6 extends immediately to this connection withoutdifficulties

Holonomies of the YangndashMills field can be defined as operators on Kprecisely as gravitational holonomies Surface integrals of the YangndashMillselectric field can be defined precisely as for the E gravitational field

The diffeomorphism-invariant quantum states of GR + YangndashMills arethen given by s-knot states labeled by abstract knotted graphs carrying

72 Matter kinematics 287

irreducible representations of the group G on the links and the corre-sponding intertwiners on the nodes Since G is a direct product its irre-ducibles are simply given by products of irreducibles of SU(2) and GYMIn other words each link is labeled with a spin jl and an irreduciblerepresentation of GYM

Notice how the quantum theory realizes the relational localization char-acteristic of GR the position of the YangndashMills field is well defined withrespect to the quantum state of spacetime defined by the gravitationalpart of the spin network or equivalently vice versa

Notice also that in the absence of diffeomorphism invariance the aboveconstruction would not yield a sensible quantum state space of the YangndashMills field because it would yield a nonseparable Hilbert space

722 Fermions

Let η(x) be a Grassman-valued fermion field It transforms under a rep-resentation k of the YangndashMills group GYM and under the fundamentalrepresentation of SU(2) It is more convenient for the quantum theoryto take the densitized field ξ equiv

radic|detE| η as the basic field variable

The Grassman-valued field ξ and its complex conjugate take value in a(finite-dimensional) superspace S An integral over S is defined by theBerezin symbolic integral dξ

Define a cylindrical functional Ψ[Aψ] of the connection and the fermionfield as follows Given (i) a collection Γ of a finite number L of paths γl(ii) a finite number N of points xn and (iii) a function f of L groupelements and N Grassman variables a cylindrical functional is defined by

ΨΓf [Aψ] = f(U(A γ1) U(A γL) ξ(x1) ξ(xN )) (727)

Since Grassman variables anticommute cylindrical functionals can be atmost linear in each (component of the) fermion fields in each point n

A scalar product is defined on the space of these functions as followsGiven two cylindrical functionals defined by the same Γ define

(ΨΓf ΨΓg) equivint

GL

dUl

int

SN

dξn g(U1 UN ξ1 ξN )

times f(U1 UN ξ1 ξN ) (728)

The extension of this scalar product to any two cylindrical functions isthen completely analogous to the purely gravitational case This definesthe extension of K to fermions

Basis states are easily constructed by fixing the degree Fn of the mono-mials in ξ at each node n which determine the fermion number in theregion of the node


In the absence of fermions we constructed gauge-invariant function-als by contracting the indices of the holonomies among themselves withintertwiners In the presence of fermions we can also contract SU(2)and YangndashMills indices with the indices of the fermions In other wordsfermions live on the nodes of the graph Γ and the gravitational andYangndashMills lines of flux can end at a fermion For this to happen ofcourse generalized ClebschndashGordan conditions must be satisfied For in-stance a single fermion cannot sit at the open end of a link in the trivialrepresentations of SU(2) because its SU(2) index cannot be saturatedThis physical picture is well known for instance in canonical lattice gaugetheory In fact each Hilbert subspace KΓ can be identified as the Hilbertspace of a lattice YangndashMills theory with fermions defined on a lattice ΓWe are back to the original intuition of Faraday the lines of force canemerge from the charged particles

723 Scalars

The present formulation of the standard model requires also a certainnumber of scalar fields so far unobserved Whether these fields ndash theHiggs fields ndash are in fact present in Nature and observable or whetherthey represent a phenomenological description of some aspect of Naturewe havenrsquot yet fully understood is still unclear Scalar fields can be incor-porated in LQG but in a less natural manner than YangndashMills fields andfermions

Let φ(x) be a suitable multiplet of scalar fields (that we can alwaystake as real) transforming in a representation k of the gauge group GYMand therefore taking value in the corresponding vector space Hk

The complication is that Hk is noncompact ndash it has infinite volumeunder natural invariant measures This makes the definition of the scalarproduct of the theory more difficult (because the Hilbert space associatedto subgraphs is not a subspace of the Hilbert space associated to a graph)One way out of this difficulty is the following Assume k is the adjointrepresentation of GYM We can then exponentiate the field φ(x) definingU(x) = expφ(x) The field U(x) then takes values in GYM which iscompact and carries the Haar invariant measure

Define a cylindrical functional Ψ[Aψ φ] of the connection fermionand scalar fields as follows Given (i) a collection Γ of a finite number Lof paths γl (ii) a finite number N of points xn and (iii) a function f ofL group elements N fermion variables and N other group elements acylindrical functional is defined by

ΨΓf [Aψ φ] = f(U(A γ1) U(A γL) ψ(x1) ψ(xN ) eφ(x1)

eφ(xN )) (729)

73 Matter dynamics and finiteness 289

A scalar product is defined on the space of these functions as followsGiven two cylindrical functionals defined by the same Γ we define


GL

dUl

int

SN

dξnint

GN

dU primen

times f(U1 UL ξ1 ξN U prime1 U

primeN )

times g(U1 UL ξ1 ξN U prime1 U

primeN ) (730)

and extend this to any graph as usual

724 The quantum states of space and matter

A state |s〉 in Kdiff can then be labeled by the following quantum numbers

bull An abstract knotted graph Γ with links l and nodes n

bull A spin jl associated with each link l

bull An irreducible representation kl of the YangndashMills group GYM asso-ciated with each link l

bull An integer Fn associated with each node

bull An irreducible representation Sn of the YangndashMills group GYM as-sociated with each node n

bull An SU(2) intertwiner in associated with each node n

bull A GYM intertwiner wn associated with each node n

Thus we can write

|s〉 = |Γ jl kl Fn Sn in wn〉 (731)

This state describes a quantum excitation of the system that has a simpleinterpretation as follows There are N regions n that have volume andwhere fermions and Higgs scalars can be located These are separated byL surfaces l that have area and are crossed by flux of the (electric) gaugefield The quantum numbers are related to observable quantities as inTable 71 This completes the definition of the kinematics of the coupledgravity+matter system

73 Matter dynamics and finiteness

The dynamics of the coupled gravity+matter system is simply definedby adding the terms defining the matter dynamics to the gravitational


Table 71 Quantum numbers of the spin networkstates for gravity and matter

Quantum number Physical quantity

Γ adjacency between the regionsin volume of the node njl area of the surface lFn number of fermions at node nSn number of scalars at node nwn field strength at node nkl electric flux across the surface l

relativistic hamiltonian The hamiltonian for the fields described is givenby

H = HEinstein + HYangndashMills + HDirac + HHiggs (732)

HEinstein is the gravity hamiltonian described in the previous chaptersThe other terms are

HYangndashMills =1

2g2YMe

3tr[EaEb] Tr[EaEb + BaBb]

HDirac =12e

Eai (iπτ iDaξ + Da(πτ iξ) +

i2Ki

aπξ + cc)

HHiggs =12e

(p2 + tr[EaEb] Tr[(Daφ)(Dbφ)] + e2V (φ2)

) (733)

Here π is the momentum conjugate to the fermion field p the momentumconjugate to the scalar field E is the momentum conjugate to the YangndashMills potential (the electric field) B the curvature of the YangndashMillspotential (the magnetic field) D is the SU(2)timesGYM covariant derivativetr is the trace in the SU(2) Lie algebra and Tr is the trace in the Liealgebra of GYM e equiv

radic|detE| V is the Higgs potential p2 = Tr[pp]

and φ2 = Tr[φφ] and gYM is the YangndashMills coupling constant For aderivation and a discussion of these expressions see [211]

To define the quantum hamiltonian the expressions in (733) must beregulated and expressed in terms of the operators well defined on K Thiscan be done following the same strategy we used for the gravitationalquantum hamiltonian in Section 71 I do not present the detailed con-struction here for which I refer the reader to Thiemannrsquos work [211]

The essential result of this construction is that the total hamiltonian(732) can be constructed as a well-defined operator on the Hilbertspace of gravity and matter K The operator acts on nodes as does thegravitational part but its action is more complex than the pure gravity


operator and codes the entire dynamics of the standard model andgeneral relativity

The fact that the total hamiltonian turns out to be finite is extremely re-markable It is perhaps the major pay-off of the background-independentquantization strategy on which LQG is based

I advise the reader to read the beautiful account by Thiemann in[211] and especially in [11] for an explanation of the internal reasonsof this finiteness Thiemann illustrates how the ultraviolet divergencesof ordinary quantum field theory can be directly interpreted as a con-sequence of the approximation that disregards the quantized discretenature of quantum geometry For instance Thiemann shows how the op-erator 1

2β2e3tr[EaEb] Tr[EaEb] the kinetic term of the YangndashMills hamil-

tonian is well defined so long as we treat E as an operator but becomesinfinite as soon as we replace E with a smooth background field


With the definition of the operator H for gravity and matter finite onKdiff the formal definition of the quantum theory of gravity is completedThe theory is finite provides a compelling intuitive description of thePlanck-scale structure of space has definite predictions such as the eigen-values of area and volume and reduces to classical GR in the naive rarr 0limit The transition amplitudes of the theory are defined by


where s and sprime are two s-knot states and P is the projector on the space ofthe solutions of the equation HΨ = 0 The quantity W (s sprime) is interpretedas the probability amplitude of observing the discretized geometry withmatter determined by the s-knot s if the geometry with matter determinedby sprime was observed

If we consider a region of spacetime bounded by two disconnected sur-faces the diff-invariant boundary space is Kdiff = Klowast

diff otimesKdiff and we canrewrite (734) in terms of the covariant vacuum state

〈0|sout sin〉 = 〈sout|P |sin〉 (735)

where |sout sin〉 = 〈sout| otimes |sin〉 isin KdiffMore generally we can consider a finite region of spacetime bounded

by a 3d surface Σ If s represents the outcome of the measurement of thegravitational field and matter fields on Σ then

W (s) = 〈0|s〉 (736)


gives the correlation probability amplitude of the measurement of thestate s This can also be viewed as the transition amplitude from theempty set to the full s hence

W (s) = 〈empty|P |s〉 (737)

or

|0〉 = P |empty〉 (738)

in K This is loop quantum gravityMuch remains to be done Here are some issues that I have not ad-

dressed

(i) Lorentzian theory So far I have dealt only with the euclidean the-ory As already mentioned in Section 422 the lorentzian theorycan be expressed in terms of the same kinematics as the euclideantheory only adding a second term to the hamiltonian I shall notdiscuss the quantization of this second term here This is done indetail in [20] Alternatively the quantization has to be defined us-ing the complex connection but the full quantum state space witha complete operator algebra has not yet been constructed for thecomplex connection as far as I know Another alternative is to de-rive the amplitudes of the lorentzian theory from the amplitudesof the euclidean theory as one can do in flat-space QFT As men-tioned a naive reproduction of the flat-space technique is not viablein quantum gravity but a suitable extension of this might work

(ii) Transition amplitudes The matrix elements of the projector P arenot easy to compute

(iii) Scattering A general technique to connect the transition amplitudesW (s sprime) to particle observables such as gravitonndashgraviton scatteringmust be developed

(iv) Classical limit Can we prove explicitly that classical GR can berecovered from LQG

(v) Form of the dynamics Is the proposed form of the hamiltonian con-straint correct or does it have to be corrected

(vi) Physical consequences What does the theory say about the stan-dard physical problems where quantum gravity is expected to berelevant such as black-hole thermodynamics and early cosmology

(vii) Observable predictions Are there any

Much is known on several of these issues Some of them are discussedbelow In particular Chapter 8 deals with (vi) and (vii) Chapter 9 with(i) (ii) (iii) and (v)


Nevertheless I emphasize the fact that whatever its consequences andits physical correctness the theory developed thus far provides a finiteand consistent general covariant and background-independent quantumfield theory for the gravitational field and the matter fields In it thecore physical insights of GR and QFT merge beautifully Finding such atheory was our major aim

741 Variants

The LQG theory that I have described above is a standard version of thetheory There are a number of possible variants that have been consideredin the literature

Different regularization of H I have described this possibility abovein Section 71

q-deformed spin networks An intriguing possibility is to replace thegroup SU(2) with the quantum group SU(2)q in the quantum the-ory and choose q to be given by qN = 1 where N is a large numberIt is possible to define q-deformed spin networks labeled by repre-sentations of SU(2)q and build the rest of the theory as above Thisis an interesting possibility for several reasons Several of the spin-foam models studied in Chapter 9 are defined using quantum groupsand their states are q-deformed spin networks The spinfoam modelsshow that the use of q-deformed spin networks is naturally connectedwith a cosmological constant λ The quantum group SU(2)q has afinite number of irreducible representations which grows with N This implies that the quantum of area has a maximum value de-termined by N and related to the (large) length determined by thecosmological constant Finally N works like a natural infrared cut-off which is likely to cure any eventuality of infrared divergencesq-deformed LQG has been studied in the literature but a system-atic construction of LQG in terms of q-deformed spin networks isstill missing

Different ordering of the area operator We can quantize the har-monic oscillator choosing an ordering of the hamiltonian such thatthe vacuum energy is zero instead of 1

2ω Similarly we can choosedifferent orderings for the area operator and obtain a different spec-trum The ordering used in this book and in most of the literatureis the natural one for the Casimir operator but alternatives havebeen considered and produce some intriguing effects In particu-lar it is possible to order the operator to obtain an equally spacedspectrum This would reintroduce the BekensteinndashMukhanov effectstudied below in Section 824 and apparently would automati-cally give a dominance of spin-1 quanta for a black hole bringing


the value of the Immirzi parameter to match the frequency of theblack-hole ringing modes (see Section 823)

Different regularization of the area operator The full spectrum ofthe area operator given in (6125) contains the main sequence(675) If we think that a diffeomorphism-invariant notion of a sur-face is truly the boundary of a region and the region is an ensembleof quanta of volume then we are led to the idea that physical sur-faces are described by the mathematical surfaces that cut the linkswithout touching the nodes These surfaces have area given by themain sequence (675) Thus the degenerate sector might be phys-ically spurious To have the eigenvalues in the degenerate sectorwe need a surface that cuts precisely through the node and this isagainst the intuition that the location of the surfaces is only definedup to Planck scale A different regularization of the area operatormight get rid of the degenerate sector

Unknotted spin networks A very interesting possibility is to modifythe definition of the spin network states of the theory droppingthe information on the knotting and linking of the graphs Thatis to define the graphs Γ that form the spin networks solely interms of the adjacency relations between nodes as is usually donein graph theory and not as is done above as equivalence classesof embedded graphs under extended diffeomorphisms The physicaldifferences implied by the two definitions are not clear at present

Different regularization of the volume Two definitions of the vol-ume have been given in the literature The two turn out to be slightlydifferent Originally the two were given in different mathematicallanguages and it was thought that the difference had to do with thedifferent formulations of the theory Later it became clear that bothoperators can be defined in either formulation The volume opera-tor defined here does not distinguish a node in which some tangentsof the adjacent links are coplanar from a node in which they arenot coplanar The other version of the volume operator used forinstance in [20] makes this distinction its action on a node withcoplanar links differs from the one given here This second operatoris covariant under Diff but not under Diff lowast

Extended loop representation Gambini and Pullin have developeda version of LQG in which loop states are not normalizableNormalizable states are obtained smearing loop states The mainmotivation is the fact that in flat-space QFT this is the case I referthe reader to their book [7] for a discussion and details

Lorentz spin networks In the hamiltonian theory on which LQG isbased there is a partial gauge-fixing One of the consequences of


this gauge-fixing is that the connection with the covariant formalismused in the spinfoam models becomes technically more cumbersomeTo avoid this difficulty Sergei Alexandrov has studied the possibilityof defining LQG without making this gauge-fixing and keeping thefull Lorentz group in the hamiltonian formalism [127 212] Thismight give a different regularization of the area operator as well[213]

mdashmdash


The WheelerndashDeWitt equation appeared in [214] The first version of thehamiltonian operator of LQG and its first solutions were constructed in[177] Various other solutions were found see for instance [215] A re-view of early solutions of the hamiltonian operator is in [216] The resultthat diffeomorphism-invariant states make the operator finite appearedin [217] The general structure of the hamiltonian constraint and the op-erator D are illustrated in [218]

The idea of expressing the hamiltonian as a commutator and there-fore the first fully well-defined version of the hamiltonian operator wasobtained by Thomas Thiemann in [133] and systematically developed byThiemann in the remarkable ldquoQSDrdquo series of papers [201 209] Matrix el-ements of this operator were systematically studied in [208] On the hamil-tonian operator with positive cosmological constant and the possibilityof defining the theory in this case see [220] Thiemann and collaboratorsare developing an original and promessing approach to the definition ofthe quantum dynamics called the ldquoMaster Programrdquo The idea is to con-densate a full set of constraints into a single one For an introduction andreferences see [221]

Fermions were introduced in LQG in [222] and in [223] The key stepfor the present formulation of the fermionndashLQG coupling was taken byThomas Thiemann using half-density spinor fields [224] A complete studyof the matter hamiltonian in LQG is due to Thiemann See his [20] andcomplete references therein The fermionic contribution to the spectrumof the area operator was considered in [225] The intriguing possibilitythat fermions are described by the linking of the spin networks has beenrecently explored in [226]

On the area operator with equispaced eigenvalues and its effect on theblack-hole entropy see [227] A q-deformed version LQG was consideredin [228] see [220] and references therein For q-deformed spin networkssee also [229] [230] and [231]

8Applications

In this chapter I briefly mention some of the most successful applications of LQG toconcrete physical problems I have no ambition of completeness and I will not presentany detailed derivation For these I refer to original papers and review articles I onlyillustrate the main ideas and the main results

The two traditional applications of quantum gravity are early cosmology and black-hole physics In both these fields LQG has obtained interesting results In addition acertain number of tentative calculations concerning other domains where Planck-scalephysical effects could perhaps be observable have also been performed

81 Loop quantum cosmology

A remarkable application of LQG is to early cosmology A direct treat-ment of semiclassical states in Kdiff representing cosmological solutionsof the Einstein equations is not yet available However it is possible toimpose homogeneity and isotropy on the basis states and operators of thetheory and in this way restrict the theory to a finite-dimensional systemdescribing a quantum version of the cosmological dynamics that can bestudied in detail

The result is different from the traditional WheelerndashDeWitt minisuper-space quantization of the dynamics of Friedmann models The key to thedifference is the fact that the system inherits certain physical aspects ofthe full theory In particular the quantization of the geometry These havea major effect on the dynamics of the early Universe The main resultsare the following

(i) Absence of singularities Dynamics is well defined at the Big Bangwith no singular behavior In particular the inverse scale factor isbounded In this sense the Universe has a minimal size

(ii) Semiclassical behavior Cosmological evolution approximates thestandard Friedmann dynamics for large values of the scale factora(t) but differs from it at small values of a(t)

296

81 Loop quantum cosmology 297

(iii) Quantization of the scale factor The scale factor ndash and the volumeof the Universe ndash are quantized

(iv) Discrete cosmological evolution We can view the scale factor as acosmological time parameter Then we can say that cosmologicaltime is quantized Accordingly the WheelerndashDeWitt equation is adifference equation and not a differential equation in a

(v) Inflation Just after the Big Bang the Universe underwent an infla-tionary phase d2a(t)dt2 gt 0 This is driven not by a scalar inflatonfield but by quantum properties of the gravitational field itself

These are all remarkable results but of different kinds Results (i) and(ii) are what one would expect from a quantum theory of gravity giving aconsistent description of the early Universe Results (iii) and (iv) reflectthe most characteristic aspect of LQG the quantization of the geometryResult (v) came as a big surprise Let me briefly illustrate how theseresults are derived

Consider a homogeneous and isotropic Universe Its gravitational fieldis given by the well-known line element

ds2 = minusdt2 + a2(t)

(dr2

1 minus kr2+ r2(dθ2 + sin θ dφ2)

)

(81)

where a(t) is the scale factor and k is equal to zero or plusmn1 (see for instance[75]) The Einstein equations reduce to the Friedmann equation

(a

a

)2

=8πG

3ρminus k

a2 (82)

where ρ is the time-dependent matter energy density Let us represent thematter content of the Universe in terms of a single field that for simplicitycan be taken as a scalar field φ Homogeneity then demands that φ is afunction of the sole time coordinate The system is therefore describedby a(t) and φ(t) Assume for simplicity that φ(t) has a simple quadraticself-interaction (potential) term namely its hamiltonian is

Hφ =12(p2

φ + ω2φ2) (83)

where pφ is the momentum conjugate to φ Therefore

φ(t) = A sin(ωt + φ0) (84)

Do not confuse the field φ with the inflaton it has no inflationary po-tential The energy density is related to the conserved matter energy Hφ

by

ρ = aminus3Hφ = aminus3ρ0 =12aminus3ω2A2 (85)

298 Applications

The constant ρ0 is the density at a = 1 The Friedmann equation (82)can be derived from the hamiltonian

H = minus(p2a

8a+ 2ka

)

+ 16πGHφ (86)

by simply computing a = dHdpa and using H = 0 In the simplestspatially flat case k = 0 the Friedmann equation (82) reduces to

(a

a

)2

=8πG

3ρ0

a3 (87)

by taking a derivative we obtain

a = minus43πGρ0

1a2

(88)

which is precisely equation (2112) that we had obtained in the con-text of newtonian cosmology The equation is solved by the well-knownFriedmann evolution

a(t) = a0(tminus t0)23 (89)

Interpretation These equations are written in a particular gauge-choicefor the variable t but the full theory is invariant under reparametriza-tion in t The relativistic configuration space is coordinatized by a and φThe physical content of the theory is not in the dependence of these twoquantities on t but in their dependence on each other The proper mean-ing of (84)ndash(89) concerns the relation between φ and a For instancewe can interpret φ as a clock That is we can define its oscillations asisochronous This defines a physical time variable Then the scale factorgrows in this time variable as described by (89) Alternatively we canuse the scale factor as a measure of time In this cosmological time allmaterial physical processes slow down as in

φ(a) = A sin(ω a32 + φ0) (810)

In other words Friedmann evolution is the relative evolution of the rateof change of the material processes and the rate of change of the scalefactor it is the evolution of the ratio between the two rates of change Asolution of the Friedmann equation describes therefore the values thatthe scale factor can take for a given value φ of the matter variable orequivalently the values φ(a) that the matter variable can take at a givenvalue a of the scale factor (equation (810))

Notice that there is no need to think in terms of ldquoevolution in trdquo forthese relations to make sense As discussed in Section 34 time evolu-tion namely the idea of a physical ldquoflowrdquo of time with respect to which


a increases and φ oscillates may simply derive from the physics of ther-modynamical processes that happen in the presence of many (gravity andmatter) variables Therefore the question ldquoWhat happened before the BigBangrdquo might be as empty as the question ldquoWhat is there on the Earthrsquossurface one meter north of the North polerdquo

Traditional quantum cosmology In the traditional approach to quantumcosmology one introduces a wave function ψ(a φ) This is governed bythe WheelerndashDeWitt equation obtained from (86) Up to factor orderingthis can be written as

(h2

8apart

parta

part

parta+ 2ka

)

ψ(a φ) = 16πG H0φ ψ(a φ) (811)

where H0φ is the hamiltonian of an harmonic oscillator with angular fre-

quency ω This equation has semiclassical solutions which are wave pack-ets that approximate the Friedmann evolution for large a For small a onthe other hand the singular behavior of the classical theory persists

Loop quantum cosmology What changes if we use LQG The essentialnovelty is the quantization of the geometry Recall that up to a constant

a sim 3radicV (812)

where V is the volume of the compact universe But the volume has adiscrete spectrum in the theory Therefore we should expect a to have adiscrete spectrum In fact the detailed construction carried out in [232]shows that this is precisely the case The observable a has a discretespectrum with eigenstates |n〉 labeled by an integer n and eigenvalues

an = a1

radicn (813)

where the constant a1 is

a1 =radic

43γπhG (814)

Therefore in LQG the size of the universe is quantized If a is quantizedwe cannot represent states as functions ψ(a φ) (for the same reason thatwe do not represent states of the harmonic oscillators as continuous func-tions of the energy) Rather we can represent states in the form

ψn(φ) = 〈n φ|ψ〉 (815)

where |n φ〉 is an eigenstate of a and φ Accordingly the partial derivativeswith respect to a in the WheelerndashDeWitt equation (811) are replaced

300 Applications

in loop quantum cosmology with finite-difference operators In fact theWheelerndashDeWitt equation is explicitly derived in [232] It has the form

αnψn+4(φ) minus 2βnψn(φ) + γnψn+4(φ) = 16πG aminus3Hφψn(φ) (816)

where the constants αn βn γn are given in [232] Notice the volume den-sity factor aminus3 on the right-hand side It appears because the quantumconstraint must be obtained in LQG from the densitized hamiltonianaminus3H

The key point is now the meaning of the operator aminus3 in the right-hand side of this equation Recall that in the definition of the hamiltonianoperator of the full theory it was essential to use a proper definition ofthe inverse volume element 1detE in order to define a well-behavedoperator This was obtained by expressing it via a Poisson bracket inthe classical theory and via a commutator in the quantum theory Thisprocedure circumvents technical difficulties associated with the definitionof the inverse of the volume element operator The inverse scale factor aminus3

in (816) is what remains of that term in the cosmological theory But ifthe cosmological theory has to approximate the full theory we have tobetter define this operator in the same way the inverse volume elementwas defined in the full theory In fact it is not hard to do so writing

d = aminus3 (817)

as a commutator of well-defined quantum operators The resulting oper-ator d is well defined Its spectrum is however more complicated than thesimple inverse of the spectrum of a3

dn =

⎛

⎝ 12j(j + 1)(2j + 1)

sum

k=minusjj

kradicVn

⎞

⎠

6

(818)

In the definition of the operator d there is a quantization ambiguity Thisis because d is defined using a holonomy and this can be taken in anyrepresentation j This is precisely the ambiguity in the definition of thehamiltonian operator that was discussed above in Section 713 Remark-ably its spectrum turns out to be bounded In fact for large n we have

dn sim aminus3n (819)

but for small ndn sim a12

n (820)

There is a maximum value of dn whose value and location are determinedby the free quantization parameter j Using this operator in the WheelerndashDeWitt equation (816) yields a perfectly well-behaved evolution on and

82 Black-hole thermodynamics 301

around n = 0 A numerical study of this equation shows easily that it givesthe standard semiclassical behavior and therefore standard Friedmannevolution for large n

811 Inflation

The most surprising and intriguing aspect of LQC is the fact that itpredicts an inflationary phase in the expansion of the early UniverseThis can be seen by explicit numerical solutions of the WheelerndashDeWittequation (816) or more simply as follows For small n the behavior ofthe operator d is governed by (820) instead of (819) The correspond-ing cosmological evolution can therefore be effectively approximated bya modification of the Friedmann equation in which Hφ is proportional toa12 instead of aminus3 This yields an accelerated initial expansion of the form

a(t) sim (t0 minus t)minus29 (821)

A numerical solution of the WheelerndashDeWitt equation confirms this re-sult The initial acceleration subsequently decreases smoothly and con-verges to a standard decelerating Friedmann solution The duration ofthis inflationary expansion is governed by j

What goes on physically can be understood as follows The kinetic termof the matter hamiltonian contains effectively a coupling with gravityWhen the gravitational field is strong near the initial singularity thematter field feels the quantum structure of the gravitational field whichaffects its dynamics

This scenario deserves to be explored in more detail and better under-stood

82 Black-hole thermodynamics

The first hint that a black hole can have thermal properties came fromclassical GR In 1972 Hawking proved a theorem stating that the Einsteinequations imply that the area of the event horizon of a black hole cannotdecrease Shortly after Bardeen Carter and Hawking showed that in GRblack holes obey a set of laws that strongly resembles the principles ofthermodynamics impressed by this analogy Bekenstein suggested thatwe should associate an entropy

SBH = akB

hGA (822)

to a Schwarzschild black hole of surface area A (In this chapter wherethe connection does not appear the area is denoted A not A) Here a is a

302 Applications

constant of the order of unity kB the Boltzmann constant and the speedof light is taken to be 1 The reason for the appearance of h in this formulais essentially to get dimensions right Bekensteinrsquos suggestion was that thesecond law of thermodynamics should be extended in the presence of blackholes the total entropy that does not decrease in time is the sum of theordinary entropy with the black-hole entropy SBH Bekenstein presentedseveral physical arguments supporting this idea but the reaction of thephysics community was very cold mainly for the following reason Thearea A of a Schwarzschild black hole is related to its energy M by

M =

radicA

16πG2 (823)

If (822) was correct the standard thermodynamical relation Tminus1 =dSdE would imply the existence of a black-hole temperature

T =h

a32πkBGM (824)

and therefore a black hole would emit thermal radiation at this tempera-ture a consequence difficult to believe However shortly after Bekensteinrsquossuggestion Hawking derived precisely such a black-hole thermal emissionfrom a completely different perspective Using conventional methods ofquantum field theory in curved spacetime Hawking studied a quantumfield in a gravitational background in which a black hole forms (say a starcollapses) and found that if the quantum field is initially in the vacuumstate after the star collapse we find it in a state that has properties of athermal state This can be interpreted by saying that the black hole emitsthermal radiation Hawking computed the emission temperature to be

T =h

8πkBGM (825)

which beautifully supports Bekensteinrsquos speculation and fixes the con-stant a at

a =14 (826)

so that (822) becomes

SBH =kBA

4hG (827)

Since then the subscript BH in SBH does not mean ldquoblack holerdquo itmeans ldquoBekensteinndashHawkingrdquo Hawkingrsquos theoretical discovery of black-hole emission has since been rederived in a number of different ways andis today generally accepted as very credible1

1Although perhaps some doubts remain about its interpretation One can write a purequantum state in which the energy distribution of the quanta is planckian Is the


Hawkingrsquos beautiful result raises a number of questions First in Hawk-ingrsquos derivation the quantum properties of gravity are neglected Are thesegoing to affect the result Second we understand macroscopical entropyin statistical mechanical terms as an effect of the microscopical degrees offreedom What are the microscopical degrees of freedom responsible forthe entropy (822) Can we derive (822) from first principles Becauseof the appearance of h in (822) it is clear that the answer to these ques-tions requires a quantum theory of gravity The capability of answeringthese questions has since become a standard benchmark against which aquantum theory of gravity can be tested

A detailed description of black-hole thermodynamics has been devel-oped using LQG and research is active in this direction The major re-sult is the derivation of (822) from first principles for Schwarzschild andfor other black holes with a well-defined calculation where no infinitiesappear As far as I know LQG is the only detailed quantum theory ofgravity where this result can be achieved2

As I illustrate below the result of LQG calculations gives (822) with

a asymp 023754γ

(828)

where γ is the Immirzi parameter This agrees with Hawkingrsquos value (826)provided that the Immirzi parameter has the value

γ asymp 02375 (829)

In fact this is the way the value of γ is fixed in the theory nowadaysThe calculation can be performed for different kinds of black holes andthe same value of γ is found assuring consistency An independent wayof determining γ would make this result much stronger

In what follows I present the main ideas that underlie the derivationof this result

821 The statistical ensemble

The degrees of freedom responsible for the entropy Consider a black holewith no charge and no angular momentum Its entropy (822) can originate

state of the quantum field after the collapse truly a thermal state or a pure state thathas the energy distribution of a thermal state Namely are the relative phases of thedifferent energy components truly random or are they fixed deterministically by theinitial state Do the components of the planckian distribution form a thermal or aquantum superposition In the second case the transition to a mixed state is just thenormal result of the difficulty of measuring hidden correlations

2So far string theory can only deal with the highly unphysical extreme or nearlyextreme black holes

304 Applications

from horizon microstates corresponding to a macrostate described bythe Schwarzschild metric Intuitively we can think of this as an effect offluctuations of the shape of the horizon

One can raise an immediate objection to this idea a black hole has ldquonohairrdquo namely a black hole with no charge and no angular momentum isnecessarily a spherically symmetric Schwarzschild black hole leaving nofree degrees of freedom to fluctuate

This objection however is not correct It is the consequence of a com-mon confusion about the meaning of the term ldquoblack holerdquo The confusionderives from the fact that the expression ldquoblack holerdquo is used with twodifferent meanings in the literature In its first meaning a ldquoblack holerdquo is aregion of spacetime hidden beyond an horizon such as a collapsed star Inits second meaning ldquoblack holerdquo is used as a synonym of ldquostationary blackholerdquo When one says that ldquoa black hole is uniquely characterized by massangular momentum and chargerdquo one refers to stationary black holes notto arbitrary black holes In particular a black hole with no charge and noangular momentum is not necessarily a Schwarzschild black hole and isnot necessarily spherical Its rich dynamics is illustrated for instance bythe beautiful images of the rapidly varying shapes of the horizon obtainedin numerical calculations of say the merging of two holes Generally ablack hole has a large number of degrees of freedom and its event horizoncan take arbitrary shapes These degrees of freedom of the horizon canbe the origin of the entropy

To be sure in the classical theory a realistic black hole with vanishingcharge and vanishing angular momentum evolves very rapidly towards theSchwarzschild solution by rapidly radiating away all excess energy Its os-cillations are strongly damped by the emission of gravitational radiationBut we cannot infer from this fact that the same is true in the quantumtheory or in a thermal context In the quantum theory the Heisenbergprinciple prevents the hole from converging exactly to a Schwarzschildmetric and fluctuations may remain In fact we will see that this is thecase

Recall that in the context of statistical mechanics we must distinguishbetween the macroscopic state of a system and its microstates Obviouslythe symmetry of the macrostate does not imply that the relevant micro-states are symmetric For instance in the statistical mechanics of a sphereof gas the individual motions of the gas molecules are certainly not con-fined by spherical symmetry When the macrostate is spherically symmet-ric and stationary the microstates are not necessarily spherically symmet-ric or stationary

When we study the thermodynamical behavior of a Schwarzschild blackhole it is therefore important to remember that the Schwarzschild sol-ution is just the macrostate Microstates can be nonstationary and


non-spherically symmetric Indeed trying to explain black-hole thermo-dynamics from properties of stationary or spherically symmetric metricsalone is a nonsense such as trying to derive the thermodynamics of anideal gas in a spherical box just from spherically symmetric motions ofthe molecules

Thermal fluctuations of the geometry To make the case concrete con-sider a realistic physical system containing a nonrotating and nonchargedblack hole as well as other physical components such as dust gas or radia-tion which I denote collectively as ldquomatterrdquo We are interested in the sta-tistical thermodynamics of such a system Because of Einsteinrsquos equationsat finite temperature the microscopic time-dependent inhomogeneities ofthe matter distribution due to its thermal motion must generate time-dependent microscopic thermal inhomogeneities in the gravitational fieldas well One usually safely disregards these ripples of the geometry Forinstance we say that the geometry over the Earthrsquos surface is given by theMinkowski metric (or the Schwarzschild metric due to the Earthrsquos grav-itational field) disregarding the inhomogeneous time-dependent gravita-tional field generated by each individual fast-moving air molecule TheMinkowski geometry is therefore a ldquomacroscopicrdquo coarse-grained aver-age of the microscopic gravitational field surrounding us These thermalfluctuations of the gravitational field are small and can be disregardedfor most purposes but not when we are interested in the statisticalndashthermodynamical properties of gravity these fluctuations are preciselythe sources of the thermal behavior of the gravitational field as is thecase for any other thermal behavior

In a thermal context the Schwarzschild metric represents therefore onlythe coarse-grained description of a microscopically fluctuating geometryMicroscopically the gravitational field is nonstationary (because it inter-acts with nonstationary matter) and nonspherically symmetric (becausematter distribution is spherically symmetric on average only and noton individual microstates) Its microstate therefore is not given by theSchwarzschild metric but by some complicated time-dependent nonsym-metric metric

Horizon fluctuations Let us make the considerations above slightly moreprecise Consider first the classical description of a system at finite tem-perature in which there is matter the gravitational field and a black holeFoliate spacetime into a family of spacelike surfaces Σt labeled with atime coordinate t The intersection ht between the spacelike surface Σt

and the event horizon (the boundary of the past of future null-infinity)defines the instantaneous microscopic configuration of the event horizonat coordinate time t I loosely call ht the surface of the hole or the hori-zon Thus ht is a closed 2d surface in Σt As argued above generally this

306 Applications

microscopic configuration of the event horizon is not spherically symmet-ric Denote by gt the intrinsic and extrinsic geometry of the horizon htLet M be the space of all possible (intrinsic and extrinsic) geometriesof a 2d surface As t changes the (microscopic) geometry of the horizonchanges Thus gt wanders in M as t changes

Since the Einstein evolution drives the black hole towards theSchwarzschild solution (we can choose the foliation in such a way that)gt will converge towards a point gA of M representing a sphere of a givenradius A However as mentioned before exact convergence may be for-bidden by quantum theory and quantum effects may keep gt oscillatingin a finite region around gA

Which microstates are responsible for SBH Let us assume that (822)represents a true thermodynamical entropy associated with the black holeThat is let us assume that heat exchanges between the hole and theexterior are governed by SBH Where are the microscopical degrees offreedom responsible for this entropy located The microstates that arerelevant for the entropy are only the ones that can affect energy exchangeswith the exterior That is only the ones that can be distinguished fromthe exterior If I have a system containing a perfectly isolated box theinternal states of the box do not contribute to the entropy of the systemas far as the heat exchange of the system with the exterior is concernedThe state of matter and gravity inside a black hole has no effect on theexterior Therefore the states of the interior of the black hole are irrelevantfor SBH

To put it vividly the black-hole interior may be in one out of an in-finite number of states indistinguishable from the outside For instancethe black-hole interior may in principle be given by an infinite Kruskalspacetime on the other side of the hole there may be billions of galaxiesthat do not affect the side detectable by us The potentially infinite num-ber of internal states does not affect the interaction of the hole with itssurroundings and is irrelevant here because it cannot affect the energeticexchanges between the hole and its exterior which are the ones that de-termine the entropy We are only interested in configurations of the holethat have distinct effects on the exterior of the hole

Observed from outside the hole is completely determined by the geo-metric properties of its surface Therefore the entropy (relevant for thethermodynamical description of the thermal interaction of the hole withits surroundings) is entirely determined by the geometry of the black-holesurface namely by gt

The statistical ensemble We have to determine the ensemble of the micro-states gt over which the hole may fluctuate In conventional statistical


thermodynamics the statistical ensemble is the region of phase space overwhich the system could wander if it were isolated namely if it did notexchange energy with its surroundings Can we translate this condition tothe case of a black hole The answer is yes because we know that in GRenergy exchanges of the black hole are accompanied by a change in itsarea Therefore we must define the statistical ensemble as the ensembleof gt with a given value A of the area

To support the choice of this ensemble consider the following3 Theensemble must contain reversible paths only In the classical theory re-versible paths conserve the area because of the Hawking theorem Quan-tum theory does not change this because it allows area decrease only byemitting energy (Hawking radiance) namely violating the (counterfac-tual) assumption that defines the statistical ensemble that the systemdoes not exchange energy

We can conclude that the entropy of a black hole is given by the numberN(A) of states of the geometry gt of a 2d surface ht of area A The quantityS(A) = kB lnN(A) is the entropy we should associate with the horizon inorder to describe its thermal interactions with its surroundings

Quantum theory This number N(A) is obviously infinite in the classicaltheory But not in the quantum theory The situation is similar to the caseof the entropy of the electromagnetic field in a cavity which is infiniteclassically and finite in quantum theory To compute it we have to countthe number of (orthogonal) quantum states of the geometry of a two-dimensional surface with total area A The problem is now well definedand can be translated into a direct computation

Two objections I have concluded that the entropy of a black hole is de-termined by the number of the possible states of a 2d surface with area AThe reader may wonder if something has got lost in the argument doesthis imply that any surface has an associated entropy just because it hasan area Where has the information about the fact that this is a blackhole gone And where has the information about the Einstein equationsgone These objections have often been raised to the argument aboveHere is the answer

The first objection can be answered as follows Given any arbitrarysurface we can of course ask the mathematical question of how manystates exist that have a given area But there is no reason generally tosay that there is an entropy associated with the surface In a general sit-uation energy or more generally information can flow across a surface

3In this context it is perhaps worthwhile recalling that difficulties to rigorously justify-ing a priori the choice of the ensemble plague conventional thermodynamics anyway

308 Applications

The surface may emit heat without changing its geometry Therefore ingeneral the geometry of the surface and the number of its states havenothing to do with heat exchange or with entropy But in the specialcase of a black hole the horizon screens us from the interior and any heatexchange that we can have with the hole must be entirely determined bythe geometry of the surface It is only in this case that the counting ismeaningful because it is only in this case that the number of states ofa geometry of a given area corresponds precisely to the number of statesof a region which are distinguishable from the exterior To put it moreprecisely the future evolution of the surface of a black hole is completelydetermined by its geometry and by the exterior this is not true for anarbitrary surface It is because of these special properties of the horizonthat the number of states of its geometry determine an entropy

You can find out how much money you own by summing up the numberswritten on your bank account This does not imply that if you sum upthe numbers written on an arbitrary piece of paper you get the amountof money you own The calculation may be the same but an arbitrarypiece of paper is not a bank account and only for a bank account doesthe result of the calculation have that meaning Similarly you can makethe same calculation for any surface but only for a black hole because ofits special properties is the result of the calculation an entropy

The second objection concerns the role of the Einstein equations thatis the role of the dynamics This objection has been raised often but Ihave never understood it The role of the Einstein equations is preciselythe usual role that the dynamical equations always play in statisticalmechanics Generally the only role of the dynamics is that of definingthe energy of the system which is the quantity which is conserved if thesystem is isolated and exchanged when heat is exchanged The statisticalensemble is then determined by the value of the energy In the case ofa black hole it is the Einstein equations that determine the fact thatthe area governs heat exchange with the exterior of the hole If it wasnrsquotfor the specific dynamics of general relativity the area would not increasefor an energy inflow or decrease for energy loss Thus it is the Einsteinequations that determine the statistical ensemble

822 Derivation of the BekensteinndashHawking entropy

Above we have found on physical grounds what the entropy of a blackhole should be It is given by

SBH = kB lnN(A) (830)

where N(A) is the number of states that the geometry of a surface witharea A can assume It is now time to compute it


Let the quantum state of the geometry of an equal-time spacelike 3dΣt be given by a state |s〉 determined by an s-knot s The horizon is a 2dsurface S immersed in Σt Its geometry is determined by its intersectionswith the s-knot s

Intersections can be of three types (a) an edge crosses the surface (b) avertex lies on the surface (c) a finite part of the s-knot lies on the surfaceHere we are interested in the geometry as seen from the exterior of thesurface therefore the geometry we consider is more properly the limitof the geometry of a surface surrounding S as this approaches S Thislimit cannot detect intersections of the type (b) and (c) and we thereforedisregard such intersections

Let i = 1 n label the intersections of the s-knot with the horizonS Let j1 jn be the spins of the links intersecting the surface Thearea of the horizon is

A = 8πγhGsum

i


The s-knot is cut into two parts by the horizon S Call sext the externalpart The s-knot sext has n open ends that end on the horizon Fromthe point of view of an external observer a possible geometry of thesurface is a possible way of ldquoendingrdquo the s-knot A possible ldquoendrdquo ofa link with spin j is simply a vector in the representation space Hj Therefore a possible end of the external s-knot is a vector in otimesiHji Thus seen from the exterior the degrees of freedom of the hole appear asa vector in this space In the limit in which the area is large any furtherconstraint on these vectors becomes irrelevant The possible states areobtained by considering all sets of ji that give the area A and for each setthe dimension of otimesiHji Let us first assume that the number of possiblestates is dominated by the case ji = 12 In this case the area of a singlelink is

A0 = 4πγhGradic

3 (832)

This is the first value that was derived for the Immirzi parameter fromback hole states counting Later Domagala and Lewandowski realized[233] that the assumption that the entropy is dominated by spin 12 iswrong and found a higher value that was evaluated by Meissner [234]giving (829)

Hence there are

n =A

A0=

A

4πγhGradic

3(833)

intersections and the dimension of H12 is 2 so the number of states ofthe black hole is

N = 2n = 2A4πγhGradic

3 (834)

310 Applications

and the entropy is

SBH = kB lnN =1γ

ln 24π

radic3

kB

hGA (835)

This is the BekensteinndashHawking entropy (822) The numerical factoragrees with the Hawking value (826) and we get (827) if the Immirziparameter is fixed at the value

γ =ln 2πradic

3 (836)

A far more detailed account of this derivation is given in [238] where the Hilbertspace of the states of the black-hole surface is carefully constructed by quantizing atheory that has an isolated horizon as a boundary

Following [238] however Thiemann [20] derives an equation that looks like (836)but in fact differs from it by a factor of 2 Why this discrepancy The reason is thatThiemann observes that if the horizon is a boundary then in the absence of the ldquootherside of the horizonrdquo in the formalism we have jd = 0 in (6125) With jiu = ji andjid = 0 we have by definition of jt j

it = ji and therefore (taking γ = 1) the area of a

link with spin j entering the horizon contributes a quantum of area

A = 4πGhradic

j(j + 1) (837)

which is one-half the contribution to the area of a link of spin j that cuts a surface inthe bulk What is happening is that the area operator in a sense counts the area of asurface by summing contributions of links entering the two sides in the absence of oneside this does not work Therefore the area of the boundary is one-half the area of abulk surface infinitesimally close to it This is not very convincing on physical groundsof course The authors of [238] correct this discrepancy by effectively doubling the areaof the horizon This gives the same final result as in this book Thiemann has observedthat the need of this correction ldquoby handrdquo is an inconvenience of a formalism thattreats the horizon as a boundary One way out is to define the horizon area as the limitof the area of a bulk surface approaching the horizon

Notice that the black hole turns out to carry one bit of informationper quantum of area A0 This is precisely the ldquoit from bitrdquo picture thatJohn Wheeler suggested should be at the basis of black-hole physics in[165 166]

One remark before concluding The reader may object to the derivationabove (and the one in [238]) as follows the states that we have countedare transformed into each other by a gauge transformation Why thendo I consider them distinct in the entropy counting The answer to thisobjection is the following When we break the system into componentsgauge degrees of freedom may become physical degrees of freedom on theboundary The reason is that if we let the gauge group act independentlyon the two components it will act twice on the boundary A holonomyof a connection across the boundary for instance will become ill definedTherefore there are degrees of freedom on the boundary that are notgauge they tie the two sides to each other so to say


To illustrate this point let us consider two sets A and B and a groupG that acts (freely) on A and on B Then G acts on AtimesB What is thespace (AtimesB)G One might be tempted to say that it is (isomorphicto) AGtimesBG but a moment of reflection shows that this is not correctand the correct answer is

AtimesB

Gsim A

GtimesB (838)

(If G does not act freely over A we have to divide B by the stability groupsof the elements of A) Now imagine that A is the space of the states ofthe exterior of the black hole B the space of the states of the black holeand G the gauge group of the theory Then we see that we must not divideB by the gauge group of the surface but only by those internal gaugesand diffeomorphisms that leave the rest of the spin network invariant4

823 Ringing modes frequencies

Is there a way of understanding the peculiar numerical value (836) Theminimal quantum of area that plays a central role in black-hole thermo-dynamics is the one with spin j = 12 which using (836) is

A12 = 8πhGγ

radic12

(12 + 1

)= 4 ln 2 hG (839)

Because of (823) a change of one such quantum of area implies a changeof energy

ΔE = ΔM =A12

32πGM=

ln 2 h

8πM (840)

If we use the Bohr relation ΔE = hω to interpret this as a quantumemitted by an oscillator with angular frequency ω then the quantumgravity theory indicates that in the system there should be somethingoscillating with a proper frequency

ω =ln 2

8πM (841)

As far as I know this frequency plays no role in the classical theoryHowever suppose that for some reason the minimal quantum of area thatplays a central role in black-hole thermodynamics was due not to j=12spins as above but to j = 1 spins Then the calculation above would beslightly different The minimal area is

A1 = 8πhGγradic

1(1 + 1) (842)

4Actually in [239] only the boundary degrees of freedom due to diff invariance weretaken into account while in [238] and here only the boundary degrees of freedomdue to internal gauge invariance are taken into account Perhaps by taking both intoaccount (836) could change

312 Applications

and since the spin-1 representation has dimension 3 the entropy is

S = ln(3AA1) =ln 3

8πhGγradic

2A (843)

This agrees with the BekensteinndashHawking entropy if

γ =ln 3

2πradic

2 (844)

which in turn fixes the minimal relevant quantum of area to be

A1 = 4 ln 3hG (845)

Using (823) and the Bohr relation we obtain the proper frequency

ω =ln 3

8πM (846)

Now very remarkably there is something oscillating precisely with thisfrequency in a classical Schwarzschild black hole In fact the frequency(846) is precisely the frequency of the most damped ringing mode ofa Schwarzschild black hole The calculation of this frequency from theEinstein equations is complicated It was first computed numerically thenguessed to be (846) on the basis of the numerical value and only recentlyderived analytically Quite remarkably the quantum theory of gravityappears to know rather directly about this frequency hidden inside thenonlinearity of the Einstein equations

This fact seems to support the idea that the ringing modes of the blackhole are at the roots of its thermodynamics On the other hand it isnot clear why we should not consider the spin j = 12 Several possi-bilities have been suggested including dynamical selection rules and anequispaced area spectrum Overall this intriguing observation raises morequestions than providing fully satisfactory answers

824 The BekensteinndashMukhanov effect

In 1995 Bekenstein and Mukhanov suggested that the thermal nature ofHawkingrsquos radiation may be affected by quantum properties of gravityquite dramatically They observed that in some approaches to quantumgravity the area can take only quantized values Since the area of theblack-hole surface is connected to the black-hole energy the latter is likelyto be quantized as well The energy of the black hole decreases whenradiation is emitted Therefore emission happens when the black holemakes a quantum leap from one quantized value of the energy to a lowerquantized value very much as atoms do A consequence of this picture


is that radiation is emitted at quantized frequencies corresponding tothe differences between energy levels Thus quantum gravity implies adiscrete emission spectrum for the black-hole radiation

This result is not physically in contradiction with Hawkingrsquos predictionof an effectively continuous thermal spectrum To understand this con-sider the black-body radiation of a gas in a cavity at high temperatureThis radiation has a thermal planckian emission spectrum essentially con-tinuous However radiation is emitted by elementary quantum emissionprocesses yielding a discrete spectrum The solution of the apparent con-tradiction is that the spectral lines are so dense in the range of frequenciesof interest that they give rise ndash effectively ndash to a continuous spectrum

However Bekenstein and Mukhanov suggest that the case of a blackhole may be quite different from the case of the radiation of a cavityThey consider a simple ansatz for the spectrum of the area that the areais quantized in multiple integers of an elementary area A0 Namely thatthe area can take the values

An = nA0 (847)

where n is a positive integer and A0 is an elementary area of the orderof the Planck area

A0 = αhG (848)

where α is a number of the order of unity The ansatz (847) agrees withthe idea of a quantum picture of a geometry made up of elementaryldquoquanta of areardquo Since the black-hole mass (energy) is related to thearea by (823) it follows from this relation and the ansatz (847) that theenergy spectrum of the black hole is given by

Mn =

radicnαh

16πG (849)

Consider an emission process in which the emitted energy is much smallerthan the mass M of the black hole From (849) the spacing between theenergy levels is

ΔM =αh

32πGM (850)

From the quantum mechanical relation E = hω we conclude that energyis emitted in frequencies that are integer multiples of the fundamentalemission frequency

ω =α

32πGM (851)

This is the fundamental emission frequency of Bekenstein and MukhanovLet us now assume that the emission amplitude is correctly given by

314 Applications

Hawkingrsquos thermal spectrum Then the full emission spectrum is givenby spectral lines at frequencies that are multiples of ω whose envelopeis Hawkingrsquos thermal spectrum Now this spectrum is drastically differ-ent than the Hawking spectrum Indeed the maximum of the planckianemission spectrum of Hawkingrsquos thermal radiation is around

ωH sim 282kBTH

h=

2828πGM

=282 times 4

αω asymp ω (852)

The fundamental emission frequency ω is of the same order of magnitudeas the maximum of the Planck distribution of the emitted radiation Itfollows that there are only a few spectral lines in the regions where emis-sion is appreciable The BekensteinndashMukhanov spectrum and the Hawk-ing spectrum have the same envelope but while the Hawking spectrum iscontinuous the BekensteinndashMukhanov spectrum is formed by just a fewlines in the interval of frequencies where emission is appreciable This isthe BekensteinndashMukhanov effect

Is this BekensteinndashMukhanov effect truly realized in LQG At firstsight one is tempted to say yes since the spectrum of the area in LQGgiven in (678) is quite similar to the ansatz (847) If we disregard the+1 under the square root in (678) we obtain the ansatz (847) and thusthe BekensteinndashMukhanov effect But the +1 is there and the differenceturns out to be crucial

Let us study the consequences of the presence of the +1 Consider asurface Σ in the present case the event horizon of the black hole Thearea of Σ can take only a set of quantized values These quantized valuesare labeled by unordered n-tuples of positive half-integers j = (j1 jn)of arbitrary length n

We estimate the number of area eigenvalues between the value A hG and the value A + dA of the area where we take dA much smallerthan A but still much larger than hG Since the +1 in (678) affects ina considerable way only the terms with low spin ji we can neglect it fora rough estimate It is more convenient to use integers rather than half-integers Let us therefore define pi = 2ji We must estimate the numberof unordered strings of integers p = (p1 pn) such that

sum

i=1n

pi =A

8πγhG 1 (853)

This is a well-known problem in number theory It is called the partitionproblem It is the problem of computing the number N of ways in whichan integer I can be written as a sum of other integers The solution forlarge I is a classic result by Hardy and Ramanujan [240] According tothe HardyndashRamanujan formula N grows as the exponent of the square


root of I More precisely we have for large I that

N(I) sim 14radic

3Ieπ

radic23I (854)

Applying this result in our case we have that the number of eigenvaluesbetween A and A + dA is

ρ(A) asymp e

radicπA

12γhG (855)

Using (823) we have that the density of the states is

ρ(M) asymp e

radic4G3γh

πM (856)

Now without the +1 term there is a high degeneracy due to the fact thatall states with the same value of

sumn pn are degenerate The presence of the

+1 term kills this degeneracy and eigenvalues can overlap only acciden-tally generically all eigenvalues will be distinct Therefore the averagespacing between eigenvalues will be the inverse of the density of statesand will decrease exponentially with the inverse of the square of the areaThis result is to be contrasted with the fact that this spacing is constantand of the order of the Planck area in the case of the ansatz (847) Itfollows that for a macroscopic black hole the spacing between energy lev-els is infinitesimal and the spectral lines are virtually dense in frequencyWe effectively recover in this way Hawkingrsquos thermal spectrum5

The conclusion is that the BekensteinndashMukhanov effect disappears if wereplace the naive ansatz (847) with the area spectrum (678) computedfrom LQG

83 Observable effects

Possible low-energy effects of LQG have been studied using a semiclassicalapproximation Gambini and Pullin have introduced the idea to study thepropagation of matter fields over a weave state taking expectation valuesof smeared geometrical operators For suitable weave states this may leadto the possibility of having quantum gravitational effects on the dispersionrelations In particular they have studied light propagation and pointedout the possibility of an intriguing birefringence effect Alfaro Morales-Tecotl and Urrita have developed this technique In particular for a

5Mukhanov subsequently suggested that discretization could still occur as a conse-quence of dynamics For instance transitions in which a single Planck unit of area islost could be strongly favored by the dynamics

316 Applications

fermion of mass m they derived dispersion relations between energy Eand momentum p of the general form

E2 = p2 + m2 + f(p lP) (857)

where the last Lorentz-violating term may be helicity dependent Thepossibility that LQG could yield observable effects indeed has raised muchinterest Suggestions that these effects may be connected to observableor even already observed effects have been put forward These regardcosmic-ray energy thresholds gamma-ray bursts pulsar velocities andothers

In fact the old idea that quantum gravitational effects are certainly un-observable at present has been strongly questioned in recent years Somehave even expressed the hope that we could be ldquoat the dawn of quantumgravity phenomenologyrdquo [241] It is too soon to understand if these hopeswill be realized but the possibility is fascinating and the developmentof LQG calculations that could relate to these possible observations is animportant direction of development

Lorentz invariance in LQG Lorentz-violating effects might not bepresent in LQG There are two reasons for expecting Lorentz violations inLQG One is that the short-scale structure of a macroscopically Lorentz-invariant weave might break Lorentz invariance However it is not clearwhether all weave states break Lorentz invariance or not A single spinnetwork state cannot be Lorentz invariant but this does not imply thata state which is a quantum superposition of spin network states cannoteither

The second reason which is often mentioned is the observation that aminimal length (or a minimal area) necessarily breaks Lorentz invarianceThe reason would be the following if an observer measures the minimallength 13P then a boosted observer will observe the Lorentz-contractedlength 13prime = γminus113P which is shorter than 13P and therefore 13P cannotbe a minimal length Here γ = 1

radic1 minus v2c2 is the LorentzndashFitzgerald

contraction factor This observation is wrong because it ignores quantummechanics

Length area and volume are not classical quantities They are quantumobservables If an observer measures the length 13P of some system thismeans that the system is in an eigenstate of the length operator A boostedobserver who measures the length of the same system is measuring adifferent observable Lprime which generally does not commute with L If thesystem is in an eigenstate of L generally it will not be in an eigenstateof Lprime Therefore there will be a distribution of probabilities of observingdifferent eigenvalues of Lprime The eigenvalues of Lprime will be the same as the


Σ

Σprime

Fig 81 The grey region represents the world-history of an object The twoarrows represent the worldlines of two observers The bold segments Σ and Σprime

are the intersections between the object world-history and the observersrsquo equal-time surfaces Two observers in relative motion measuring the lengths of thesame object measure the gravitational field on these two distinct surfaces Thegravitational field on Σ does not commute with the gravitational field on ΣprimeHence the two lengths do not commute

eigenvalues of L it is the expectation value of Lprime that will be Lorentzcontracted

The situation is the same as for the Lz component of angular momen-tum Consider a quantum system with total spin = 1 Say an observermeasures Lz and obtains the eigenvalue Lz = h Does this mean thata second observer rotated by an angle α will observe the eigenvalueLprimez = cosα h Of course not The second observer will still measure

Lprimez = 0plusmnh with a probability distribution such that the mean value

is Lprimez = cosα h States and mean values transform continuously with a

rotation but eigenvalues stay the same In the same fashion in a Lorentzboost states and values transform continuously while the eigenvalues staythe same

To understand why a length L and boosted length Lprime do not commuteconsider Figure 81 It represents the world-history of an object and thetwo lengths measured by two observers in relative motion Notice that thetwo observers measure the gravitational field on two distinct segments Σ

318 Applications

and Σprime with a time separation Since no quantum field operator com-mutes with itself at timelike separations clearly the two functions of thegravitational field e(x)

L =int

Σ

radic|e| and Lprime =

int

Σprime

radic|e| (858)

do not commute For a detailed discussion of this point and the actualconstruction of boosted geometrical operators see [242]

mdashmdash


Loop quantum cosmology has been mostly developed by Martin BojowaldThe absence of the initial singularity was derived in [243] and the sug-gestion about the possibility of a quantum-driven inflation presented in[244] For a review see [232] Loop cosmology is rapidly developing arecent introduction with up-to-date (2007) results and references is [235]

Another powerful recent application of loop quantum gravity is to re-solve the r = 0 singularity at the center of a black hole The idea hasbeen proposed by Leonardo Modesto [236] and has been independentlyconsidered and developed by Ashtekar and Bojowald [237]

The Hawking theorem on the growth of the black-hole area is in [245]Bardeen Carter and Hawking presented their ldquofour laws of black-hole mechanicsrdquo in [246] Bekenstein entropy was presented in [247] andHawking black-hole radiance in [248] For a review of the field see [28]

The first suggestions on the possibility of using the counting of thequanta of area of LQG to describe black-hole thermodynamics were pro-posed by Kirill Krasnov [249] For discussions on black-hole entropy inLQG I have followed here [239] The derivation with the horizon as aboundary is in [238] The idea that black-hole entropy originates fromthe fluctuations of the shape of the horizon was suggested by York [250]The relevance of the horizon surface degrees of freedom for the entropyhas since been emphasized from different perspectives see for instance[251] On the notion of an isolated horizon see [252] The relevance of theChernndashSimon boundary theory for the description of the horizon surfacedegrees of freedom was noticed in [253] the importance of the gauge de-grees of freedom on the boundary has been emphasized in [254] A recentreview including the new value of the Immirzi parameter is []

The relation between the ringing-mode frequencies and the spectrumhas been studied by S Hod in [255] and the fact that spin-1 area excitationsin LQG are related to the ringing-mode frequency has been pointed out


by Olaf Dreyer in [256] On the reason why spin-1 could contribute tothe horizon area more than spin-12 see [257] for a possible dynamicalselection rule and [227] for the role of the ordering of the area operator

The BekensteinndashMukhanov effect was presented in [258] (with α =4 ln 2) For a review of earlier suggestions in this direction see [259] Theargument presented here on the absence of this effect in loop gravityappeared in [260] The same result was derived in [173]

The possibility of a semiclassical description of the propagation over aweave that leads to estimates of quantum gravity effects was introducedin [261] for photons and developed in [262] for photons and neutrinosFor a review see [232] On the fact that violations of Lorentz invarianceare not necessarily implied by LQG see [242] and [263] On the difficultiesimplied by the breaking of Lorentz invariance in quantum gravity see also[264] On the possibility that Planck-scale observation might be withinobservation reach see [241 265 266]

9Quantum spacetime spinfoams

Classical mechanics admits two different kinds of formulations hamiltonian and la-grangian (I never understood why) Feynman realized that so does quantum mechanicsit can be formulated canonically with Hilbert spaces and operators or covariantly asa sum-over-paths The two formulations have different virtues and calculations thatare simple in one can be hard in the other Generically the lagrangian formalism issimpler more transparent and intuitive and keeps symmetries and covariance manifestThe hamiltonian formalism is more general more powerful and in the case of quantumtheory far more rigorous The ideal situation of course is to be able to master a theoryin both formalisms

So far I have discussed the hamiltonian formulation of LQG In this chapter I discussthe possibility of a lagrangian sum-over-paths formulation of the same theory Thisformulation has been variously called sum-over-surfaces state-sum and goes todayunder the name of ldquospinfoam formalismrdquo The spinfoam formalism can be viewed as amathematically well-defined and possibly divergence-free version of Stephen Hawkingrsquosformulation of quantum gravity as a sum-over-geometries

The spinfoam formalism is less developed than the hamiltonian version of loop the-ory Also while the general structure of the spinfoam models matches nicely with thehamiltonian loop theory the precise relation between the two formalisms (each of whichexists in several versions) has been rigorously established only in 3d But research iscurrently moving fast in this direction

The aim of the spinfoam formalism is to provide an explicit tool to compute tran-sition amplitudes in quantum gravity These are expressed as a sum-over-paths Theldquopathsrdquo summed over are ldquospinfoamsrdquo A spinfoam can be thought of as the worldsur-face swept out by a spin network Spinfoams are background-independent combinatorialobjects and do not need a spacetime to live in A spinfoam itself represents a spacetimein the same sense in which a spin network represents a space

The most remarkable aspect of the spinfoam approach is the fact that a surprisingnumber of independent research directions converge towards the same formalism Iillustrate some of these converging research paths below This convergence seems toindicate that the spinfoam formalism is a sort of natural general language for sum-over-paths formulations of general-relativistic quantum field theories

There are several good review articles on this subject ndash see the bibliographical notesat the end of the chapter I do not repeat here what is done in detail in other reviewsand I recommend the studious reader to refer to these reviews to complement the

320

91 From loops to spinfoams 321

introduction provided here since this is a subject which can be approached from avariety of points of view Here I focus on the overall significance of these models forour quest for a complete theory of quantum gravity

91 From loops to spinfoams

Sum-over-paths Consider a nonrelativistic one-dimensional quantum sys-tem Let x be its dynamical variable The propagator W (x t xprime tprime) of thesystem is defined in (510) As emphasized by Feynman W (x t xprime tprime)contains the full dynamical information about the quantum system It issimply related to the exponential of the Hamilton function S(x t xprime tprime)which as illustrated in Chapter 3 codes the classical dynamics of the sys-tem As illustrated in Chapter 5 in the relativistic formalism W (x t xprime tprime)can be obtained as the matrix element of the projection operator P be-tween states |x t〉 that are eigenstates of the operators corresponding tothe partial observables x and t

W (x txprime tprime) = 〈x t|P |xprime tprime〉K (91)

where K = L2[R2 dxdt] is the kinematical Hilbert space on which theoperators corresponding to the partial observables x and t are defined

A key intuition of Richard Feynman was that the propagator can beexpressed as a path integral

W (x t xprime tprime) simint

x(t)=x

x(tprime)=xprimeD[x(t)] eiS[x] (92)

in which the sum is over the paths x(t) that start at (xprime tprime) and endat (x t) and S[x] =

int ttprime L(x(t) x(t))dt is the action of this path There

are several techniques to define and manipulate this integral Feynmanand many after him have suggested that (92) can be taken as the basicdefinition of the quantum formalism This can therefore be based on asum of complex amplitudes eiS[x] over the paths x(t)

The idea of utilizing a sum-over-paths formalism in quantum gravity isold (see Appendix B) and has been studied extensively The idea is to tryto define a path integral over 4d metrics

intD[gμν(x)] eiSGR[g] (93)

In particular we can consider 3d metric gprime and a final 3d metric g andstudy the transition amplitude between these defined by

W [g gprime] =int

g|t=1=g

g|t=0=gprimeD[gμν(x)] eiSGR[g] (94)

322 Quantum spacetime spinfoams

where SGR is the action of the strip between t = 0 and t = 1 The specificvalue chosen for the coordinate t is irrelevant if the functional integralis defined in a diffeomorphism-invariant manner The techniques used inquantum mechanics and quantum field theory to give meaning to the func-tional integral however fail in gravity and the effectiveness of the pathintegral approach has long remained confined to crude approximationsThe reason is that nonperturbative definitions of the measure D[gμν(x)]leading to a complete theory are not known and perturbative definitionsaround a background metric lead to nonrenormalizable divergences

The situation has changed with loop gravity because of the discoveryof the discreteness of physical space To understand why a few generalconsiderations are in order

From the hamiltonian theory to the sum-over-paths Generally we do nothave a very good control of the direct definition of the Feynman sum-over-paths For many issues such as the choice of the measure the canonicaltheory often provides the best route toward the correct definition of the in-tegral Feynman of course derived the functional integral from the canon-ical theory in the first place He did so by writing the evolution operatoreminusiH0t as a product of small time-step evolution operators inserting res-olutions of the identity 1 =

intdx |x〉〈x| and taking the limit for the time

interval dt = (tminus tprime)N rarr 0 The functional integral is then defined asint

x(t)=x

x(tprime)=xprimeD[x(t)] eiS[x] equiv lim

Nrarrinfin

intdx1 dxNminus1

〈x| eminusiH0(tminustprime)

N |xNminus1〉〈xNminus1| eminusiH0(tminustprime)

N |xNminus2〉

〈x2|eminusiH0(tminustprime)

N |x1〉〈x1| eminusiH0(tminustprime)

N |xprime〉(95)

The canonical theory can therefore be used to construct a sum-over-pathsCan this be done in quantum gravity The problem is to obtain in quan-tum gravity the analog of the FeynmanndashKac formula which provides aprecise definition of the path integral So far this problem is unsolvedand a sum-over-paths formalism has not yet been rigorously derived fromthe canonical theory However we can still proceed in this direction usingthe following strategy

First we may write formal equations that indicate the general structurethat a sum-over-paths in quantum gravity should have That is we canderive the basics of the gravitational sum-over-paths formalism from thecanonical theory Indeed as we shall see in a moment a few loose for-mal manipulations show immediately that a generally covariant sum-over-paths formalism for quantum gravity must have quite peculiar properties


because of the discreteness of space This is done below Second we canstudy specific theoretical models having this general structure This isdone in the subsequent sections

Transition amplitudes are between spin networks A key observation isthat the quantity x in the argument of the propagator is not the classicalvariable but rather a label of an eigenstate of this variable The differenceis irrelevant so long as x is an observable with a continuous spectrum suchas the position But it becomes relevant if the spectrum of x is nontrivialFor instance consider a harmonic oscillator subjected to an external forceor to a small nonlinear perturbation Instead of asking for the amplitudeW (x txprime tprime) of measuring x given xprime ask for the probability amplitudeW (E t Eprime tprime) of measuring the (unperturbed) energy E This is given by

W (E t Eprime tprime) = 〈E|eminusiH0(tminustprime)|Eprime〉 (96)

where |E〉 is the eigenstate of the unperturbed energy with eigenvalueE (H0 is the nonrelativistic hamiltonian not the unperturbed one) Butthe eigenvalues E are quantized E = En and |n〉 = |En〉 ThereforeW (E t Eprime tprime) is defined only for the values of E = En in the spectrumThe argument of the propagator must be discrete energy levels not clas-sical energies Thus (96) only makes sense in the special form

W (n t nprime tprime) equiv W (En t Enprime tprime) = 〈n|eminusiH0(tminustprime)|nprime〉 (97)

Consider now the integral (94) This must define transition amplitudesbetween eigenstates of the 3-geometry We have seen in Chapter 6 that theeigenvalues of the 3-geometry are not 3d continuous metrics Rather theyhave a discrete structure and are labeled by spin networks Therefore thepropagator in quantum gravity must be a function of spin networks Forinstance we should study instead of (94) a quantity W (s sprime) giving theprobability amplitude of measuring the quantized 3-geometry describedby the spin network s if the quantized 3-geometry described by the spinnetwork sprime has been measured

Histories of spin networks Consider the propagator W (s sprime) As dis-cussed at the end of Section 74 we can express W (s sprime) in the form

W (s sprime) = 〈s|P |sprime〉K (98)

I now write some formal expressions that may indicate the way to expressW (s sprime) as a sum-over-paths Loosely speaking the operator P is theprojector on the kernel of the hamiltonian operator H If we tentatively


assume that H has a nonnegative spectrum we can formally write thisprojector as

P = limtrarrinfin

eminusHt (99)

because if |n〉 is a basis that diagonalizes H and En are the correspondingeigenvalues then

P = limtrarrinfin

sum

n

|n〉eminusEnt〈n| =sum

n

δ0En |n〉〈n| (910)

Since H is a function of a spatial coordinate x we can formally write

P = limtrarrinfin

prod

x

eminusH(x)t = limtrarrinfin

eminusint

d3xH(x)t (911)

Hence

W (s sprime) = limtrarrinfin

〈s|eminusint

d3xH(x)t|sprime〉K (912)

If we can define the 4d propagation generated by H in a diff-invariantmanner the limit is irrelevant and we can write again quite loosely

W (s sprime) = 〈s|eminusint 1

0dt

intd3xH(x)|sprime〉K (913)

We can now expand this expression in the same manner as in the right-hand side of (95) Inserting resolutions of the identity 1 =

sums |s〉〈s| we

obtain an expression of the form

W (s sprime) = limNrarrinfin

sum

s1sN

〈s|eminusint

d3xH(x) dt|sN 〉K 〈sN |eminusint

d3x H(x) dt|sNminus1〉K

〈s1|eminusint

d3xH(x) dt|sprime〉K (914)

For small dt we can expand the exponentials At fixed N the first termin the expansion (ex = 1 + 0(x)) produces a term equivalent to historieswith lower N Therefore we can view the sum as a sum-over-sequences ofspin networks where the sequences can have arbitrary lengths The resultis that the transition amplitude is not expressed as an integral over 4dfields but rather as a discrete sum-over-histories σ of spin networks

W (s sprime) =sum

σ

A(σ) (915)

A history is a discrete sequence σ = (s sN s1 sprime) of spin networks

The amplitude associated with a single history is a product of terms

A(σ) =prod

v

Av(σ) (916)



where v labels the steps of the history and Av(σ) is determined by thematrix elements

〈sn+1 |eminusint

d3xH(x)dt|sn〉K (917)

For small dt we can keep only the linear term in H in these matrix el-ements as Feynman did in (95) Now the action of the hamiltonianoperator H is given in (721) It is a sum over individual terms actingon the nodes of the spin network Therefore it has nonvanishing matrixelements only between spin networks sn+1 and sn that differ at one nodeby the action of H A typical term in the action (for instance) of H ona trivalent node is illustrated in Figure 72 or more schematically inFigure 91

Summarizing we can write W (s sprime) as a sum-over-paths of spinnetworks the paths are generated by individual steps such as the oneillustrated in Figure 72 the amplitude of each step is determined by thecorresponding matrix element of H the amplitude of the history is theproduct of the individual amplitudes of the steps

Spinfoams A history σ = (s sN s1 sprime) of spin networks is called a

spinfoam A more precise definition is given below A spinfoam admits anatural representation as follows Imagine a 4d space (representing coor-dinate spacetime) in which the graph of a spin network s is embeddedNow imagine that this graph moves upward along a ldquotimerdquo coordinateof the 4d space sweeping a worldsheet changing at each step under theaction of H Call ldquofacesrdquo the worldsurfaces of the links of the graph anddenote them f Call ldquoedgesrdquo the worldlines of the nodes of the graph anddenote them as e Figure 92 illustrates the worldsheet of a theta-shapedspin network

Since the hamiltonian acts on nodes the individual steps in the historyof a spin network can be represented as the branching of the edges whichlocally changes the number of nodes We call ldquoverticesrdquo the points whereedges branch and denote them as v

For instance an edge can branch to form three edges as in Figure 93representing the action of the hamiltonian constraint illustrated in Figure91


i

fs

is

f

Σ

Σ

Fig 92 The worldsheet of a spin network si on an initial surface Σi evolvingwithout intersections into the spin networks sf on the final surface Σf forminga spinfoam

Fig 93 A vertex of a spinfoam

The resulting worldsheet is illustrated in Figure 94 which represents aspinfoam with a single vertex A spinfoam with two vertices is representedin Figure 95 What we obtain in this manner is a collection of faces f meeting at edges e which in turn meet at vertices v The combinatorialobject Γ defined by the set of these elements and their adjacency relationsis called a ldquotwo-complexrdquo

A spin network is not defined solely by its graph but also by the coloringof its links (representations) and nodes (intertwiners) Accordingly thetwo-complex Γ determined by a sequence of spin networks is colored withirreducible representations jf associated with faces and intertwiners ieassociated with edges A spinfoam σ = (Γ jf ie) is a two-complex Γ withcolored faces and edges That is it is a two-complex with a representationjf associated with each face f and an intertwiner ie associated with eachedge e

92 Spinfoam formalism 327

v

si

sf

Σi

Σf

Fig 94 A spinfoam with one vertex

v2

v1

5

56

7

8

8

1

3

7

63

3

si

sf

s1

Σi

Σf

Fig 95 A spinfoam with two vertices

92 Spinfoam formalism

We are now ready to give a general definition of a spinfoam theory Thediscussion in the previous section indicates that a sum-over-paths formu-lation of quantum gravity can be cast in the form of a sum-over-spinfoamsof amplitudes given by products of individual vertex amplitudes


Consider a sum defined as follows

Z =sum

Γ

w(Γ)sum

jf ie

prod

v

Av(jf ie) (918)

The sum is over a set of two-complexes Γ and a set of representations andintertwiners j and i The function Av(jf ie) called the vertex amplitudeis an amplitude associated with each vertex v It is a function of the colorsadjacent to that vertex w(Γ) is a weight factor that depends only on thetwo-complex

It is often convenient to rewrite expression (918) in the extended form

Z =sum

Γ

w(Γ)sum

jf ie

prod

f

Af (jf )prod

e

Ae(jf ie)prod

v

Av(jf ie) (919)

with amplitudes Af and Ae associated to faces and edges as well al-though these can in principle be included in a redefinition of Av Formost models so far considered Af (jf ) is simply the dimension dim(jf ) ofthe representation jf Hence we have (using the notation σ = (Γ jf ie))

Z =sum

σ

w(Γ(σ))prod

f

dim(jf )prod

e

Ae(jf ie)prod

v

Av(jf ie) (920)

This is the general expression that we take as the definition of the spin-foam formalism

A choice of(i) a set of two-complexes Γ and associated weight w(Γ)(ii) a set of representations and intertwiners j and i(iii) a vertex amplitude Av(jf ie) and an edge amplitude Ae(jf ie)

defines a ldquospinfoam modelrdquo We shall study several of these models andtheir relation with gravity in the next section Generally speaking thechoice (iii) of the vertex amplitude corresponds to the choice of a specificform of the hamiltonian operator in the canonical theory

What is remarkable about expression (920) is that many very differentapproaches and techniques have converged precisely to this formula aswe shall see later on Perhaps an expression of this sort can be taken as ageneral definition of a background-independent covariant QFT formalism

In Table 91 I have summarized the terminology used to denote theelements of spin networks spinfoams and triangulations (which play arole later on)

921 Boundaries

The boundary of a spinfoam σ is a spin network s This follows easilyfrom the very way we have constructed spinfoams If σ is bounded by

93 Models 329

Table 91 Terminology

0d 1d 2d 3d 4d

spin network node link

spinfoam vertex edge face

triangulation point segment triangle tetrahedron four-simplex

the spin network s we write this as partσ = s The relation between (920)and transition amplitudes is obtained by summing over spinfoams with agiven boundary

W (s) =sum

partσ=s

w(Γ(σ))prod

f

dim(jf )prod

e

Ae(jf ie)prod

v

Av(jf ie) (921)

In particular if the spin network s is connected it can be interpreted asthe state of the gravitational field on the connected boundary of a space-time region For instance the boundary of a finite region of spacetime

If the boundary spin network is composed of two connected componentss and sprime we write

W (s sprime) =sum

partσ=scupsprimew(Γ(σ))

prod

f

dim(jf )prod

e

Ae(jf ie)prod

v

Av(jf ie) (922)

and we interpret the spinfoam model sum as a sum-over-paths definition ofthe transition amplitude between two quantum states of the gravitationalfield in analogy with (92)

So far the situation is not that we can compute W (s sprime) in the two for-malisms and prove the two to be equal In the spinfoam framework thereis uncertainty in the definition of the model but as we shall see tran-sition amplitudes can be computed (order by order) In the hamiltonianformalism on the other hand even disregarding the uncertainty in thedefinition of the hamiltonian we are not yet able to compute transitionamplitudes

93 Models

I shall now illustrate a few key examples of spinfoam models Each of theseis a realization of equation (920) Namely each of these is obtained from(920) by choosing a set of two-complexes a set of representations and in-tertwiners and vertex and edge amplitudes These models are relatedto different theories general relativity without and with cosmological


Table 92 Spinfoam models (λ cosmological constant Tr triangulation)

Model Class theory Two-complexes Representation Vertex

PonzanondashRegge 3d GR fixed dual 3d Tr SU(2) 6j

TuraevndashViro 3d GR +λ fixed dual 3d Tr SU(2)q 6jq

Ooguri (TOCY) 4d BF fixed dual 4d Tr SO(4) 15j

CranendashYetter 4d BF +λ fixed dual 4d Tr SO(4)q 15jq

BarrettndashCrane A cut-off 4d GR fixed dual 4d Tr simple SO(4) 15j

BarrettndashCrane B cut-off 4d GR fixed dual 4d Tr simple SO(4) 10jBC

GFT A 4d GR Feynman graphs simple SO(4) 15j

GFT B 4d GR Feynman graphs simple SO(4) 10jBC

constant in 3d and 4d and in BF theory (see Section 932) They arelisted in Table 92

These models form a natural sequence that has historically led to thepresent formulation of spinfoam quantum gravity This is represented bythe last of them constructed using an auxiliary field theory defined ona group manifold This represents a complete tentative sum-over-pathsformalism for quantum gravity in 4d The sequence of these models doesnot have just historical interest rather it allows me to introduce stepby step the ingredients that enter the complete model The models haveincreasing complexity Each of them has introduced and illustrates animportant peculiar aspect of the complete theory Here is a condensedpreview of the way each model has contributed to the construction ofthe formalism (terms and concepts will be clarified in the course of thechapter)

(i) The PonzanondashRegge theory is a quantization of gravity in 3d Itillustrates how a sum-over-paths in quantum gravity naturally takesthe form of a spinfoam model and why the gravitational vertexamplitude can be expressed in terms of simple invariant objects fromgroup representation theory For a long time it was assumed thatthese simple features were characteristic of 3d andor reflected thefact that the theory has no local degrees of freedom (is topological)The other models show that this assumption was wrong

93 Models 331

(ii) The Ooguri or TOCY (TuraevndashOogurindashCranendashYetter) model ex-tends the formalism to 4d

(iii) The BarrettndashCrane models extend the formalism to a theory withlocal degrees of freedom The key for doing this is the realizationthat GR can be obtained from a topological theory by adding certainconstraints and these constraints can be implemented in the modelas a restriction on the set of representations summed over

(iv) As soon as the model is no longer topological the sum over two-complexes becomes nontrivial A way to implement it is providedby the Group Field Theory (GFT )

931 3d quantum gravity

3d GR Consider riemannian general relativity in three dimensions Thiscan be defined by minor modifications of the definition of 4d GR given atthe beginning of Chapter 2 In 3d the gravitational field e is a one-form

ei(x) = eia(x)dxa (923)

with values in R3 The spin connection ω is a one-form with values in theso(3) Lie algebra

ωi(x) = ωia(x)dxa (924)

and we denote its curvature two-form as Ri The action that defines thetheory is

S[e ω] =int

ei andR[ω]i (925)

Varying ω gives the Cartan structure equation De = 0 Varying e givesthe equation of motion R = 0 which implies that spacetime is flat1 Thatis locally in spacetime there is a single solution of the equations of motionup to gauge

The theory is nevertheless nontrivial if the space manifold has nontrivialglobal topology For instance there are distinct nonisometric flat tori Aflat torus is characterized say by its volume and its two radii which areglobal variables The dynamics of 3d general relativity is reduced to thedynamics of this kind of global variables A theory of this sort that has

1The EinsteinndashHilbert action S[g] =int

d3xradicgR where gab is the 3d metric and R its

Ricci scalar gives the same equations of motion as (925) but differs from (925) by asign when the determinant of eia is negative Hence quantizations of the two actionsmight lead to inequivalent theories Which of the two is to be called 3d GR is a matterof taste as they have the same classical solutions


Table 93 Relation between a triangulation Δ and its dual Δ in 3d (left) and4d (right) In italic the two-complex In parentheses adjacent elements

Δ3 Δlowast3

tetrahedron vertex (4 edges 6 faces)

triangle edge (3 faces)

segment facepoint 3d region

Δ4 Δlowast4

4-simplex vertex (5 edges 10 faces)

tetrahedron edge (4 faces)

triangle facesegment 3d regionpoint 4d region

no local degrees of freedom but only global ones is called a topologicaltheory2

Discretization I now give a concrete definition of the functional integral(93) for 3d GR by discretizing the theory To this end fix a triangulationΔ of the spacetime manifold It is more convenient to work with the dualΔlowast of the triangulation and in particular with the two-skeleton of ΔlowastThese are defined as follows see Table 93

To obtain Δlowast we place a vertex v inside each tetrahedron of Δ if two tetrahedrabound the same triangle e we connect the two corresponding vertices by an edge edual to the triangle e for each segment f of the triangulation we have a face f of Δlowastbounded by the edges corresponding to the triangles of Δ that are bounded by thesegment f finally to each point of Δ we have a 3d region of Δlowast bounded by the facesdual to the segments that are bounded by the point In 4d Δlowast is obtained by placinga vertex in each four-simplex and so on The collection of the sole vertices edges andfaces of Δlowast (with their boundary relations) is called the two-skeleton of Δlowast and isprecisely a two-complex

Let ge be the holonomy of ω along each edge e of Δlowast (The connectionω is in the algebra and the algebra of SO(3) is the same as the algebra ofSU(2) In defining the holonomy we have to decide whether we interpretω as an SO(3) or an SU(2) connection Let us choose the SU(2) inter-pretation That is we define ge = P expinte ωiτi isin SU(2) where τi arePauli matrices)

Let lif be the line integral of ei along the segment f of Δ Wechoose these as basic variables for the discretization The variables of the

2The expression ldquotopological field theoryrdquo is used with different meanings in the litera-ture In [267] for instance it is used to designate any diffeomorphism-invariant theorywith a finite or infinite number of degrees of freedom This is done to emphasize thesimilarities among all these theories and their difference from QFT on a backgroundHere there is no risk of underemphasizing this difference which is stressed throughoutthis book and I prefer to follow the common usage

93 Models 333

discretized theory will therefore be an SU(2) group element ge associatedwith each edge e of Δlowast and a variable lif in R3 associated with each seg-ment of Δ or equivalently to each face f of Δlowast Accordingly we candiscretize the action as

S[lf ge] =sum

f

lif tr[gfτi] (926)

wheregf = g

ef1 g

efn(927)

is the product of the group elements associated with the edges ef1 efn

that bound the face f If we vary this action with respect to lif we obtainthe equation of motion gf = 1 namely the lattice connection is flat Usingthis if we vary this action with respect to ge we obtain the equation ofmotion lif1 + lif2 + lif3 = 0 for the three sides f1 f2 f3 of each triangle Thisis the discretized version of the Cartan structure equation De = 0

Path integral Using this discretization we can define the path integralas

Z =int

dlif dge eiS[lf ge] (928)

where the measure on SU(2) is the invariant Haar measure The integralover lf gives immediately (up to an overall normalization factor that weabsorb in the definition of the measure)

Z =int

dgeprod

f

δ(gef1

gefn

) (929)

We can now expand the delta function over the group manifold using theexpansion

δ(g) =sum

j

dim(j) trRj(g) (930)

where the sum is over all unitary irreducible representations of SU(2)Inserting this in (929) and exchanging the sum and the product wehave

Z =sum

j1jN

prod

f

dim(jf )int

dgeprod

f

trRjf (gef1

gefn

) (931)

It is not difficult to perform the integrations over the group There isone integral per edge Since every edge bounds precisely three faces eachintegral is of the form

intdURj1(U)ααprime Rj2(U)ββprime R

j3(U)γγprime = vαβγ vαprimeβprimeγprime (932)


where vαβγ is the (unique) normalized intertwiner between the represen-tations of spin j1 j2 j3 The reader should not confuse the symbol v thatdenotes vertices with the tensor vα1middotmiddotmiddotαn used to denote intertwiners

Each of the two invariant tensors on the right-hand side is associatedwith one of the two vertices that bound the edge (whose group elementis integrated over) Its indices get contracted with those coming from theother edges at this vertex A moment of reflection shows that at eachvertex we have four of these tensors that contract giving a function ofthe six spins associated with the six faces that bound the vertex

6j equiv(j1 j2 j6j4 j3 j5

)equiv

sum

α1α6

vα3α6α2 vα2α1α5 vα6α4α1 vα4α3α5 (933)

The pattern of the contraction of the indices reproduces the structure ofa tetrahedron if we (i) represent each 3-tensor vα1α2α3 using the repre-sentation (686) namely we write it as a trivalent vertex (ii) representindex contraction by joining legs (open ends) and (iii) indicate the rep-resentation to which the index belongs the 6j symbol is represented as

j1

j2 j3

j4 j6

j5

(934)

This function denoted 6j is a well-known function in the representa-tion theory of SU(2) It is a natural object that one can construct givensix irreducible representations It is called the Wigner 6j symbol see Ap-pendix A1

Bringing all this together we obtain the following form for the partitionfunction of 3d GR

ZPR =sum

j1jN

prod

f

dim(jf )prod

v

6jv (935)

This is the PonzanondashRegge spinfoam model Using the notation (686)we can write it in the form

ZPR =

sum

j

prod

f

dim(jf )prod

v

j1

j2

j3

j4 j6j5

(936)

93 Models 335

This expression has the general form (921) with the following choicesbull The set of two-complexes summed over is formed by a single two-

complex This is chosen to be the two-skeleton of the dual of a 3dtriangulation

bull The representations summed over are the unitary irreducibles ofSU(2) The intertwiners are trivial

bull The vertex amplitude is Av = 6jRemarkably these simple choices define the PonzanondashRegge quantizationof 3d GR

The fact that the vertex amplitude is simply the Wigner 6j symbol isperhaps surprising The Wigner 6j symbol is a simple algebraic constructof SU(2) representation theory The action of general relativity is a com-plicated expression coding the complexity of the gravitational interactionEven more remarkable is the fact that as we shall see this is not a strangecoincidence of the particularly simple form of GR in 3d rather the sameconnection between simple algebraic group theoretical quantities and thegravitational action holds in four dimensions as well This connection isone of the ldquomiraclesrdquo that nurtures the interest in the spinfoam approach

The original derivation the PonzanondashRegge ansatz The derivation aboveis not the original one of Regge and Ponzano It is instructive to men-tion also the general lines of the original derivation Regge introduced adiscretization of classical general relativity called Regge calculus definedover a fixed triangulation Consider the triangulation Δ of the spacetimemanifold and denote as f its segments (the choice of the letter f will beclear below) A gravitational field associates length lf to each segment f In turn these lengths lf can be taken as a discrete set of variables thatreplace the continuous metric The action of a given gravitational fieldcan be approximated by an action functional SRegge(lf ) of these lengthsAs the continuous action is an integral over spacetime the Regge action isa sum over the n-simplices v of the triangulation of the action of a singlen-simplex

S =sum

v

Sv (937)

We can then write a discretized version of (93) in the form

Z =int

dl1 dlN eiSRegge(lf ) (938)

Hence we can write

Z =int

dl1 dlNprod

v

eiSv(lf ) (939)

where Sv(lf ) is the Regge action of an individual n-simplex


Ponzano and Regge studied the integral (939) in the case of generalrelativity in 3d under one additional assumption that the length of eachlink can take only the discrete values

ln = jn jn =12 1

32 2 (940)

in units in which the Planck length lP = 1 This assumption is calledthe PonzanondashRegge ansatz Ponzano and Regge did not provide any jus-tification for it They introduced it just as discretization of the multipleintegral over lengths Notice that this is a second discretization in addi-tion to the triangulation of spacetime Its physical meaning was clarifiedmuch later and I will come back to it later on Under the PonzanondashReggeansatz (940) equation (939) becomes

ZPR =

sum

j1jN

prod

v

eiSv(jn) (941)

In 3d the Regge action of a 3-simplex (a tetrahedron) v can be writtenas a sum over the segments f in v as

Sv =sum

f

lfθf (lf ) (942)

where θf is the dihedral angle of the segment f that is the angle betweenthe outward normals of the triangles incident to the segment One canshow that this action is an approximation to the integral of the Ricciscalar curvature Under the PonzanondashRegge ansatz therefore Sv is afunction Sv(jn) of six spins j1 j6 (A tetrahedron has six edges)

The ldquomiraclerdquo GR dynamics in a symbol The surprising discovery ofPonzano and Regge was that the Wigner 6j symbol approximates theaction of general relativity More precisely they were able to show thatin the limit of large js we have the asymptotic formula

6j sim(eiSv(jn) + eminusiSv(jn)

)+

π

4 (943)

The term π4 does not affect classical dynamics The two exponentialterms in (943) are analogous to the two terms that we found in Sec-tion 523 see in particular the discussion at the end of that section Theclassical theory does not distinguish between forward and backward prop-agation in coordinate time and the path integral sums over the two Thetwo terms in (943) correspond to these two propagations Inserting (943)in (941) and fixing the normalization factors by imposing triangulationindependence (see below) we get (935) which is the expression that Pon-zano and Regge proposed as a discretized path integral for 3d quantumgravity

93 Models 337

Physical meaning of the PonzanondashRegge ansatz As noted at the begin-ning of this chapter the path integral defines transition amplitudes be-tween eigenstates of field operators not between classical fields ThePonzanondashRegge path integral (935) defines transition amplitudes betweentriangulated 2d surfaces where the links of the 2d triangulation have alength lf These lengths under the PonzanondashRegge ansatz are quan-tized Therefore the PonzanondashRegge ansatz is equivalent to the physicalassumption that length is quantized in 3d quantum gravity Now thislength quantization is not an ansatz but a result in loop quantum gravityIndeed it is not hard to see that in 3d the result that the area is quantizeddescribed in Chapter 6 translates into the quantization of length There-fore the key additional input provided by the PonzanondashRegge ansatz isphysically justified by loop gravity

In the previous derivation the discretization of the length derives fromthe following manipulation that we did over the integral First we inte-grated over the continuous variable lif obtaining a delta function (equa-tion (929)) Then second we expressed this delta function as a sum inequation (930) To understand the sense of this back and forward trans-formation consider the following example Let x be a variable in theinterval (minusπ π) We have

intdp eipx = 2πδ(x) (944)

and also sum

n

eipnx = 2πδ(x) (945)

where pn = n Here the distribution δ(x) is over functions on the compact(minusπ π) interval Therefore as long as we deal with functions on thiscompact interval we can replace an integral with a sum This is preciselywhat we did between (929) and (931) The compact space is the groupmanifold over which the holonomy takes values

The mathematical fact that the ldquoFourier componentsrdquo over a compactinterval are discrete is of course strictly connected to the physical factthat quantities conjugate to variables that take value in a compact spaceare quantized In fact this is the origin of the quantization of the area in4d and the quantization of length in 3d

Indeed we can view pn in (945) as the quantized value of the continuousvariable p in (944) conjugate to the compact variable x Similarly therepresentation je can be considered as quantization of the continuousvariable le More precisely we can identify lif with the generator of thespin-jf representation and identify the length |lf | of the segment f withthe square root of the quadratic Casimir operator of the representationjf


Sum-over-surfaces The sum over representations in (935) has a nice in-terpretation as a sum-over-surfaces Consider a triangulation Δ and aspecific assignment of representations jf to the faces of Δlowast Consider anedge e that bounds the three faces f f prime f primeprime The intertwiner on e is non-vanishing (and therefore the amplitude of the spinfoam is nonzero) onlyif the three representations jf jf prime jf primeprime satisfy the ClebschndashGordan rela-tions (A10)ndash(A11) Assuming this is the case associate 2jf elementaryparallel surfaces with each face f Join these elementary surfaces acrosseach edge The constraint (A10)ndash(A11) is precisely the condition underwhich the surfaces can be joined and there is only one way of joiningthem across each edge (There are a b and c surfaces crossing over fromjf to jf prime from jf prime jf primeprime and from jf to jf primeprime respectively and

2 jf = a + c 2 jf prime = a + b 2 jf primeprime = b + c) (946)

In this way we obtain surfaces without boundaries that wrap around thetwo-skeleton of Δlowast Each such surface carries a spin = 12

The sum over representations in (935) can therefore be viewed as asum over all the ways of wrapping these spin-12 surfaces around thetwo-skeleton of Δlowast The coloring jf of the face f is the total spin on theface that is half the number of spin-12 surfaces passing on f

Divergences bubbles The PonzanondashRegge model suffers from infrared di-vergences These have a peculiar structure which is reproduced in allspinfoam models

In the Feynman diagrams of ordinary QFT divergences are associatedwith loops that is closed curves within the Feynman diagram In theabsence of loops divergences do not appear because momentum conser-vation at the vertices constrains the value of the momenta on the internalpropagators In a spinfoam model the role of the internal momenta (in-tegrated over) is played by the representations (summed over) These areconstrained at edges by the ClebschndashGordan conditions at the edge whichplay the role of momentum conservation Accordingly divergences are notassociated with loops as in ordinary Feynman diagrams but rather withldquobubblesrdquo A bubble is a collection of faces f in Δlowast that form a closedtwo-surface If we increase each of the representations associated with thefaces forming the bubble by the same amount j then the relations (A10)ndash(A11) remain satisfied because if jf jf prime jf primeprime satisfy (A10)ndash(A11) so dojf + j jf prime + j jf primeprime

The sum-over-surfaces picture described in the previous section givesus a clear understanding of the structure of the resulting divergences wecan always add an arbitrary number of spin-12 surfaces wrapped aroundthe bubble

93 Models 339

The minimal bubble configuration is the ldquoelementary bubblerdquo This isformed by four triangular faces connected to each other as in a tetrahe-dron This elementary bubble appears if four vertices are connected toone another in Δlowast and equivalently if four tetrahedra are connectedto one another in Δ This configuration can be obtained for instance bydecomposing a single tetrahedron into four tetrahedra by picking an inte-rior point and connecting it to the four vertices Within the representation(929) this configuration gives the contribution

Abubble =int

dg1 dg6 δ(g1g2gminus16 )δ(g3g4g6)δ(g4g1g

minus15 )δ(g2g3g5) (947)

to Z (see (934)) Integration is immediate giving the divergent expression

Abubble = δ(0) (948)

The same result can be obtained in the sum over representations Forinstance assume the spins of all faces connected to the elementary bubblevanish Then

Abubble =sum

j

(dim(j))4 (6j(j j j 0 0 0))4 (949)

From the definition (933)

6j(j j j 0 0 0) = vα1α2 vα1α3 vα2α3 (950)

The normalized intertwiner vα1α2 is vα1α2 = δα1α2(dim(j))12 giving

6j(j j j 0 0 0) = dim(j)minus32 dim(j) = dim(j)minus12 (951)

Inserting this in (949) yields

Abubble =sum

j

(dim(j))2 (952)

which is equal to (948) as is clear from (930)In the Regge triangulation picture this divergence corresponds to the

case in which there is a point of the Regge lattice connected to the restof the lattice by four segments We can then make the lengths of thesefour segments arbitrarily large Geometrically this describes a long andnarrow ldquospikerdquo emerging from the 3d (discretized) manifold

Notice that this is an infrared divergence since it regards large jf snamely large lengths It is not related to ultraviolet divergences absentin the theory Ponzano and Regge have developed a renormalization pro-cedure to divide away this divergence


TuraevndashViro An appealing way to get rid of the divergence of the modelis to replace the representation theory of the group SU(2) with the repre-sentation theory of the quantum group SU(2)q with q chosen to be a rootof unity Both the dimension and the Wigner 6j symbol are well definedfor this quantum group The irreducible representations of this quantumgroup are finite in number and therefore the sum is finite This sum iscalled the TuraevndashViro invariant Furthermore it can be argued that thedeformation of the group from SU(2) to SU(2)q corresponds simply tothe addition of a cosmological term to the classical action Namely to theaction

S[e ω] =int

εijk ei and (R[ω]jk minus λ

3ej and ek) (953)

A good discussion on this issue can be found in [231]

Triangulation independence A remarkable result by Ponzano and Reggeis triangulation independence the quantity Z defined by (935) dependson the global topology of the 3-manifold chosen but not on Δ namelynot on the way the manifold is triangulated In particular whether wechoose a minimal triangulation or a very fine triangulation the partitionfunction does not change

The proof of triangulation independence is tricky in the PonzanondashReggecase where a renormalization procedure is needed On the other handit is a clean theorem in the TuraevndashViro case where everything is finiteThe TuraevndashViro sum is defined in terms of a triangulation of a compact3-manifold but it is a well-defined 3-manifold invariant

Triangulation independence is the consequence of the fact that 3d GRis a topological theory Since the theory has no local degrees of freedomwe are not really losing degrees of freedom in the discretization Usuallydiscretization loses the short-scale degrees of freedom but there are noshort-scale degrees of freedom in this theory Hence the triangulated ver-sion of the theory has the same number of degrees of freedom as the fullfield theory and refining the triangulation does not change this number

The triangulation independence of the expression (935) is an importantmathematical property It has inspired much mathematical work But wedo not expect triangulation independence to hold for 4d GR which isnot a topological theory since it has local degrees of freedom Thereforefrom the point of view of the problem of quantum gravity triangulationindependence is a less interesting aspect of the PonzanondashRegge theory Itwill not survive the generalization that we will study later on

932 BF theory

Let us extend the above construction to four dimensions As a first stepwe do not consider GR but a much simpler 4d theory called BF theory

93 Models 341

which is topological and is a simple extension to 4d of the topological3d GR Consider BF theory for the group SO(4) This is defined by twofields a two-form BIJ with values in the Lie algebra of SO(4) and anSO(4) connection ωIJ The action is a direct generalization of (925)

S[Bω] =intBIJ and F IJ [ω] (954)

where F is the curvature two-form of ω I use here the notation F insteadof R as this is standard in this context (and is the origin of the nameldquoBFrdquo of theory)

We can discretize this theory and define a path integral following thevery same steps we took for 3d GR We obtain precisely equation (929)again and from this equation (931) with the sole difference that the sumis over irreducible representations of SO(4) and that the two-complex isthe two-skeleton of the dual of a four-dimensional triangulation

Again it is not difficult to perform the integrations over the group Butnow every edge bounds four faces not three We have then instead of(932) the integral


j3(U)γγprime Rj4(U)δδprime =

sum

i

vαβγδi viαprimeβprimeγprimeδprime (955)

Here the index i labels the orthonormal basis vαβγδi in the space of the in-tertwiners between the representations of spin j1 j2 j3 j4 We have there-fore a sum over intertwiners for each edge in addition to the sums overrepresentations for each face At each vertex we have now ten represen-tations (because the vertex bounds ten faces) and five intertwiners Thesedefine the function

15j equiv A(j1 j10 i1 i5)

equivsum

α1α10

vα1α6α9α5i1

vα2α7α10α1i2

vα3α7α8α2i3

vα4α9α7α3i4

vα5α10α8α4i5

(956)

where the indices an are in the representation jn The pattern of thecontraction of the indices reproduces the structure of (the one-skeletonof ) a four-simplex

131313

vi1

vi2

vi3

vi5

vi4

α1

α2

α3

α4

α5

α6

α8

α9 α7

α10

(957)


The function (956) is denoted 15j The name comes from the factthat if the group is SU(2) the intertwiners can be labeled with the repre-sentation of the internal virtual link hence this function depends on 15spins

Combining everything we obtain the following form for the partitionfunction of 4d BF theory

ZTOCY =sum

jf ie

prod

f

dim(jf )prod

v

15jv (958)

which we can write as

ZTOCY =sum

jf ie

prod

f

dim(jf )prod

v

1313

i1

i2

i3

i5

i4

j1

j2

j3

j4

j5

j6j8j9 j7

j10(959)

This expression is called the TOCY model (from Turaev Ooguri Craneand Yetter) or the Ooguri model It has the general form (921) with thefollowing choices

bull The set of two-complexes summed over is formed by a single two-complex This is chosen to be the two-skeleton of the dual of a 4dtriangulation

bull The representations summed over are the unitary irreducibles ofSO(4)

bull The vertex amplitude is Av = 15jThese choices define the quantization of 4d BF theory

Divergences and CranendashYetter model As for the PonzanondashRegge modelthe sum (958) suffers from infrared divergences The typical divergence isagain associated with a ldquobubblerdquo The elementary bubble is now formedby five vertices of Δlowast connected to each other In Δ this corresponds tofive four-simplices connected to each other Namely the configuration ob-tained by subdividing a single four-simplex into five four-simplices addinga single point inside and connecting it to the vertices We can computethe degree of this divergence as we did for the PonzanondashRegge modelstarting from the expression (929) In the present case the pattern ofthe integration variables and the delta functions are given by (957) Thisgives

Abubble =int

dg1 dg10 δ(g1g2gminus16 )δ(g2g3g

minus17 )δ(g3g4g

minus18 )δ(g4g5g

minus19 )

times δ(g5g1gminus110 )δ(g1g7g8)δ(g2g8g9)δ(g3g9g10)δ(g4g10g7)δ(g5g6g8)

(960)

93 Models 343

Integration is immediate giving the divergent expression

Abubble = δ4(0) (961)

The divergences can be cured by passing to the quantum group SO(4)qThe definition of the quantum 15j symbol requires care but can begiven The resulting model is finite and triangulation independent It iscalled the CranendashYetter model Its classical limit can be shown to berelated to BF theory plus a cosmological term

933 The spinfoamGFT duality

There is a surprising duality between the PonzanondashRegge and TOCYmodels on the one hand and certain peculiar QFTs defined over a group(Group Field Theory or GFT) on the other This duality will play animportant role in what follows I illustrate it here in the 4d case

Consider a real field φ(g1 g2 g3 g4) over the cartesian product of fourcopies of G = SO(4) Require that φ is symmetric and SO(4) invariantin the sense

φ(g1 g2 g3 g4) = φ(g1g g2g g3g g4g) (forall g isin SO(4)) (962)

Consider the QFT defined by the action

S[φ] =12

int 4prod

i=1

dgi φ2(g1 g2 g3 g4)

+λ

5

int 10prod

i=1

dgi φ(g1 g2 g3 g4)φ(g4 g5 g6 g7)φ(g7 g3 g8 g9)

timesφ(g9 g6 g2 g10)φ(g10 g8 g5 g1) (963)

The potential (fifth-order) term has the structure of a 4-simplex if werepresent each of the five fields in the product as a node with 4 legs ndashone for each gi ndash and connect pairs of legs corresponding to the sameargument we obtain (the one-skeleton of) a 4-simplex see Figure 96

The remarkable fact about this field theory is the following The Feyn-man expansion of the partition function of the GFT

Z =int

Dφ eminusS[φ] (964)

turns out to be given by a sum over Feynman graphs

Z =sum

Γ

λv[Γ]

sym[Γ]Z[Γ] (965)

where the amplitude of a Feynman graph is

Z[Γ] =sum

jf ie

prod

f

dim(jf )prod

v

15jv (966)


6

g

g

g10 8

9

g2g3

1

5

7

f

g

f

g2

g3

4g

4g

1

g g

f

f

g

f f

f

g

Fig 96 The structure of the kinetic and potential terms in the action

Here Γ is a Feynman graph v[Γ] the number of its vertices and sym[Γ]its symmetry factor The Feynman graphs Γ of the theory have a naturaladditional structure as two-complexes The Feynman integrals over mo-menta are discrete sums (because the space on which the QFT is definedis discrete) over SO(4) representations jf and over intertwiners ie associ-ated with faces and edges of the two-complex Furthermore for each giventwo-complex Γ the Feynman sum over momenta is precisely the TOCYmodel defined on that two-complex Indeed the right-hand side of (966)is equal to the right-hand side of (958) That is

Z[Γ] = ZTOCY (967)

The proof of these results is a straightforward application of perturbativeexpansion methods in QFT and the use of the PeterndashWeyl theorem thatallows us to mode-expand functions on a group in terms of a basis givenby the unitary irreducible representations of the group This is done insome detail below

Mode expansion First expand the field φ(g1 g2 g3 g4) into modes and rewrite theaction in terms of these modes (in ldquomomentum spacerdquo) Consider a square integrablefunction φ(g) over SO(4) The PeterndashWeyl theorem tells us that we can expand this

function in the matrix elements R(j)αβ(g) of the unitary irreducible representations j

φ(g) =sum

j

φjαβ R

(j)αβ(g) (968)

The indices α β label basis vectors in the corresponding representation space Accord-ingly the field can be expanded into modes as

φ(g1 g4) =sum

j1j4

φj1j4α1β1α4β4

R(j1)α1β1

(g1) R(j4)α4β4

(g4) (969)

Using the invariance (962) under the left group action we can write

φ(g1 g4) =

int

SO(4)

dg φ(gg1 gg4) (970)

93 Models 345

Substituting here the mode expansion (969) and using the expression (955) for theintegral of the product of four group elements we can write

φ(g1 g4) =sum

j1j4

φj1j4α1α4i

R(j1)α1β1

(g1) R(j4)α4β4

(g4) viβ1β4 (971)

where we have definedφj1j4α1α4i

= φj1j4α1β1α4β4

viβ1β4 (972)

We use the quantities φj1j4α1α4i

as the Fourier components of the field Written in termsof these the kinetic term of the action reads

1

2

int 4prod

i=1

dgi φ2(g1 g4) =

1

2

sum

j1j4

sum

i

φj1j4 iφj1j4 i (973)

The interaction term becomes

λ

5

int 10prod

i=1

dgiφ(g1 g2 g3 g4)φ(g4 g5 g6 g7)φ(g7 g3 g8 g9)

timesφ(g9 g6 g2 g10)φ(g10 g8 g5 g1)

=λ

5

sum

j1j10

sum

i1i5

φj1j2j3j4 i1φj4j5j6j7 i2φj7j3j8j9 i3φj9j6j2j10 i4φj10j8j5j1 i5

timesA(j1 j10 i1 i5) (974)

where A is given in (956)

Feynman graphs The partition function is given by the integral over modes

Z =

int [Dφj1j4 i

]eminusS[φA] (975)

We expand Z in powers of λ The gaussian integrals are easily computed giving thepropagator

P j1j4 i jprime1jprime4 iprime equiv 〈φj1j4 i φjprime1j

prime4 iprime〉 =

1

4

sum

σ

δj1j

primeσ(1) δ

j4jprimeσ(4) δiiprime (976)

where σ are the permutations of 1 2 3 4 There is a single vertex of order five whichis

〈φj1j2j3j4 i1 φj10j8j5j1 i5〉 = λ A(j1 j10 i1 i5) (977)

The set of Feynman rules one gets is as follows First we obtain the usual overallfactor λv[Γ]sym[Γ] (see for instance [268] page 93) Second we represent each of theterms in the right-hand side of the definition (976) of the propagator by four parallelstrands as in Figure 97 carrying the indices at their ends We can represent thepropagator itself by the symmetrization of the four strands In addition edges e arelabeled by a representation je

The Feynman graphs we get are all possible ldquo4-strandrdquo five-valent graphs wherea ldquo4-strand graphrdquo is a graph whose edges are collections of four strands and whosevertices are those shown in Figure 98 Each strand of the propagator can be connectedto a single strand in each of the five ldquoopeningsrdquo of the vertex Orientations in thevertex and in the propagators should match (this can always be achieved by changinga representation to its conjugate) Each strand of the 4-strand graph goes throughseveral vertices and several propagators and then closes forming a cycle A particular


Fig 97 The propagator can be represented by a collection of four strands eachcarrying a representation

Fig 98 The structure of the vertex generated by the Feynman expansion

strand can go through a particular edge of the 4-strand graph more than once Cyclesget labeled by the simple representations of the indices For each graph the abstractset formed by the vertices the edges and the cycles forms a two-complex in whichthe faces are the cycles The labeling of the cycles by simple representations of SO(4)determines a coloring of the faces by spins Thus we obtain a colored two-simplexnamely a spinfoam

Edges e are labeled by an intertwiner with index ie Vertices v contribute a factorλ times A which depends on the ten simple representations labeling the cycles thatgo through the vertex and on the five intertwiners basis elements in Kie

labeling theedges that meet at v The weight of two-complex Γ is then given by (966)3

3The following remarks may be useful for a reader who wants to compare the formulasgiven with others in the literature For each given permutation namely for each two-complex and for each edge e I have chosen a fixed orthonormal basis in the spaceof the intertwiners associated with the edge e Alternatively one can choose a basisassociated with a decomposition of the four faces adjacent to e in two couples If wedo so the propagator (976) contains also a matrix of change of basis It reads

P j1j4 i jprime1jprime4 iprime equiv 〈φj1j4 Λ φjprime1j

prime4 iprime〉 =

1

4

sum

σ

δj1j

primeσ(1) δ

j4jprimeσ(4) M j1j4

σiiprime

(978)

where the matrix Mσ is given by a 6j symbol Each edge contracts two vertices sayv and vprime and contributes a matrix Mσ This is the matrix of the change of basis fromthe intertwiner basis used at v and that used at vprime Since here I have fixed a basis ofintertwiners for every e once and for all for each fixed two-complex the matrix Mσ isautomatically included in the vertex amplitude and the propagator is the identity

93 Models 347

Transition amplitudes Next consider SO(4)-invariant transition ampli-tudes in the GFT That is let f [φ] be an SO(4)-invariant polynomialfunctional of the field and consider the amplitude

W (f) =int

Dφ f [φ] eminusS[φ] (979)

and its expansion in Feynman graphs

W (f) =sum

Γ

λv[Γ]

sym[Γ]Zf [Γ] (980)

It is simplest to construct all SO(4)-invariant polynomial functionals ofthe field in momentum space namely as functions of the Fourier modesφj1j4α1α4i

defined in (969)ndash(971) To obtain an SO(4) scalar we must con-tract the indices αn We start with n field variables φj1j4

α1α4i and contract

the indices pairwise in all possible manners The resulting functional isdetermined by a four-valent graph Γ giving the pattern of the indices con-traction colored with representations jl on the links and the intertwinersin on the nodes The set of data s = (Γ jl in) forms precisely a spinnetwork In other words the SO(4)-invariant observables of the GFT arelabeled by spin networks

Writing n1 n4 to indicate four links adjacent to the node n wehave

fs[φ] =prod

n

φjn1 jn4αn1 αn4 in

prod

l

δl1l2 (981)

where ni = l1 (or ni = l2) if the ith link of the node n is the outgoing (oringoing) link l

For instance the spin network s = (Γ j1 j4 i1i2) on a graph with two nodesconnected by four links determines the function of the field

fs[φ] =sum

α1α4

φj1j4α1α4i1

φj1j4α1α4i2

(982)

I leave to the reader the simple exercise to show that expressions of this kind correspondto coordinate space expressions such as

fs[φ] =

int prod

l

dglprod

n

φ(gn1 gn4)fs(gni) (983)

where the spin network function is

fs(gni) =prod

n

vαn1 αn4in

prod

l

Rjl(gl)αl1αl2 (984)


All transition amplitudes of the GFT can therefore be expressed interms of the spin network amplitudes

W (s) =int

Dφ fs[φ] eminusS[φ] (985)

Consider the Feynman expansion of these As usual in feynmanology theexpectation value of a polynomial of order n in the fields has n externallegs In the Feynman expansion of the GFT we have in addition toconsider the faces I leave to the reader the simple and instructive exerciseto show that these turn out to be bounded precisely by the links of thespin network In other words the Feynman expansion of W (s) is given by

W (s) =sum

partΓ=s

λv[Γ]

sym[Γ]Zs[Γ] (986)

where the sum is over all two-complexes bounded by s and the amplitudeof the Feynman graph is

Zs[Γ] =sum

jf ie

prod

f

dim(jf )prod

v

15jv (987)

The coloring on the external nodes and links is determined by s and notsummed over

Expressing this the other way around the spinfoam sum at a fixed spinnetwork boundary s is determined by the GFT expectation value (985)

As far as the TOCY model is concerned the duality I have just illus-trated is not particularly useful BF theory has a large invariance groupthat implies that the theory is topological This implies that the corre-sponding spinfoam model is triangulation invariant up to a divergentfactor Therefore the GFT amplitudes are given by divergent sums ofequal terms On the other hand the spinfoamGFT duality will play acrucial role in the context of the BC models

934 BC models

It is time to begin the return towards 4d GR There is a strict relationbetween SO(4) BF theory and euclidean GR If we replace BIJ in (954)by

BIJ = εIJKL eK and eL (988)

we get precisely the GR action We can therefore identify the B field ofBF theory with the gravitational field e and e The constraint (988) on Bsometimes called the Plebanski constraint transforms BF theory into GRCan we implement the constraint (988) directly in the quantum theory

93 Models 349

An immediate consequence of (988) is

εIJKL BIJBKL = 0 (989)

In 3d the continuous variable lif can be identified with the generators ofthe representation jf In 4d it is the variable BIJ

f that we can identifywith a generator of the SO(4) representation If we do so (989) becomessimply a restriction on the representations summed over

Recall that the Lie algebra of SO(4) is sim su(2)oplussu(2) The irreduciblerepresentations of SO(4) are therefore labeled by couples of representa-tions of SU(2) namely by two spins j = (j+ jminus) If BIJ is the generatorof SO(4) the generators of the two SU(2) groups are

Biplusmn = P i

plusmnIJBIJ (990)

where the projectors P iplusmnIJ are the euclidean analogs of the projectors

defined in (217) That is

P iplusmnjk =

12εijk P i

plusmn0j = minusP iplusmnj0 = plusmn1

2δij (991)

SO(4) has two Casimirs the scalar Casimir

C = BIJBIJ = |B|2 (992)

and the pseudo-scalar Casimir

C = εIJKL BIJBKL (993)

This last one is precisely the quantity which is constrained to zero by(989) The SO(4) representations in which the pseudo-scalar Casimir(989) vanishes are called ldquosimplerdquo or ldquobalancedrdquo The value of C in therepresentation (j+ jminus) is easy to compute because

εIJKL BIJBKL = Bi+B+i minusBi

minusBminusi = j+(j+ + 1) minus jminus(jminus + 1) (994)

From (989) and (994) we have j+ = jminus The representations that satisfythis constraint namely those of the kind (j+ jminus) = (j j) are the simplerepresentations they are labeled by a single spin j Some mathematicalfacts about representation theory of SO(4) and about simple representa-tions are included in Appendix A3

This suggests that quantum GR can be obtained by restricting thesum over representations in (958) to the simple representations Thisprocedure defines a class of models denoted the BC models (from JohnBarrett and Louis Crane who introduced the use of simple representationsin the spinfoam formalism)


Relation with loop gravity In the discretization of BF theory we havediscretized the B field which is a two-form by assigning a variable Bf

to each triangle f of the triangulation Bf can be taken to be the surfaceintegral of B on f We can discretize the gravitational field e which is aone-form by assigning a variable es to each segment s of the triangulationes can be taken to be the line integral of e along the segment s Equation(988) then relates the variable Bf on a triangle with the variables es ontwo sides of the triangle The scalar Casimir (992) can be expressed interms of the gravitational field using (988) obtaining

C = |e and e|2 (995)

which is the square of the area of the triangle Hence the Casimir of therepresentation jf gives the area of the triangle f The spin jf can thereforebe interpreted as the quantum number of the area of the triangle f

Consider the case in which the triangulated manifold has a boundaryand the triangle f belongs to the boundary In the two-complex picture fis a face dual to the f triangle and jf is associated with this face The facef cuts the boundary along a link which is one of the links of the boundaryspin network The color of the link is jf This link intersects once and onlyonce the triangle f Hence we conclude that the representation associatedwith the link that intersects the triangle f is the quantum number thatdetermines the area of the triangle f But this is precisely the result thatwe have obtained in hamiltonian theory in Chapter 6

The boundary states of a BC spinfoam model is a spin network whoselinks carry quanta of area labeled by a spin j This is precisely the struc-ture of the states of hamiltonian loop quantum gravity Spinfoam modelsand hamiltonian LQG ldquotalkrdquo very nicely to each other

The other constraints Equation (989) does not imply (988) On theother hand the system formed by (989) and the two equations

εIJKL BIJμνB

KLνρ = 0 (996a)

4εIJKL BIJμνB

KLρσ

εμνρσεIJKL BIJμνB

KLρσ

= εμνρσ (996b)

(no sum over repeated spacetime indices in the first equation) does Itfollows that if we consider BF theory plus the additional constraint (996)the resulting theory has precisely all the solutions of GR

Two remarks First GR has far more solutions than BF theory which is topologi-cal The fact that adding a constraint on B increases the number of solutions shouldnot surprise us In BF theory B plays the role of Lagrange multiplier enforcing thevanishing of the curvature By constraining B in the action we reduce the number of

93 Models 351

independent components of the Lagrange multiplier Hence we get less constraints onthe curvature Hence more solutions If you jail some guards more thieves go free

Second the system (996) has other classes of solutions beside (988) In particular

BIJ = minusεIJKL eK and eL and BIJ = plusmneI and eJ (997)

The first simply redefines orientation The others give a topological term in the actionthat has no effect in the classical equations of motion I refer the reader to the literaturefor a full discussion of this point

The constraint (996a) can be implemented in the quantum theory byidentifying the couple of variables BIJ

μνBKLνρ that share an index with gen-

erators of the representations associated with faces that share an edgeIn turn for the constraint (996b) we identify the two BKL

ρσ BKLνρ variables

without common indices with generators of the representations associatedto opposite faces of a tetrahedron

There are different ways of discretizing these constraints that have beenconsidered in the literature yielding different BC models There are alsodifferent possible choices of the face and edge amplitudes Af and Ae thathave been considered in the literature The situation is still unclear as towhich variants correspond to discretized GR I shall not enter into thedetails of motivations of the different choices here (see [19] for a detaileddiscussion) Rather I simply illustrate some of the models referring tothe literature for their motivation

BCA The simplest of the BC models denoted BCA model (BarrettndashCrane model version A) is simply obtained by choosing Ae = Af = 1 Itis given by

ZBCA =sum

simple jf

sum

ie

prod

v

131313

i1

i2

i3

i5

i4

j1

j2

j3

j4

j5

j6j8j9

j7

j10

(998)

BCB A second model denoted BCB (BarrettndashCrane model version B)assumes that the intertwiners are constrained by (996) to the form

i(aaprime)(bbprime)(ccprime)(ddprime)BC =

sum

j

(2j + 1) vabfvfcd vaprimebprimef prime

vfprimecprimedprime (999)

where indices in an SO(4) representation are given by couples of indices inan SU(2) representation and the indices f and f prime are in the representationj This is called the BC intertwiner See [269] for details on the relation


between this intertwiner and (996) The BC intertwiner has the propertyof being formed by a simple virtual link in any decomposition It is theunique state with this property Using the representation (686) it is givenby

ibc =sum

j (2j + 1)

j

j (9100)

Choosing Ae = 1 we obtain the sum

ZBCB =sum

simple jf

prod

f

dim(jf )prod

v

131313

iBC

iBC

iBC

iBC

iBC

j1

j2

j3

j4

j5

j6j8j9

j7

j10

(9101)

The vertex amplitude

A(j1 j10) =

1313

iBC

iBC

iBC

iBC

iBC

j1

j2

j3

j4

j5

j6j8j9 j7

j10

(9102)

=sum

i1i5

1313

i1

i2

i3

i5

i4

j1

j2

j3

j4

j5

j6j8j9 j7

j10

1313

i1

i2

i3

i5

i4

j1

j2

j3

j4

j5

j6j8j9 j7

j10

depends on ten spins and is called a 10j symbol Thus we write (9101)as

ZBCB =sum

simple jf

prod

f

dim(jf )prod

v

10j (9103)

Therefore the BCB model has the general form (921) with thefollowing choices

93 Models 353


bull The representations summed over are the simple unitary irreduciblerepresentations of SO(4) All intertwiners are fixed to be iBC

bull The vertex amplitude is Av = 10j

These choices define a tentative quantization of 4d riemannian GR on afixed two-complex namely with a cut-off at the high-frequency modesClearly a fixed two-complex can accommodate only a finite number ofdegrees of freedom and cannot capture all the degrees of freedom of thetheory which are infinite

BCC A variant of the model B is of particular interest since as we shallsee it is perturbatively finite This is obtained by adding an edge ampli-tude to the BCB model

ZBCC =sum

simple jf

prod

f

dim(jf )prod

e

Ae(je1 je4)prod

v

10j (9104)

where Ae is a function of the four representations je1 je4 associatedto the four faces bounded by the edge e and is defined as follows LetHj1j4 be the tensor product of the four representations j1 j4 andH0

j1j4its invariant subspace Then Ae is the ratio of the dimensions of

these spaces

Ae(j1 j4) =dimH0

j1j4

dimHj1j4

(9105)

As we shall see below this amplitude emerges naturally in the group fieldtheory (GFT) context

At present it is not clear which of these variants or others is the mostphysically interesting one Interpreting the choice as a choice of sum-over-paths measure and imposing diff invariance Bojowald and Perez haveobtained indications in favor of certain models [270] In [271] the differentstatistical properties of these models have been analyzed numericallyhowever it is not yet clear which are the ldquocorrectrdquo statistical propertiesto be expected

Geometrical interpretation of the Plebanski constraints The constraintson the representations and intertwiners that define the BC models canbe given a geometrical interpretation In fact they were first obtainedfrom an independent set of considerations based on this geometrical in-terpretation Consider a tetrahedron embedded in R4 Denote vectors in


R4 as v = (vI) I = 1 4 Label the four vertices of the tetrahedron asvi i = 1 4 A vector

v(ij) = vj minus vi (9106)

describes the edge (ij) of the tetrahedron The triangle (ijk) can be de-scribed by the ldquobivectorrdquo

vIJ(ijk) = vI(ij) vJ(jk) minus vJ(ij) v

I(jk) (9107)

often written asv(ijk) = v(ij) and v(jk) (9108)

For instance the triangle determined by the three vertices (123) is de-scribed by the bivector

vIJ(123) = vI1 vJ2 minus vI2 vJ1 + vI2 vJ3 minus vI3 vJ2 + vI3 vJ1 minus vI1 vJ3 =sum

i

εijkvIj vJk

(9109)

obtained by inserting (9106) in (9107) The area of the triangle (ijk) isthe norm of the bivector

AIJ(ijk) = vIJ(ijk)v(ijk)IJ (9110)

From the definition

εIJKL vIJ(ijk)vKL(ijk) = 0 (9111)

εIJKL vIJ(ijk)vKL(ijl) = 0 (9112)

Furthermore consider a four-simplex embedded in R4 with vertices viwhere i = 1 5 This defines the four tetrahedra (ijkl) and the tentriangles (ijk) Then for two triangles sharing a vertex i

εIJKL vIJ(ijk)vKL(ilm) =

sum

i

εijklmεIJKL vIj vJk v

Kl vLm (9113)

which is independent of i Hence we can write

εIJKL vIJ(ijk)vKL(ilm)

sumi ε

ijklm εIJKL vIj vJk v

Kl vLm

= εjklm (9114)

Notice the remarkable similarity of the bivector equations (9111)(9112) and (9114) with the Plebanski constraints (989)ndash(996) Thebivector equations can be interpreted as a discretization of the Plebanskiconstraints Expressed the other way around the Plebanski constraintscan be interpreted as the requirement that the B field of the BF theoryis an infinitesimal area element of elementary triangles in spacetime

93 Models 355

Quantum tetrahedron The historical path to the BC models has been the onedescribed above I remember long hours of discussion with Louis Crane searching un-successfully for a way to implement the Plebanski constraint (989) within his TOCYmodel But in the seminal work [269] which brilliantly solves the problem in terms ofsimple representations no reference is made to the Plebanski constraint and the rela-tion between BF and GR The paper is based on a description of the intrinsic metricdegrees of freedom of a single tetrahedron embedded in R4 and a ldquoquantizationrdquo ofthese degrees of freedom as I now describe

Consider a single tetrahedron embedded in R4 View it as a physical sys-tem whose dynamical variables are given by its geometry We can expectthat the quantum properties of this system are described by a quantumstate space H and dynamical variables be represented by operators in thisstate space The geometry of the tetrahedron can be described in terms ofthe bivectors vIJ(ijk) defined in the previous section Thus bivectors will berepresented by operators on H This construction defines a sort of ldquoquan-tum tetrahedronrdquo Since SO(4) acts naturally on the bivectors we expectthat H carries a representation of SO(4) Since classical bivectors trans-form in the adjoint representation we expect the quantum operators todo the same It follows that bivector operators vIJ(ijk) are the infinitesimalgenerators of a representation j of SO(4) in H The quadratic expression(9110) giving the area of the triangle is precisely one of the two Casimirsof j Hence H will carry a representation jijk such that its Casimir isthe area of the triangle (ijk) The other Casimir of SO(4) is given by(9111) constrained to vanish Hence H will contain only representationsin which this Casimir vanishes These are the simple representations Thesum (9101) can then be constructed as a sum-over-states in the Hilbertspace where the bivector operators live

This method of arriving at the BC models has the shortcoming of hidingits relation with GR as well as with conventional quantization proceduresOn the other hand it has the virtue of opening an entirely new interestingperspective on quantum spacetime The convergence of different ways ofthinking of a model which shed light on each other is always valuable

In fact the idea of the quantum tetrahedron as an elementary systemcan perhaps be taken seriously on physical grounds Ordinary QFT on abackground has two natural physical interpretations The two correspondto two different choices of families of observables For instance in free elec-tromagnetism we can measure the electric field or the magnetic field andinterpret the theory as a theory for a continuous field Alternatively wecan measure energies and momenta and interpret the theory as describ-ing particles moving in spacetime the photons There is no contradictionof course between the two descriptions for the same reason that there isno contradiction between the continuity of the elongation of a harmonicoscillator and the discreteness of its energy


Similarly we can construct the QFT starting from the quantization ofthe classical field theory and deriving the existence of the particles Or wecan construct the QFT starting from the particles define the quantumtheory of a single particle then the ldquomany-particlerdquo quantum theory ofan arbitrary number of particles and so on The two roads yield the sametheory as is well known In an interacting QFT nontrivial dynamics canbe expressed by simple interaction vertices between the particle states

In GR loop quantization shows that space has a granular structureat short scale Space can be thought of as made up of individual quantaof space (which can be connected to each other) These quanta of spaceare like the particles eigenstates of certain measurable quantities It isthen not unreasonable to think that we can reinterpret quantum GR asa ldquomany-particlerdquo theory built up from a quantum theory of a singlequantum of space The mathematics of the ldquoquantum tetrahedronrdquo canperhaps be seen as a first step in this direction If we take this pointof view then dynamics can be expressed by simple interaction verticesbetween these ldquoparticlerdquo states For instance an elementary vertex suchas the one in Figure 93 can be interpreted as one quantum of space(connected to three other quanta not represented) decaying into threequanta of space (connected to each other and to the three other quantanot represented) and so on Spacetime is then an history of interactionsof a variable number of quanta of space

935 Group field theory

The remaining step to arrive at a model with some chance of describingquantum GR is to implement a sum over two-complexes so that theinfinite number of degrees of freedom of the theory could be captured

Notice that in order to capture all degrees of freedom we do not have the option ofrefining the triangulation (or the two-complex) as one does in lattice QCD The reasonis that there is in fact nothing to refine no parameter such as the lattice spacing oflattice QCD which can be set to zero

In fact the cut-off introduced by the BarrettndashCrane models is not an ultravioletcut-off The theory does not have an ultraviolet sector because there are no degreesof freedom beyond Planck scale Rather it is a sort of infrared cut-off in the sensethat a fixed triangulation cannot capture configurations that can be written on a largertriangulation (a triangulation with more n-simplices) The sum includes arbitrary largegeometries because it includes arbitrary high jf (not in the quantum group case) Buton a fixed Δ a large geometry can be represented only by large jf and not by smalljf over a larger triangulation Clearly this restriction reduces dramatically the class ofcontinuous fields that the spinfoam can approximate

We could think of defining the model in the limit of large Δ This is certainly aninteresting direction to explore On the other hand in summing over colorings of a largeΔ we have to include configurations in which the representations are trivial except ona subset Δprime of Δ The amplitude of this configuration can be viewed as associated withΔprime rather than Δ Hence we naturally fall back to a sum-over-triangulations

93 Models 357

For fixed boundary conditions yielding a classical geometry of volume Nl4P it may bereasonable to assume that triangulations with a number of four-simplices much largerthan N would not contribute much Hence the expansion in the size of the triangulationmight be of physical interest

How do we sum over two-complexes The problem is to select a classof two-complexes to sum over and to fix the relative weight Now theduality that I have illustrated above in Section 933 provides precisely aprescription for summing over two-complexes It is therefore natural totake a dual formulation of the BC models as a natural ansatz for thecomplete sum over two-complexes But is there a dual formulation of theBC models or is duality a feature of the much simpler topological BFmodel

Remarkably a dual formulation of the BC models exists BC modelsare obtained from the TOCY models by restricting representations to thesimple ones This restriction implements the constraints that transformBF theory into GR In the dual picture the sum over representations isobtained as an expansion of the field over the group in modes A genericfield can be expanded in a sum over all unitary irreducible representationsHow can we pick a field whose expansion contains only simple represen-tations The answer turns out to be easy

Pick a fixed SO(3) subgroup H of SO(4) Then the following holds Afield φ(g) on SO(4) is invariant under the action of H namely satisfies

φ(g) = φ(gh) forallh isin H (9115)

if and only if its mode expansion contains only simple irreducible repre-sentations This is an elementary result described in Appendix A3

Consider the field theory defined in Section 933 It is useful to slightlysimplify this notation First write the action in the shorthand notation

S[φ] =12

intφ2 +

λ

5

intφ5 (9116)

Second instead of demanding the field to satisfy (962) we can take anarbitrary field φ not necessarily satisfying (963) and use the projectionoperator PG defined by

PGφ(g1 g2 g3 g4) =int

SO(4)dg φ(g1g g2g g3g g4g) (9117)

We can also define the projector GP acting on the left

GPφ(g1 g2 g3 g4) =int

SO(4)dg φ(gg1 gg2 gg3 gg4) (9118)

Define now the projector PH on the simple representations

PHφ(g1 g2 g3 g4) =int

H4dh1 dh4 φ(g1h1 g2h2 g3h3 g4h4) (9119)


GFTTOCY The action

S[φ] =12

int(PGφ)2 +

λ

5

int(PGφ)5 (9120)

for a generic field is equivalent to the action (963) and yields the TOCYmodel as discussed in Section 933 Now by simply inserting the projec-tor PH into this action we obtain the following surprising results

GFTA Consider the action

SA [φ] =12

int(GPPHφ)2 +

λ

5

int(GPPHφ)5 (9121)

The Feynman expansion of the partition function of this theory gives

ZA =int

Dφ eminusS[φ] =sum

Γ

λv[Γ]

sym[Γ]ZA [Γ] (9122)

The amplitude of a Feynman graph turns out to be precisely the partitionfunction (998) of the BCA model where the model is over the two-complex determined by the Feynman graph That is

ZA [Γ] =sum

simple jf

prod

v

15j = ZBCA (9123)

GFTB The action

SB [φ] =12

int(PGPHφ)2 +

λ

5

int(PGPHφ)5 (9124)

gives the partition function (9101) of the BCB model

ZB [Γ] =sum

simple jf

prod

f

dim(jf )prod

v

10j = ZBCB (9125)

GFTC The action

SC [φ] =12

int(PGφ)2 +

λ

5

int(PHPGφ)5 (9126)

yields the partition function (9104) of the BCC model

ZC [Γ] =sum

simple jf

prod

f

dim(jf )prod

e

Ae(je1 je4)prod

v

10j

= ZBCC (9127)

The derivation of these relations is a rather straightforward applicationof the mode expansion described in Section 933 I leave it as a good

93 Models 359

exercise for the reader It can be found in the original papers quoted atthe end of the chapter

Now in the TOCY case the sum over Feynman graphs is trivial di-vergent and without physical motivation It is trivial in the sense that allterms are equal due to triangulation invariance It is divergent becausewe sum an infinite number of equal terms It has no physical motiva-tion because all the degrees of freedom of the classical theory are alreadycaptured by a finite triangulation

In the case of the BC models on the other hand a choice of a fixedtwo-complex reduces the number of degrees of freedom of the theoryTherefore a sum over two-complexes is necessary if we hope to definequantum GR and the Feynman expansion provides precisely such a sumSince the BC models are not triangulation invariant the sum is not triv-ial each two-complex contributes in a different manner to the sum Thesum over two-complexes defined by the GFT defines in these cases a newspinfoam model where the number of degrees of freedom are not cut offWe denote these as GFT spinfoam models In particular we denote thesum defined in (9122) as the group field theory version A or GFTA andthe corresponding ones in cases B and C as the GFTB and GFTC

What about finiteness Remarkably the GFTC appears to be finiteat all orders in λ The proof has been completed up to certain degeneratetwo-complexes which however are likely not to spoil the result Thereforethe GFTC model or some variant of it can be taken as a tentative ansatzfor a covariant definition of transition amplitudes in euclidean quantumgravity

In particular we can consider expectation values of spin networks as in(985)

W (s) =int

Dφ fs[φ] eminus12

int(P

Gφ)2 + λ

5

int(P

GPHφ)5 (9128)

These quantities are well defined and are likely to be convergent at everyorder in λ We can tentatively interpret them as transition amplitudes ofeuclidean quantum gravity

936 Lorentzian models

The fact that expression (9128) provides a finite tentative definition ofquantum gravitational transition amplitudes is certainly exciting but theGFT models described above are all euclidean There are two possibledirections to recover the physical lorentzian theory from here

SO(3) and SO(21) lorentzian GFT One direction for defining lorentzianamplitudes is to study the lorentzian analogs of these models These canbe obtained by simply replacing SO(4) with the Lorentz group SO(3 1)Let φ(g1 g4) be a field on [SO(3 1)]4 Define the projectors PG and


PH as in (9117) and (9119) To define the projector H there are twonatural choices for the subgroup H sub SO(3 1) The first is to take it tobe a fixed H = SO(3) subgroup of SO(3 1) This is the subgroup thatkeeps a chosen timelike vector invariant The second is to take it as anH = SO(2 1) subgroup of SO(3 1) This is the subgroup that keeps achosen spacelike vector invariant Consider the action

SH [φ] =12

int(PGφ)2 +

λ

5

int(PGPHφ)5 (9129)

for the two cases The Feynman expansion of the partition function de-fines two lorentzian spinfoam sums denoted the SO(3) and the SO(2 1)lorentzian GFT respectively

The set of the representations that appear in these spinfoam sums isdetermined by the mode expansion of the field This implies that therepresentations summed over are the unitary irreducible representationsThese are infinite-dimensional for SO(3 1) which is noncompact and la-beled also by a continuous variable Therefore in the lorentzian spinfoammodels sums over internal indices are replaced by integrals and the spin-foam sum contains an integral over a continuous set of representationsStill the technology developed above extends to this case quite well Thereis an extensive literature on this to which I refer the interested readersee bibliographical notes at the end of the chapter Most of the featuresof the euclidean theory persist in the lorentzian case Most remarkablyfiniteness results have been extended to the lorentzian SO(3) GFT

Unitary representations of SO(3 1) fall naturally into two classesspacelike and timelike ones distinguished by the sign of the Casimir rep-resenting the square of the area The action S

SO(3)above gives rise to a

sum over just the timelike representations while the action SSO(21)

givesrise to a sum over both kinds of representations The kind of the represen-tation associated with a triangle f gives a spacelike or timelike characterto the triangles of the triangulation This is physically appealing but sev-eral aspects of this issue are unclear at this time For instance the sign ofthe square of the area appears to be opposite to what one would expecton the basis of the hamiltonian theory This is an intriguing open problemthat I signal to the reader

Analytic continuation The second possible direction for the constructionof the physical theory is to define the lorentzian transition amplitudesfrom the euclidean ones by analytic continuation as is done in conven-tional QFT Standard theorems relating euclidean transition amplitudesto a lorentzian QFT are grounded on Poincare invariance and do not ex-tend to gravity However this does not imply that the project of definingphysical transition amplitudes from the euclidean theory by some form of

94 Physics from spinfoams 361

analytic continuation must necessarily fail In particular analytic continu-ation in the time coordinate is likely to be completely nonappropriate in abackground-independent context but analytic continuation in a physicaltime might be viable I discuss this possibility below in Section 941

94 Physics from spinfoams

A spinfoam model can be used to compute an amplitude W (s) associatedto any boundary spin network s How can we relate these amplitudes tophysical measurements

Relation with hamiltonian theory The spinfoam formalism has formalsimilarities with lattice gauge theory The interpretation of the two for-malisms however is quite different In the case of lattice theories thediscretized action depends on a parameter the lattice spacing a Thephysical theory is recovered as a is taken to zero In this limit the dis-cretization introduced by the lattice is removed In particular we canapproximate continuous boundary fields in terms of sequences of latticediscretization In gravity on the other hand there is no lattice spacingparameter a in the discretized action Therefore there is no sense in thea rarr 0 limit The discrete structure of the spinfoams must reflect actualfeatures of the physical theory In particular boundary states are givenby spin networks not by continuous field configurations approximated bysequences of discretized fields

The hamiltonian theory of Chapter 6 provides a physical interpreta-tion for the boundary spin network s Spin networks are eigenstates ofarea and volume operators and can therefore be interpreted as describ-ing the result of measurements of the geometry of a 3d surface Suchmeasurements do not correspond to complete observables since they donot commute with the hamiltonian operator However they correspondto partial observables a notion explained in Chapter 3 Therefore theyare still described by operators in the kinematical quantum state spaceas argued in Chapter 5 In particular the discrete spectrum of these oper-ators can be interpreted as a physical prediction on the possible outcomeof a physical measurement of these observables As recalled at the be-ginning of this chapter transition amplitudes do not in general dependon classical configurations they depend on quantum eigenstates This iswhy we expect quantum gravity transition amplitudes to be functions ofspin networks and not of continuous three-geometries The fact that thespinfoam formalism yields precisely such functions of spinfoams is there-fore satisfying There is thus a strong and encouraging consistency be-tween the physical picture of nonperturbative quantum gravity providedby spinfoam theory and by hamiltonian LQG


It would be very good if we were able to translate directly between thetwo formalisms A sketch of a formal derivation of a spinfoam sum fromthe loop formalism was given at the beginning of this chapter Expressingthis the other way around it would be very interesting to reconstruct indetail the hamiltonian Hilbert space as well as kinematical and dynam-ical operators of the loop theory starting from the covariant spinfoamdefinition of the theory At present neither of these two paths is undercomplete control The first would amount to a derivation of a FeynmanndashKac formula (see for instance [272]) valid in the diffeomorphism-invariantcontext

The second would amount at an extension to the diffeomorphism-invariant context of the Wightman and OsterwalderndashSchrader reconstruc-tion theorems [273] There are two ways in which we can derive the Hilbertstate space from a spinfoam model One is to identify K with a linear clo-sure of the set of the boundary data This is the philosophy I have usedabove The other is to directly reconstruct H from the amplitudes W (s)using the GelfandndashNeimarkndashSiegal construction This approach has beendeveloped in [274] (see also [275])

At an even more naive level there are several gaps between the hamil-tonian loop theory and the spinfoam models that have been discussedso far These concern the role of the selfdual connection the role of theSO(3 1) rarr SO(3) gauge-fixing used in the hamiltonian framework thefact that in the spinfoam models so far considered there are only four-valent nodes the fact that in the GFTB and C models there is no freeboundary intertwiner variable associated with the node the eventual roleof the quantum group deformation in the hamiltonian theory [228] andothers All these aspects of the relation between the two formalisms needto be clarified before being able to cleanly translate between the two

On the one hand there is much latitude in the definition of the hamil-tonian operator as explained in Chapter 7 In general covariant methodsdeal more easily with symmetries and interaction vertices have a simpleform in the covariant picture For instance compare the complexity of thefull QED hamiltonian with the simplicity of the QED single vertex whichcompactly summarizes all hamiltonian interaction terms The spinfoamformalism could suggest the correct form of the hamiltonian operator

On the other hand hamiltonian methods are more precise and rigorousthan covariant sums-over-paths As we have seen the hamiltonian pic-ture provides a clean and well-motivated physical interpretation for theboundary spin network states as well as a general justification of the dis-creteness of spacetime Therefore the two formalisms shed light on eachother and their relation needs to be studied in detail

(Note added in the paperback edition The precise equivalence of thespinfoam and hamiltonian LQG formalisms has been proven rigorouslyby Karim Noui and Alejandro Perez See Bibliographical notes below)


941 Particlesrsquo scattering and Minkowski vacuum

Finally I sketch here a way to compute the Minkowski vacuum statefrom the spinfoam formalism following the general theory described inSection 54 This is the first step to define particle states Consider athree-sphere formed by two ldquopolarrdquo in and out regions and one ldquoequato-rialrdquo side region Let the matter + gravity field on the three-sphere besplit as ϕ = (ϕout ϕin ϕside) Fix the equatorial field ϕside to take thespecial value ϕRT defined as follows Consider a cylindrical surface ΣRT

of radius R and height T in R4 as defined above Let Σin (and Σout) be a(3d) disk located within the lower (and upper) basis of ΣRT and let Σside

be the part of ΣRT outside these disks so that ΣRT = ΣincupΣoutcupΣside LetgRT be the metric of Σside and let ϕRT = (gRT 0) be the boundary field onΣside determined by the metric being gRT and all other fields being zeroGiven arbitrary values ϕout and ϕin of all the fields including the metricin the two disks consider W [(ϕout ϕin ϕRT )] In writing the boundaryfield as composed of three parts ϕ = (ϕout ϕin ϕside) we are in fact split-ting K as K = Hout otimesHlowast

in otimesHside Fixing ϕside = ϕRT means contractingthe covariant vacuum state |0Σ〉 in K with the bra state 〈ϕRT | in HsideFor large enough R and T we expect the resulting state in Hout otimes Hlowast

in

to reduce to the Minkowski vacuum That is (again braket mismatch isapparent only)

limRTrarrinfin

〈ϕRT |0Σ〉 = |0M〉 otimes 〈0M| (9130)

For a generic in configuration and up to normalization

ΨM[ϕ] = limRTrarrinfin

W [(ϕϕin ϕRT )] (9131)

(Below I shall use a simpler geometry for the boundary) These formu-las allow us to extract the Minkowski vacuum state from a euclideanspinfoam formalism n-particle scattering states can then be obtained bygeneralizations of the flat-space formalism and if this is well defined byanalytic continuation in the single variable T

Consider a spin network that we denote as sprime = ssT composed oftwo parts s and sT connected to each other where s is arbitrary and sTis a weave state (6143) of the three-metric gT defined as follows Take a3-sphere of radius T in R4 Remove a spherical 3-ball of unit radius gT isthe three-metric of the three-dimensional surface (with boundary) formedby the sphere with removed ball The quantity

ψM[s] = limTrarrinfin

intDΦ fssT [Φ] eminus

12

int(P

Gφ)2 + λ

5

int(P

GPHφ)5 (9132)

represents an ansatz for the Minkowski vacuum state in a ball of unitradius


(Note added in the paperback edition A technique for computing n-point functions from background independent quantum gravity has beenproposed and developed This has allowed the derivation the gravitonpropagator which is so to say the derivation of ldquoNewton lawrdquo from thetheory without space and time See Bibliographical notes below)

mdashmdash


John Baez gives a nice and readable introduction to BF theory and spin-foams in [17] which contains also an invaluable carefully annotated bib-liography A good general review is Daniele Oritirsquos [18] In [19] AlejandroPerez describes the group field theory in detail and gives an extensiveoverview on different models On the derivation from hamiltonian looptheory see [11]

The idea of describing generally covariant QFT in terms of a ldquosum-over-surfacesrdquo was initially discussed in [276ndash278] the formal derivationfrom LQG in [279ndash281] The relation spinfoamstriangulated spacetimewas clarified by Fotini Markopoulou [282]

The PonzanondashRegge model was introduced in [283] On the precise re-lation between 6j symbols and Einstein action see [284] Regge calculusis introduced in [285] the TuraevndashViro model in [286] The duality be-tween spinfoam models and QFT on groups was pointed out by Boulatovin [287] as a duality between a QFT on [SU(2)]4 and the 3d PonzanondashRegge model Boulatovrsquos aim was to extend from 2d to 3d the dualitybetween the ldquomatrix modelsrdquo and 2d quantum gravity [21] or ldquozero di-mensional string theoryrdquo [288] (For a while it was hoped that matrixmodels would provide a background-independent definition of string the-ory More recently they have been extensively developed and used in arange of applications) The result was extended by Ooguri to 4d in [289]yielding the TOCY model described in this chapter see also [290] (BFtheories were discussed in [291]) The precise construction of the quantumdeformed version of this model and the proof of its triangulation indepen-dence were given in [292] The relation between PonzanondashRegge modelLQG and length quantization was pointed out in [293]

For the construction of the BC models I have followed Roberto DePietriand Laurent Freidel [294] see also [295 296] The idea of the quantumtetrahedron was discussed by Andrea Barbieri in [297] and by John Baezand John Barrett in [298] and used to construct the spinfoam model forGR in [269] and [299] where the cute term ldquospinfoamrdquo was introducedThe models BCA and BCB were defined in [300] The model BCC wasdefined in [301] and [302] The statistical behavior of different modelsis tentatively explored in [271] Arguments for preferring some of these


models on the basis of diff invariance are considered in [270] The factthat the duality between spinfoam models and QFT on groups extendsto BC models was noticed by DePietri and is presented in [300] Theremarkable fact that it can be extended to arbitrary spinfoam modelswas noticed by Michael Reisenberger and presented in [303] see also [304]The GFT finiteness proof appeared in [305] For recent (2007) discussionson the group field theory approach and updated references see [306] and[307] An intriguing recent result is the observation by Laurent Freideland Etera Livine that a quantum theory of gravity plus matter in 3d isequivalent to a matter theory over a noncommutative spacetime [308]

The convergence of the full series in 3d is discussed in [309] LorentzianPonzanondashRegge models are discussed in [310] lorentzian BC models in[311] Quantum group versions of lorentzian models have been studiedin [231] a paper I recommend for a detailed introduction to the subjectand its extensive references As for the euclidean case the quantum defor-mation of the group controls the divergences and can be related to a cos-mological term in the classical action The lorentzian GFT model SSO(3)

is defined in [312] The lorentzian GFT finiteness proof for this modelappeared in [313] The model SSO(21) is defined in [314] On the problemof the timelikespacelike character of the representations see [315] andreferences therein Functional integral methods to define spinfoam modelsare discussed in [316]

A different approach to the lorentzian sum-over-histories is developedin [317] An interesting variant of the spinfoam formalism in which prop-agation forward and backward in time can be distinguished has beenintroduced in [318]

On the reconstruction of the Hilbert space from the amplitudes of aspinfoam model see [274] and [275] On the relation between spinfoamand canonical LQG see also [319 320] A complete proof of the equiva-lence in 3d has been obtained by Karim Noui and Alejandro Perez in [321]On the role of the Immirzi parameter in BF theory see [322] The ansatzfor the derivation of the Minkowski vacuum from spinfoam amplitudesappeared in [145] A hamiltonian approach to the construction of co-herent states representing classical solutions of the Einstein equations isbeing developed by Thiemann and Winkler [323] A causal version ofthe spinfoam formalism has been studied by Fotini Markopoulou and LeeSmolin [324] The extension of the spinfoam models to matter coupling isstill embryonic see [325326] On the PeterndashWeyl theorem and harmonicanalysis on groups in general see for instance [327328]

A background independent difinition of n-point functions has beengiven in [329] The graviton propagator has been derived in [330] Seealso [331]

10Conclusion

In this book I have tried to present a compact and unified perspective on quantumgravity its technical aspects and its conceptual problems the way I understand themThere are a great number of other aspects of quantum gravity that I would have wishedto cover but my energies are limited Just to mention a few of the major topics I haveleft out 2 + 1 gravity the Kodama state and related results quantum gravity phe-nomenology supergravity in LQG coherent states Here I briefly summarize the phys-ical picture that emerges from LQG and the solution it proposes to the characteristicconceptual issues of quantum gravity I conclude with a short summary of the mainopen problems and the main results of the theory

101 The physical picture of loop gravity

The effort to develop a quantum theory of gravity forces us to revisesome conventional physical ideas This was expected given the conceptualnovelty of the two ingredients GR and QM and the tension between thetwo The physical picture presented in this book has emerged from severaldecades of research and is a tentative solution of the puzzle Here is abrief summary of its conceptual consequences

1011 GR and QM

The first conclusion of loop gravity is that GR and QM do not contradicteach other A quantum theory that has GR as its classical limit appearsto exist In order to merge both QM and classical GR have to be suit-ably formulated and interpreted More precisely they both modify someaspects of classical prerelativistic physics and therefore some aspects ofeach other

GR changes the way we understand dynamics It changes the structureof mechanical systems classical and quantum Classical and quantummechanics admit formulations consistent with this conceptual noveltyThese formulations have been studied in Chapters 3 and 5 respectively

366

101 The physical picture of loop gravity 367

Both classical and quantum mechanics are well defined consistent andpredictive in this generalized form although the relativistic formulationof QM has aspects that need to be investigated further

The main novelty is that dynamics treats all physical variables (partialobservables) on equal footing and predicts their correlations It does notsingle out a special variable called ldquotimerdquo to describe evolution withrespect to it Dynamics is not about time evolution it is about relationsbetween partial observables

QM modifies the picture of the world of classical GR as well In quan-tum theory a physical system does not follow a trajectory A classicaltrajectory of the gravitational field is a spacetime Therefore QM impliesthat continuous spacetime is ultimately unphysical in the same sensein which the notion of the trajectory of a Schrodinger particle is mean-ingless GR remains a meaningful theory even giving up the notion ofclassical spacetime as Maxwell theory remains a physically meaningfultheory in the quantum regime even if the notion of classical Maxwellfield is lost

To make sense of GR in the absence of spacetime we must readEinsteinrsquos major discovery in a light which is different from the conven-tional ones Einsteinrsquos major discovery is that spacetime and gravitationalfield are the same object A common reading of this discovery is that thereis no gravitational field there is just a dynamical spacetime In the viewof quantum theory it is more illuminating and more useful to say thatthere is no spacetime there is just the gravitational field From this pointof view the gravitational field is very much a field like any other fieldEinsteinrsquos discovery is that the fictitious background spacetime introducedby Newton does not exist Physical fields and their relations are the onlycomponents of reality

1012 Observables and predictions

What does a physical theory predict in the absence of time evolutionand spacetime Any physical measurement that we perform is ultimatelya measurement of some local property of a quantum field These measure-ments are represented by an operator in a suitable kinematical quantumspace The theory then gives two kinds of predictions

bull First the spectral properties of the operator predict the quantiza-tion properties of the corresponding physical quantity They deter-mine the list of the values that the quantity may take

bull Second quantum dynamics predicts correlation probabilities be-tween observations That is it associates a correlation probabilityamplitude to ensembles of measurement outcomes

368 Conclusion

This conceptual structure is sufficient to formulate a meaningful and pre-dictive theory of the physical world even in the absence of a backgroundspacetime and in particular in the absence of a background time

There are no specific ldquoquantum gravity observablesrdquo Any measurementinvolving the gravitational field is also a quantum gravitational measure-ment Any measurement with which we test classical GR and whose out-comes we predict using classical GR is in principle a ldquoquantum gravitymeasurementrdquo as well The distinction is one of experimental accuracy

A measurement of a quantity that depends on the gravitational field isrepresented by an operator in the quantum theory of gravity In particulargeometric measurements such as measurements of volumes and areas areof this kind The theory illustrated in Chapter 6 constructs well-definedoperators corresponding to these measurements Their spectra are knownand provide quantitative quantum gravitational predictions

Some experiments can be viewed as ldquoscatteringrdquo experiments happen-ing in a finite region surrounded by detectors The traditional descriptionof this setting is based on two distinct kinds of measurements

(i) clocks and meters which measure the relative position of the detec-tors

(ii) particle detectors or other instruments which measure field proper-ties

In prerelativistic physics the measurements in class (i) refer to the loca-tion on background spacetime while the measurements of class (ii) referto the dynamical variables of the field theory The distinction between (i)and (ii) disappears in gravity This is because distances and time intervalsare nothing other than properties of the gravitational field and there-fore the measurements of class (i) fall into class (ii) both refer to thevalue of the field on the boundary of the experimental region Given anensemble of detectors having measured a certain ensemble of field prop-erties and their relative distances the theory should yield an associatedcorrelation probability amplitude that allows us to compute the relativefrequency of a given outcome with respect to a different outcome of thesame measurement

1013 Space time and unitarity

Space The disappearance of conventional physical space is a character-istic feature of the LQG picture There are quantum excitations of thegravitational field that have given probability amplitudes of transform-ing into each other These ldquoquanta of gravityrdquo do not live immersed in a


spacetime They are space The idea of space as the inert ldquocontainerrdquo ofthe physical world has disappeared

Instead the physical space that surrounds us is an aggregate of indi-vidual quanta of the gravitational field represented by the nodes of aspin network More precisely it is a quantum superposition of such aggre-gates

As observed in Chapter 2 the disappearance of the space-containeris not very revolutionary after all it amounts to a return to the viewof space as a relation between things which was the dominant tra-ditional way of understanding space in the Western culture beforeNewton

Perhaps I could add that in the pre-copernican world the cosmic organization wasquite hierarchical and structured Hence objects were located only with respect to oneanother but this was sufficient to grant every object a rather precise position in thegrand scheme of things This position marked the ldquostatusrdquo of each object vile objectsdown here noble objects above in the heavens With the copernican revolution thisoverall grand structure was lost Objects no longer knew ldquowhererdquo they were Newtonoffered reality a global frame It is a frame that for Newton was grounded in Godspace was the ldquosensoriumrdquo of God the World as perceived by God With or withoutsuch an explicit reference to God space has been held for three centuries as a preferredentity with respect to which all other entities are located Perhaps with the twentiethcentury and with GR we are learning that we do not need this frame to hold realityReality holds itself Objects interact with other objects and this is reality Reality isthe network of these interactions We do not need an external entity to hold the net

Time The disappearance of conventional physical time is the secondcharacteristic feature of nonperturbative quantum gravity This is per-haps a more radical step than the disappearance of space This book isas much about time as about quantum gravity A central idea defendedin this book is that in order to formulate the quantum theory of gravitywe must abandon the idea that the flow of time is an ultimate aspect ofreality We must not describe the physical world in terms of time evolu-tion of states and observables Instead we must describe it in terms ofcorrelations between observables

This shift of point of view is already forced upon us by classical GRbut in classical GR each solution of the Einstein equations still provides anotion of continuous spacetime It is only in the quantum theory of gravitywhere classical solutions disappear that we truly confront the absence oftime at the fundamental level Basic physics without time is viable Theformalism and its interpretation remain consistent In fact as soon as wegive up the idea that the ldquotimerdquo partial observable is special mechanicstakes a far more compact and elegant form as shown in Chapter 3

370 Conclusion

Unitarity In conventional QM and QFT unitarity is a consequence ofthe time translation symmetry of the dynamics In GR there isnrsquot ingeneral an analogous notion of time translation symmetry Thereforethere is no sense in which conventional unitarity is necessary in the theoryOne often hears that without unitarity a theory is inconsistent This isa misunderstanding that follows from the erroneous assumption that allphysical theories are symmetric under time translations

Some people find the absence of time difficult to accept I believe this is just a sort ofnostalgia for the old newtonian notion of the absolute ldquoTimerdquo along which everythingflows But this notion has already been shown to be inappropriate for understanding thereal world by special relativity Holding on to the idea of the necessity of unitary timeevolution or to Poincare invariance is an anchorage to a notion that is inappropriateto describe general-relativistic quantum physics

1014 Quantum gravity and other open problems

All sorts of open problems in theoretical physics (and outside it) havebeen related to quantum gravity For many of these I see no connectionwith quantum gravity In particular

bull Interpretation of quantum mechanics I see no reason why a quan-tum theory of gravity should not be sought within a standard in-terpretation of quantum mechanics (whatever one prefers) Severalarguments have been proposed to connect these two problems Acommon one is that in the Copenhagen interpretation the observermust be external but it is not possible to be external from the grav-itational field I think that this argument is wrong if it was correctit would apply to the Maxwell field as well We can consistently usethe Copenhagen interpretation to describe the interaction betweena macroscopic classical apparatus and a quantum gravitational phe-nomenon happening say in a small region of (macroscopic) space-time The fact that the notion of spacetime breaks down at shortscale within this region does not prevent us from having the regioninteracting with an external Copenhagen observer1

bull Quantum mechanical collapse Roger Penrose has proposed a subtleargument to relate GR and the QM collapse issue The argument isbased on the fact that in the Schrodinger equation there is a timevariable but the flow of physical time is affected by the gravitationalfield I think that this argument is correct but it only shows thatthe Schrodinger picture with an external time variable is not viablein quantum gravity

1However see Section 564


bull Unification of all interactions To quantize the electromagnetic fieldwe did not have to unify it with other fields And to find the quantumtheory of the strong interactions we do not have to unify them withother interactions The only vague hint that the problem of quantumgravity and the problem of the unification of all interactions mightbe related is the fact that the scale at which the running couplingconstants of the standard model meet is not very far from the Planckscale But it is not very close either

bull Particle masses cosmological constant standard modelrsquos families consciousness There are many aspects of the Universe whichwe do not understand There is no reason why all of these have tobe related to the problem of quantizing gravity or the problem ofunderstanding background-independent QFT We are far from theldquoend of physicsrdquo and there is much we do not yet understand

On the other hand there are two important open problems to whichquantum gravity is strongly connected

bull Ultraviolet divergences The disappearance of the ultraviolet diver-gences is one of the major successes of loop gravity This is achievedin a physically clear and compelling way via the short-scale quan-tization of space

bull Spacetime singularities There is no general result so far But loopquantum cosmology mentioned in Chapter 8 shows that the clas-sical initial singularity can be controlled by the theory

102 What has been achieved and what is missing

The formalism presented in Chapters 6 and 7 provides a well-definedbackground-independent quantum theory of gravity and matter The the-ory exists in euclidean and lorentzian versions In more detail

bull Background independence The main ambition of LQG was to com-bine GR and QM into a theory capable of merging the insightson Nature gathered by the two theories The problem was to un-derstand what is a general-relativistic QFT or a QFT constructedwithout using a fictitious background spacetime LQG achieves thisgoal Whether or not it is physically correct it proves that a QFTcan be general-relativistic and background independent It providesa nontrivial example of a background-independent QFT

bull A physical picture LQG offers a novel tentative unitary picture ofthe world that incorporates GR and QM In the book I have tried tospell out in detail this picture its assumptions and implications The

372 Conclusion

picture of the background-independent structure of the quantumgravitational field and matter is simple and compelling Spin net-work states describe Planck-scale quantum excitations which them-selves define localization and spatial relations as the solutions ofthe Einstein equations do Physical space is a quantum superpo-sition of spin networks Spin networks are not primary concreteldquoobjectsrdquo like particles in classical mechanics rather they describethe way the gravitational field interacts like the energy quanta ofan oscillator do The elements of the theory with a direct physicalinterpretation are elements of the algebra of the partial observablesof which spin networks characterize the spectrum

bull Quantitative physical predictions The spectra of area and volumedescribed in Chapter 6 provide a large body of precise quantitativephysical predictions These are unambiguous up to a single over-all multiplicative factor the Immirzi parameter or equivalentlythe bare value of the Newton constant Todayrsquos technology is notcapable of directly testing these spectra Indirect testing is not nec-essarily ruled out What is interesting about these predictions onthe other hand is the fact that they exist at all A theory is not ascientific theory unless it can provide a large body of precise quan-titative predictions capable at least in principle of being verifiedor falsified As far as I know no other current tentative quantumtheory of gravity provides a similar large set of predicted numbers

bull Ultraviolet divergences LQG appears to be free from ultravioletdivergences even when coupled with the standard model

bull Black-hole thermodynamics Although some aspects of the pictureare still unclear (in particular the determination of the Immirziparameter) LQG provides a compelling explanation of black-holeentropy as described in Section 82

bull Big-Bang singularity The classical initial cosmological singularityis controlled in the application of LQG to cosmology described inSection 81

The main aspects of LQG that are still missing or not sufficiently de-veloped are the following

bull Scattering amplitudes Having a well-defined physical theory is dif-ferent from knowing how to extract physics from it We can becapable of writing the full Schrodinger equation for the iron atomand be confident that this equation could predict the iron spectrumBut computing this spectrum is a different matter In a sense we arein a similar situation in LQG We have a well-defined theory but so


far we do not have a great capacity of systematic calculations of ob-servable amplitudes from the basic formalism of the theory What ismissing is a systematic formalism for doing so in some appropriateform of perturbation expansion

The difficulty in developing this formalism is of course due to thefact that a perturbative expansion around a classical solution of thegravitational field does not work The reason why this happens isclear nonperturbative effects dominate at the Planck scale yield-ing the discrete quantized structure of space We have to find analternative way for performing perturbative calculations One di-rection of research on this issue utilizes the covariant spinfoam for-malism described in Chapter 9 But this formalism is not yet at thepoint of providing a systematic technique for computing transitionamplitudes (Note added in the paperback edition Research is devel-oping rapidly in this direction General covariant n-point functionshave been defined and computed See the Bibliographical notes atthe end of the previous chapter)

bull Semiclassical limit The description of a macroscopic configurationof the electromagnetic field within quantum electrodynamics is nottrivial but it can be achieved using for instance coherent-state tech-niques Describing a macroscopic solution of the Einstein equationswith LQG is a similar problem Can we find a state in Kdiff that ap-proximates a given macroscopic solution Research programs in thisdirection are being pursued in particular by the groups of ThomasThiemann and Abhay Ashtekar I refer the reader to their worksfor a description of the state of the art in this rapidly developingdirection of research

LQG is in a peculiar specular position with respect to many tra-ditional approaches to quantum gravity The most common diffi-culty is to arrive at a description of Planck-scale physics manyformalisms tend to diverge or break down in some other way at thebackground-independent Planck-scale level LQG on the contraryprovides a formalism that gives a simple and compact descriptionof the background-independent Planck-scale physics but the recov-ery of low-energy physics appears more difficult (Note added in thepaperback edition The recent computation of the n-point functionsmentioned above provide a way of testing the large distance limit ofthe theory In particular the correct large-distance behavior of thepropagator obtained in [330] can be interpreted as the recovery ofthe Newton law from the background independent theory See theBibliographical notes at the end of the previous chapter)

374 Conclusion

(Note added in the paperback edition An important recent resultin this direction is the proof of the precise equivalence of the twoformalishs in 3d [321])

bull The Minkowski vacuum The most important state that we need isthe coherent state |0M〉 corresponding to Minkowski space This isessential to connect the theory to the usual formalism of QFT andto define particle-scattering amplitudes A direction for computingthis state was suggested at the end of Chapter 9 but it is too earlyto see if this will work

bull The form of the hamiltonian As discussed in Section 713 theprecise form of the quantum hamiltonian is not yet settled Thereare a number of quantization ambiguities and a number of possiblevariants that have been proposed The difficulty of selecting thecorrect form of the hamiltonian is not only due to the lack of directempirical guidance on the Planck scale but also to the little controlthat we have in extracting physical predictions from the theory asdiscussed above

bull Relation between the spinfoam and the hamiltonian formalism Fi-nally the relation between the lagrangian approach of Chapter 9 andthe hamiltonian approach of Chapters 6 and 7 is not yet sufficientlyclear

There are many problems that we have to solve before we can say wehave a credible and complete quantum theory of spacetime I hope thatamong the readers that have followed this book until this point there arethose that will be able to complete the journey

I close borrowing Galileorsquos marvelous prose

Ora perche e tempo di por fine ai nostri discorsi mi resta a pregarviche se nel riandar piu posatamente le cose da me arrecate incontrastedelle difficolta o dubbi non ben resoluti scusiate il mio difetto si per lanovita del pensiero si per la debolezza del mio ingegno si per la grandezzadel suggetto e si finalmente perche io non pretendo ne ho preteso da altriquellrsquoassenso chrsquoio medesimo non presto a questa fantasia2

2ldquoNow since it is time to end our discussion it remains for me to pray of you that ifin reconsidering more carefully what I have presented you find difficulties or doubtsthat were not well resolved you excuse my deficiency either because of the noveltyof the ideas the weakness of my understanding the magnitude of the subject orfinally because I neither ask nor have I ever asked from others that they attach tothis imagination that certainty which I myself do not haverdquo [332]

Part III

Appendices

Appendix A

Groups and recoupling theory

A1 SU(2) spinors intertwiners n-j symbols

SU(2) is the group of the unitary 2times2 complex matrices with determinant1 We write these matrices as UA

B where the indices A and B take thevalues AB = 0 1 The fundamental representation of the group is definedby the natural action of these matrices on C2 The representation spaceis therefore the space of complex vectors with two components These arecalled spinors and denoted

ψA =(ψ0

ψ1

) (A1)

Consider the space formed by completely symmetric spinors with n indicesψA1An This space transforms into itself under the action of SU(2) onall the indices Therefore it defines a representation of SU(2)

ψA1An rarr UA1Aprime

1 UAn

Aprimen

ψAprime1A

primen (A2)

This representation is irreducible has dimension 2j + 1 and is called thespin-j representation of SU(2) where j = 1

2n All unitary irreduciblerepresentations have this form

The antisymmetric tensor εAB (defined with ε01 = 1) is invariant underthe action of SU(2)

UAC UB

D εCD = εAB (A3)

Contracting this equation with εAB (defined with ε01 = 1) we obtain thecondition that the determinant of U is 1

detU =12εAC εBD UA

B UCD = 1 (A4)

sinceεAB εAB = 2 (A5)

377

378 Appendix A

The inverse of an SU(2) matrix can be written simply as

(Uminus1)AB = minusεBD UDC εCA (A6)

Most of SU(2) representation theory follows directly from the invari-ance of εAB For instance consider the tensor product of the fundamentalrepresentation j = 12 with itself This defines a reducible representationon the space of the two-index spinors ψAB

(ψ otimes φ)AB = ψAφB (A7)

We can decompose any two-index spinor ψAB into its symmetric and itsantisymmetric part

ψAB = ψ0εAB + ψAB

1 (A8)

whereψ0 =

12εABψ

AB (A9)

and ψAB1 is symmetric Because of the invariance of εAB this decomposi-

tion is SU(2) invariant The one-dimensional invariant subspace formedby the scalars ψ0 defines the trivial representation j = 0 The three-dimensional invariant subspace formed by the symmetric spinors ψAB

1

defines the adjoint representation j = 1 Hence the tensor product of twospin-12 representations is the sum of a spin-0 and a spin-1 representa-tion 12 otimes 12 = 0 oplus 1

In general if we tensor a representation of spin j1 with a representationof j2 we obtain the space of spinors with 2j1 + 2j2 indices symmetric inthe first 2j1 and in the last 2j2 indices By symmetrizing all the indiceswe obtain an invariant subspace transforming in the representation j1+j2Alternatively we can contract k indices of the first group with k indicesof the second using k times the tensor εAB and then symmetrize theremaining 2(j1 + j2 minus k) indices This defines an invariant subspace ofdimension 2(j1 + j2 minus k) The maximum value of k is clearly the smallestbetween 2j1 and 2j2 Hence the tensor product of the representations j1and j2 gives the sum of the representations |j1minusj2| |j1minusj2|+2 (j1 +j2)

Thus each irreducible j3 appears in the product of two representationsat most once and if and only if

j1 + j2 + j3 = N (A10)

is integer and|j1 minus j2| le j3 le (j1 + j2) (A11)

These two conditions are called the ClebshndashGordon conditions They areequivalent to the requirement that there exist three nonnegative integers

Groups and recoupling theory 379

a = 3 b = 2

c = 2

j2 = 52

2j2 = 5

j1 = 52

2 j1 = 5 2 j3 = 4

j3 = 2

=

Fig A1 ClebshndashGordon condition

a b and c such that

2j1 = a + c 2j2 = a + b 2j3 = b + c (A12)

If we have three representations j1 j2 j3 the tensor product of the threecontains the trivial representation if and only if one is in the product ofthe other two namely only if the ClebshndashGordon conditions are satisfiedThe invariant subspace in the product of the three is formed by invarianttensors with 2(j1 + j2 + j3) indices symmetric in the first 2j1 in thesecond 2j2 and in the last 2j3 indices There is only one such tensor up toscaling because it must be formed by combinations of the sole invarianttensor εAB It is given by simply taking a tensors εAB b tensors εBC andc tensors εCA that is

vA1A2j1 B1B2j2

C1C2j3

= (εA1B1 εAaBa) (εBa+1C1 εBa+bCb) (εCb+1Aa+1 εCb+cAa+c)(A13)

This is an intertwiner between the representations j1 j2 j3 We can choosea preferred intertwiner by demanding that the intertwiner is normalizednamely multiplying vA1A2j1

B1B2j2 C1C2j3 by a normalization fac-

tor K (which I give below) The normalized intertwiner is called theWigner 3j symbol

There is a simple graphical interpretation to the tensor algebra of theSU(2) irreducibles suggested by the existence of the three integers a b csee Figure A1 A representation of spin j is the symmetrized product of 2jfundamentals When three representations come together all fundamen-tals must be contracted among themselves There will be a fundamentalscontracted between j1 and j2 and so on Let us represent each irreducibleof spin j as a line formed by 2j strands An invariant tensor is a trivalent

380 Appendix A

node where three such lines meet and all strands are connected across thenode a strands flow from j1 to j2 and so on The meaning of the ClebshndashGordon conditions is then readily apparent (A10) simply demands thatthe total number of strands is even so they can pair (A11) demands thatj3 is neither larger than j1 + j2 because then some strands of j3 wouldremain unmatched nor smaller than |j1 minus j2| because then the largestamong j1 and j2 would remain unmatched Indeed this relation betweenthe lines and the strands reproduces precisely the relation between spinnetworks and loops Below this graphical representation is developed indetail

Orthonormal basis The space of the symmetric spinors with n indiceshas (complex) dimension 2j + 1 It is often convenient to choose a basisformed by 2j+1 orthonormal vectors eα1αn

α in this space For instance ifj = 1 the basis eAB

i = 1radic2σABi defined using the Pauli matrices transforms

under SU(2) in the fundamental representation of SO(3) The matricesσABi are obtained from the Pauli matrices

σAiB =

(0 11 0

)

(0 minus ii 0

)

(1 00 minus 1

)(A14)

by raising an index with εCB

σABi = σA

iCεCB =

(minus1 00 1

)

(i 00 i

)

(0 11 0

) (A15)

In general if the spin j is integer then the real section of the representationdefines a real irreducible representation

Wigner 3j symbols In an arbitrary orthonormal basis we write the nor-malized invariant tensors (A13) as

K vα1α2α3 =(j1 j2 j3α1 α2 α3

) (A16)

If we chose the basis that diagonalizes the third component of the angularmomentum (α equiv m) which we do below then these are proportional tothe Wigner 3j symbols The normalization K is fixed by

K vα1α2α3 K vα1α2α3 = 1 (A17)

For instance we have easily(

12 12 1A B i

)=

1radic6σiAB (A18)

and (1 1 1i j k

)=

1radic6εijk (A19)


Wigner 6j symbols Contracting four 3j symbols with the invariant ten-sor (minus1)jminusα defines a 6j symbol(j1 j4 j6j3 j2 j5

)=

sum

α1α6

(minus1)sum

A(j1minusα1)

(j3 j6 j2α3 α6 α2

) (j2 j1 j5α2 α1 α5

)

times(j6 j4 j1α6 α4 α1

) (j4 j3 j5α4 α3 α5

) (A20)

Since the indices are all contracted this quantity does not depend on thebasis chosen in the representation space The pattern of contraction isdictated by the geometry of a tetrahedron There is one 3j symbol foreach vertex of the tetrahedron and one representation for each edge

j1

j2

j3

j4 j6

j5 (A21)

Consider a tetrahedron with six spins j1 j6 associated with itsedges as above Denote Hj the representation space of the SU(2) irre-ducible representation of spin j Given a reducible representation H ofSU(2) denote by [H]j its component of spin j The Wigner 6j symbol isthe dimension of the intersection between the subspaces

[[Hj1 otimesHj2 ]j6 otimesHj3 ]j4 and [Hj1 [otimesHj2 otimesHj3 ]j5 ]j4 (A22)

of the space Hj1 otimesHj2 otimesHj3

Intertwiners All intertwiners can be built starting from the three-valentones For instance a four-valent intertwiner between representations j1j2 j3 and j4 can be defined (up to the normalization) by contracting twothree-valent intertwiners

vα1α2α3α4i = iα1α2α iα

α3α4 (A23)

where α is an index in a representation i The space of the intertwinersis then spanned by the tensors vi as i ranges over all representations thatsatisfy the two relevant ClebshndashGordon conditions namely such that thethree-valent intertwiners exist The representation i is said to be associ-ated with a ldquovirtual linkrdquo joining the two three-valent nodes into whichthe four-valent node has been decomposed

382 Appendix A

A different basis on this same intertwiner space is obtained by couplingthe first and the third leg instead of the first and the second That is

wα1α2α3α4i = iα1α3α iα

α2α4 (A24)

The change of basis between the vi and the wi is given by the Wigner 6jsymbols as we will show below (equation (A65))

vi =sum

j

(2j + 1)(j1 j2 jj3 j4 i

)wj (A25)

Graphically

j1

j2

j4

j3

i =sum

j

(2j + 1)(j1 j2 jj3 j4 i

)

j1

j2

j4

j3

j (A26)

Five-valent intertwiners can be constructed contracting a three-valentand a four-valent intertwiner and can thus be labeled with two irre-ducibles and so on In general we can decompose an n-valent node intonminus 2 three-valent nodes connected by nminus 2 virtual links and constructintertwiners accordingly

vα1αni1inminus3

= iα1α2β1 iβ1α3β2 iβ2

α4β3 iβnminus3αnminus1αn (A27)

where the indices βn belong to the representation in

Pauli matrices identities Define τi = minus i2 σi where σi are the Pauli ma-

trices (A14) We have the following identities

tr[τiτj ] = minus12δij (A28)

tr[τiτjτk] = minus14εijk (A29)

δijτiABτj

CD = minus1

4

(δADδ

BC minus εACεBD

) (A30)

δijtr[Aτi]tr[Bτj ] = minus14

tr[AB] minus tr[ABminus1]

(A31)

Aminus1AB = εACεBDA

DC (A32)

δABδDC = δACδ

DB + εADεBC (A33)

tr[A]tr[B] = tr[AB] + tr[ABminus1] (A34)

where A and B are SL(2 C) matrices


A2 Recoupling theory

A21 Penrose binor calculus

In his doctoral thesis Roger Penrose introduced the idea of writing ten-sor expressions in which there are sums of indices in a graphical way abeautiful idea that is at the root of spin networks1 Consider in particu-lar the calculus of spinors Penrose represents the basic element of spinorcalculus as

ψA = ψA

(A35)

ψA = ψA (A36)

δ AC =

C

A(A37)

εAC = A C (A38)

εAC = A C(A39)

and generally any tensor object as

XCAB = X

A BC (A40)

The idea is then to represent index contraction by simply joining the openends of the lines and dropping the index This convention provides thepossibility of writing the product of any two tensors in a graphical wayFor example

εABψAψB = ψ

ψ (A41)

However notice that the meaning of a diagram is not invariant if wesmoothly deform the lines For instance

εABηAηB = η

AηB

(A42)

= minusεADεBCεCDηAηB = minus η

AηB

C D (A43)

= minusεCDδDA δCBη

AηB = minusη ηA BC D

(A44)

Penrose introduced a modification of this graphical spinor calculus whichhe denoted as binor calculus that makes it invariant under deformations

1I have recently learned that Penrose called this notation ldquoloop notationrdquo

384 Appendix A

of the lines The binor calculus is obtained by adding two conventionsto the calculus above In translating a diagram into tensor notation wemust also ensure the following

(i) We assign a minus sign to each minimum and

(ii) assign a minus sign to each crossing

(iii) Maxima and minima are taken with respect to a fixed direction inthe plane (This direction is conventionally taken to be the verticaldirection on the written page)

(iv) A vertical segment represents a Kronecker delta

The advantage of these additional rules is that they make the calculustopologically invariant namely one can arbitrarily smoothly deform agraphical expression without changing its meaning

Expressed the other way around any curve can now be decomposedinto a product of δs and εs and any two curves that are ambient isotopicie that can be transformed one into the other by a sequence of Reide-meister moves represent the tensorial expression as products of epsilonsand deltas

A closed loop with this convention has value minus2 because = minusεAB εAB = minus2 (A45)

and we have the basic binor identity which reads

+ + = (minus1) δCB δDA + δCA δDB + (minus1) εAB εCD = 0 (A46)

Remarkably the two graphical identities = minus2 (A47)

= minus minus (A48)

are sufficient to generate a very rich graphical calculus

Kauffman brackets Equations (A47)ndash(A48) can be seen as a particular case of a richerstructure In the context of knot theory Lou Kauffman has defined a function of tanglesnamely planar graphical representations of knots which is now denoted the Kauffmanbrackets A planar tangle is a set of lines on a plane that overcross or undercross atintersections It represents the 2d projection of a 3d node The Kauffman brackets ofa tangle K are indicated as 〈K〉 and are completely determined by the two relations

lang

rang= A

lang rang+ Aminus1

lang rang(A49)

and lang cup Krang

= dlangK

rang (A50)


where d = minusA2 minus Aminus2 and K is any diagram that does not intersect the added loopBy applying equation (A49) to all crossings the Kauffman brackets of the tangle canbe reduced to a linear combination of Kauffman brackets of nonintersecting tangles Byrepeated application of (A50) we can then associate a number to the tangle Penrosebinor calculus and (A47)ndash(A48) are recovered for A = minus1 In this case undercrossingand overcrossing are not distinct

A22 KL recoupling theory

Following Kauffman and Linsrsquo book [174] one can pose the followingdefinitions

The antisymmetrizer Write n parallel lines as a single line labeled withn

n equiv (A51)

(The precise relation between the graphical calculus defined by this andthe following equations and the graphical calculus used in Chapter 6defined by equation (686) is discussed below in Section A23) Definethe antisymmetrizer as

n=

1n

sum

p

(minus1)|p| P (p)n (A52)

where P(p)n p = 1 n represents all the possible ways of connecting n

incoming lines with n outgoing lines obtained as n permutations and|p| is the sign of the permutation

The 3-vertex A special sum of tangles is indicated by a 3-vertex Eachline of the vertex is labeled with a positive integer n m or p

n

m

p

(A53)

and it is assumed that n = a+ b m = a+ c and p = b+ c where a b c arepositive integers This last condition is called the admissibility conditionfor the 3-vertex (mn p) The 3-vertex is then defined as

n

m

p

equiva

b c

p

n m

(A54)

386 Appendix A

Compare this definition with the discussion of the ClebshndashGordon coef-ficients and Wigner 3j symbols given above in A1 It is clear that the3-vertex in Penrose binor notation represents precisely the nonnormal-ized intertwiner (A13) In turn the Wigner 3j symbol can be obtainedby normalizing this intertwiner (Take care The KL 3-vertex representsthe nonnormalized intertwiner (A13) while the spin network vertex de-fined in (686) corresponds to the Wigner 3j symbol Therefore the twovertices differ by a normalization (See A23 below))

Chromatic evaluation If we join several trivalent vertices by their edgeswe obtain a trivalent spin network Thus in the present context a trivalentnetwork is defined as a trivalent graph with links labeled by an admissiblecoloring Notice that in this context networks are not embedded in athree-dimensional space A link of color n represents n parallel lines andan antisymmetrizer Thus a trivalent spin network determines a closedtangle The Penrose evaluation (or the Kauffman bracket with A = minus1)of this tangle is called the chromatic evaluation or network evaluation

Contractions of intertwiners and Wigner 3j symbols can therefore becomputed as chromatic evaluations of colored diagrams using only therelations (A47)ndash(A48)

As an example consider the spin network formed by two trivalent ver-tices joined to each other This is called the θ network Consider the casewith edges of color 2 1 1 Applying the definitions given we have

121

=1

1

2 13

=12

13

minus 12

=

=12(minus2)2 minus 1

2(minus2) = 3 (A55)

Therefore121

= 3 (A56)

The general formula of the chromatic evaluation of a generic θ network isgiven below in (A59)

Formulas from KL recoupling theory Direct computation using the defi-nitions above give the following formulas (See also the appendix of [195])(1) The dimension

Δn =n

= (minus1)n(n + 1) (A57)


Notice that if we write n = 2j then (n+ 1) = (2j + 1) is the dimension ofthe SU(2) spin-j representation(2) The exchange of lines in a 3-vertex

a

b

c

= λabc

ab

c

(A58)

where λabc = (minus1)(a+bminusc)2 (minus1)(a

prime+bprimeminuscprime)2 and xprime = x(x + 2)(3) The θ evaluation

θ(a b c) =

a

b

c

13

=(minus1)m+n+p(m + n + p + 1) m n p

a b c (A59)

where m = (a + bminus c)2 n = (b + cminus a)2 p = (c + aminus b)2(4) The tetrahedral net

Tet[A B EC D F

]=

B

A

C

D

F

E (A60)

=IE

sum

mleSleM

(minus1)S(S + 1)prod

i (S minus ai)prod

j (bj minus S) (A61)

where

a1 =A + D + E

2 b1 =

B + D + E + F

2

a2 =B + C + E

2 b2 =

A + C + E + F

2

a3 =A + B + F

2 b3 =

A + B + C + D

2

a4 =C + D + F

2

m = maxai M = minbj

E = ABCDEF I =prod

ij(bj minus ai)

388 Appendix A

(5) The reduction formulas

b

a

a

13c

=

abc

a

a (A62)

b

c

d

ef

a

a

=

cb

d

e

f

a

a a (A63)

Strictly speaking the identity factors on the right-hand side (the nonin-tersecting a-tangles) of both of these equations should include the anti-symmetrizer When embedded in spin networks this gets absorbed intothe nearest vertex

Also

q

p

rp

q

2 =

Tet[p p rq q 2

]

θ(p q r) q

pr middot (A64)

(6) The recoupling theorem

a

b

d

c

j =sum

i

a b ic d j

a

b

d

c

i (A65)

a b ic d j

=

Δi Tet[a b ic d j

]

θ(a d i)θ(b c i) (A66)

These formulas are sufficient for most computations performed in loopquantum gravity

A23 Normalizations

Finally it is time to relate the KauffmanndashLins recoupling theory diagramsgiven in this appendix with the spin networks recoupling diagrams thatwe have used in Chapter 6 and which are defined by equation (686)There are two main differences The first is trivial lines are labeled bythe spin j in spin network recoupling diagrams while they are labeled by


the color n = 2j in the KauffmanndashLins diagrams Thus⎛

⎝ j

⎞

⎠

spin network

=

⎛

⎝ n = 2j

⎞

⎠

KauffmanminusLins

middot (A67)

The second and more important difference is that the trivalent nodes ofthe spin network diagrams represent normalized intertwiners Thereforethey are proportional to the recoupling theory trivalent nodes and theproportionality factor is easily obtained from (A59) Thus up to possiblephase factors

⎛

⎝

j

jprime

jprimeprime

⎞

⎠

spin network

=

⎛

⎜⎜⎜⎝

1radicθ(a b c)

a=2j

b=2jprime

c=2jprimeprime

⎞

⎟⎟⎟⎠

KauffmanminusLins

middot

(A68)

These two equations provide the complete translation rules It follows forinstance that the Wigner 6j-symbol is given by

(j1 j2 j5j3 j4 j6

)=

⎛

⎜⎝

j2

j1

j3

j4

j6

j5

⎞

⎟⎠

spin network

=

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

2j2

2j1

2j3

2j4

2j6

2j5

radic2j12j2

2j6

2j32j4

2j6

2j12j4

2j5

2j22j3

2j5

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

KauffmanminusLins

=Tet

[2j1 2j2 2j52j3 2j4 2j6

]

radicθ(2j1 2j2 2j6)θ(2j3 2j4 2j6)θ(2j1 2j4 2j5)θ(2j2 2j3 2j5)

(A69)

And the recoupling theorem (A26) follows from (A65)The reason I have described two distinct normalization conventions for

the diagrams is that they are both utilized in the loop quantum gravityliterature and both turn out to be useful The diagrammatic notationdeveloped in this appendix is the one used by Kauffman and Lins in

390 Appendix A

[174] This notation has often been used for computing matrix elementsof loop operators

On the other hand one can find many results on the Wigner 3nj sym-bols and a well-developed graphical calculus for the representation theoryof SU(2) in the general literature These results are routinely used for in-stance in atomic and nuclear physics A standard reference is for instanceBrink and Satcheler [333] The BrinkndashSatcheler diagrams are written withthe convention I have used in Chapter 6 for the spin networks2

A3 SO(n) and simple representations

I collect here some facts on the representation theory of SO(n) Labelfinite-dimensional irreducible representations of SO(n) by their highestweight Λ Here Λ is a vector of length n = [d2] ([middot] is the integral part)Λ = (N1 middot middot middot Nn) where Ni are integers and N1 ge ge Nn If we areinterested in representations of Spin(n) we let the Ni be half-integers Therepresentation labeled by the highest weight Λ = (N 0 0) are calledsimple or spherical Let Xij 1 le i j le n be a basis of the Lie algebra ofSO(n) The simple representations are those for which the ldquosimplicityrdquorelations

X[ijXij] middot VN = 0 (A70)

are satisfied The representation space VN of a simple representation canbe realized as a space of spherical harmonics that is harmonic homoge-neous polynomials on Rn Any L2 function on the sphere can be uniquelydecomposed in terms of these spherical harmonics

L2(Snminus1) = oplusinfinN=0VN (A71)

In the case of SO(4) since Spin(4) = SU(2)timesSU(2) there is an alter-nate description of the representation as products of two representationsjprime and jprimeprime of SU(2) The relation with the highest weight presentation isgiven by

N1 = jprime + jprimeprime N2 = jprime minus jprimeprime (A72)

The simple representations are therefore the representation in whichjprime = jprimeprime equiv j Thus we can label simple representations with a half-integerspin j Notice that the integer ldquocolorrdquo N = 2j is also the (nonvanishingcomponent of the) highest weight of the representation

2With a minor difference in the angular momentum literature Wignerrsquos 6j symbolsare generally indicated with curl brackets while following Kauffman I have denoted6j symbols by round brackets reserving the curl brackets in (A65) to denote therecoupling matrix of a four-valent node which differs from the 6j symbol by thenormalization factor Δi


An elementary illustration of simple representations can be given asfollows The vectors vα of the representation Λ = (jprime jprimeprime) can be writ-ten as spinors ψA1Ajprime B1Bjprimeprime with jprime ldquoundottedrdquo symmetrized indicesAi = 1 2 transforming under one of the SU(2) and jprimeprime ldquodottedrdquo anti-symmetrized indices Bi=1 2 transforming under the other Consider theparticular vector wα which (in the spinor notation and in a given basis)has components ψA1Ajprime B1Bjprimeprime = ε(A1B1 εAj)Bj where εAB is the unitantisymmetric tensor and the symmetrization is over the Ai indices onlyThe subgroup of SO(4) that leaves this vector invariant is an SO(3) sub-group of SO(4) (which depends on the basis chosen) Clearly since εAB isthe only object invariant under this SU (2) a normalized SO(3)-invariantvector (and only one) exists in these simple representations only

Equivalently the simple representations of SO(4) are those defined bythe completely symmetric traceless 4d tensors of rank N The invariantvector w is then the traceless part of the tensor with (in the chosen basis)all components vanishing except w444 The SO(3) subgroup is given bythe rotations around the fourth coordinate axis The relation between thevector and spinor representation is obtained contracting the spinor indiceswith the (four) Pauli matrices vμ1μj = ψA1Ajprime B1Bjprimeprimeσμ1

A1B1 σ

μj

AjBj

Let VΛ be a representation of SO(n) we say that ω isin VΛ is a sphericalvector if it is invariant under the action of SO(nminus1) Such a vector existsif and only if the representation is simple In that case this vector is uniqueup to normalization

Let ω be a vector of VΛ and consider an orthonormal basis vi of VΛWe can construct the following functions on G

Θi(g) = 〈ω|Rminus1(g)|vi〉 (A73)

These functions span a subspace of L2(G) The group acts on this sub-space by the right regular representation and the corresponding rep-resentation is equivalent to the representation VΛ If ω is sphericalthen these functions are in fact L2 functions on the quotient spaceSO(n)SO(n minus 1) = Snminus1 and VΛ is therefore a spherical representa-tion On the other hand if the representation is spherical then we canconstruct a spherical vector Θω(g) =

sumi Θi(g)Θi(1) When ω is spheri-

cal the spherical function Θω is a function on the double-quotient spaceSO(nminus1)SO(n)SO(nminus1) = U(1) It is now a standard exercise to showthat there is a unique harmonic polynomial on Rn invariant by SO(nminus1)of a given degree hence there is a unique spherical function

The space of the intertwiners of three representations of SO(4) isat most one-dimensional The dimension nΛ1Λ2Λ3 of the space of theintertwiners between three representations Λ1Λ2Λ3 is given by the

392 Appendix A

integral

nΛ1Λ2Λ3 =int

dg χΛ1(g)χΛ2(g)χΛ3(g) (A74)

where χΛ are the characters of the representation Λ Since any repre-sentation of SO(4) is the product of two representations of SU(2) theintertwining number of three SO(4) representations is the product ofSU(2) intertwining numbers These numbers can take the values 0 or1 for SU(2)

Representations of SO(n) are real This means that it is always possibleto choose a basis of VΛ such that the representation matrices are real Forthe half-integer spin representations of Spin(n) it is still true that Λis equivalent to its complex conjugate or dual but the isomorphism isnontrivial

mdashmdash


On Penrose calculus see [181] and [334] The basic reference on recouplingtheory that I have used here is [174] where the formulas given here arederived in the general case with arbitrary A see also the appendix of[195] for more details On the relation between recoupling theory andspin networks see [335 336] Chromatic evaluation is used to computeSU(2) ClebshndashGordon coefficients in for instance [337] A widely usedgraphical method for angular momentum theory is the one of Levinson[338] developed by Yutsin Levinson and Vanagas [339] and the slightlymodified version of Brink and Satcheler [333]

Appendix B

History

In this appendix I sketch the main lines of development of the research in quantumgravity from the first explorations in the early 1930s (the thirties) to nowadays

I have no ambition of presenting complete references to all the important workson the subject some of the references are to original works others to reviews wherereferences can be found Errors and omissions are unfortunately unavoidable and Iapologize for these I have made my best efforts to be balanced but in a field that hasnot yet succeeded in finding consensus my perspective is obviously subjective Tryingto write history in the middle of the developments is hard Time will pass dust willsettle and it will slowly become clear if we are right if some of us are right or ndash apossibility never to disregard ndash if we all are wrong

I am very much indebted to the many friends that have contributed to this histor-ical perspective I am particularly grateful to John Stachel Augusto Sagnotti GaryHorowitz Ludwig Faddeev Alejandro Corichi Jorge Pullin Lee Smolin Joy ChristianBryce DeWitt Cecile DeWitt Giovanni Amelino-Camelia Daniel Grumiller NikolaosMavromatos Stanley Deser Ted Newman and Gennady Gorelik

B1 Three main directions

The quest for quantum gravity can be separated into three main lines ofresearch The relative weight of these lines has changed there have beenimportant intersections and connections between the three and there hasbeen research that does not fit into any of the three lines Neverthelessthe three lines have maintained a distinct individuality across 70 yearsof research They are often denoted ldquocovariantrdquo ldquocanonicalrdquo and ldquosum-over-historiesrdquo even if these names can be misleading and are often usedinterchangeably They cannot be characterized by a precise definitionbut within each line there is a certain methodological unity and a certainconsistency in the logic of the development of the research

bull The covariant line of research is the attempt to build the theory asa quantum field theory of the fluctuations of the metric over a flat

393

394 Appendix B

Minkowski space or some other background metric space The pro-gram was started by Rosenfeld Fierz and Pauli in the thirties TheFeynman rules of GR were laboriously found by DeWitt Feynmanand Faddeev in the sixties trsquoHooft and Veltman Deser and VanNieuwenhuizen and others found increasing evidence of nonrenor-malizability at the beginning of the seventies Then a search foran extension of GR giving a renormalizable or finite perturbationexpansion started Through high-derivative theory and supergrav-ity the search converged successfully to string theory in the lateeighties

bull The canonical line of research is the attempt to construct a quan-tum theory in which the Hilbert space carries a representation of theoperators corresponding to the full metric or some functions ofthe metric without any background metric to be fixed The pro-gram was initiated by Bergmann and Dirac in the fifties Unravel-ing the canonical structure of GR turned out to be laborious DiracBergmann and his group and Peres completed the task in the fiftiesTheir cumbersome formalism was drastically simplified by the intro-duction of new variables first by Arnowit Deser and Misner in thesixties and then by Ashtekar in the eighties The formal equationsof the quantum theory were written down by Wheeler and DeWittin the middle sixties but turned out to be too ill defined A well-defined version of the same equations was successfully found onlyin the late eighties with the formulation of LQG

bull The sum-over-histories line of research is the attempt to use someversion of Feynmanrsquos functional integral quantization to define thetheory The idea was introduced by Misner in the fifties followinga suggestion by Wheeler and developed by Hawking in the formof euclidean quantum gravity in the seventies Most of the discrete(lattice-like posets ) approaches and the spinfoam formalismintroduced more recently belong to this line as well

bull Others There are of course other ideas that have been explored

ndash Noncommutative geometry has been proposed as a key math-ematical tool for describing Planck-scale geometry and hasrecently obtained very surprising results particularly with thework of Connes and collaborators

ndash Twistor theory has been more fruitful on the mathematicalside than on the strictly physical side but it is still developing

ndash Finkelstein Sorkin and others pursue courageous and intrigu-ing independent paths

History 395

ndash Penrosersquos idea of a gravity-induced quantum state reductionhas recently found new life with the perspective of a possibleexperimental test

ndash

So far however none of these alternatives has been developed intoa detailed quantum theory of gravity

B2 Five periods

Historically the evolution of the research in quantum gravity can roughlybe divided into five periods summarized in Table B1

bull The Prehistory 1930ndash1957 The basic ideas of all three lines ofresearch appear as early as the thirties By the end of the fifties thethree research programs are clearly formulated

bull The Classical Age 1958ndash1969 The sixties see the strong develop-ment of two of the three programs the covariant and the canonicalAt the end of the decade the two programs have both achieved thebasic construction of their theory the Feynman rules for the gravi-tational field on one side and the WheelerndashDeWitt equation on theother To get to these beautiful results an impressive amount oftechnical labour and ingenuity proves necessary The sixties close ndashas they did in many regards ndash with the promise of a shining newworld

bull The Middle Ages 1970ndash1983 The seventies disappoint the hopes ofthe sixties It becomes increasingly clear that the WheelerndashDeWittequation is too ill defined for genuine field theoretical calculationsAnd evidence for the nonrenormalizability of GR piles up Both linesof attack have found their stumbling block

In 1974 Stephen Hawking derives black-hole radiation Trying todeal with the WheelerndashDeWitt equation he develops a version ofthe sum-over-histories as a sum over ldquoeuclideanrdquo (riemannian)geometries There is excitement with the idea of the wave func-tion of the Universe and the approach opens the way for thinking ofand computing topology change But for field theoretical quantitiesthe euclidean functional integral will prove as weak a calculationtool as the WheelerndashDeWitt equation

On the covariant side the main reaction to nonrenormalizabil-ity of GR is to modify the theory Strong hopes then disap-pointments motivate extensive investigations of supergravity and

Table B1 The search for a quantum theory of the gravitational field

Prehistory

1920 The gravitational field needs to be quantized

1930 ldquoFlat-space quantizationrdquo

1950 ldquoPhase spacequantizationrdquo

1957Constraint theory

ldquoFeynmanquantizationrdquo

Classical Age

1961ADM Tree-amplitudes

1962 Background fieldmethod

1963 Wave function of the 3-geometry spacetime foam

1967 Ghosts

1968 MinisuperspaceFeynman rules

completed

Middle Ages

1971 YM renormalization

1972 Twistors

1973Nonrenormalizability

1974 Black-holeradiation

1976 Asymptotic safety

1976 Supergravity

1977 High-derivative theories

1978Euclidean QG

1981

1983 Wave functionof the Universe

Renaissance

1984 String renaissance

1986 Connectionformulation of GR

TQFT

1987Superstring theory

1988Loop quantum gravity

2+1

1989 2d QG

1992 Weaves State sum models

1994 Noncommutative

geometry

Nowadays

1995Eigenvalues of area and volume

Null surfaceformulation Nonperturbative strings

1996 BH radiation from loops Spin foams BH radiation from strings

1997 ldquoQuantum gravity phenomenologyrdquo

Stringsndashnoncommutativegeometry

WheelermdashDeWitt equation

396

History 397

higher-derivative actions for GR The landscape of quantum gravityis gloomy

bull The Renaissance 1984ndash1994 Light comes back in the middle ofthe eighties In the covariant camp the various attempts to modifyGR to get rid of the infinities merge into string theory Perturbativestring theory finally delivers on the long search for a computable per-turbative theory for quantum gravitational scattering amplitudesTo be sure there are prices to pay such as the wrong dimensional-ity of spacetime and the introduction of supersymmetric particleswhich year after year are expected to be discovered but so far arenot But the result of a finite perturbation expansion long soughtafter is too good to be discarded just because the world insists inlooking different from our theoriesLight returns to shine on the canonical side as well Twenty yearsafter the WheelerndashDeWitt equation LQG finally provides a versionof the theory sufficiently well defined for performing explicit com-putations Here as well we are far from a complete and realistictheory and scattering amplitudes for the moment canrsquot be com-puted at all but the excitement for having a rigorously definednonperturbative generally covariant and background-independentquantum field theory in which physical expectation values can becomputed is strong

bull Nowadays 1995c Both string theory and LQG grow strongly fora decade until in the middle of the nineties they begin to deliverphysical results The BekensteinndashHawking black-hole entropy for-mula is derived within both approaches virtually simultaneouslyLQG leads to the computation of the first Planck-scale quantita-tive physical predictions the spectra of the eigenvalues of area andvolumeThe sum-over-histories tradition in the meantime is not dead Inspite of the difficulties of the euclidean integral it remains as a ref-erence idea and guides the development of several lines of researchfrom the discrete lattice-like approaches to the ldquostate sumrdquo for-mulation of topological theories Eventually the last motivate thespinfoam formulation a translation of LQG into a Feynman sum-over-histories formOther ideas develop in the meanwhile most notably noncommuta-tive geometry which finds intriguing points of contact with stringtheory towards the end of the decadeThe century closes with two well-developed contenders for a quan-tum theory of gravity string theory and LQG as well as a set of

398 Appendix B

intriguing novel new ideas that go from noncommutative geometryto the null surfaces formulation of GR to the attempt to mergestrings and loops And even on a very optimistic note the birth of anew line of research the self-styled ldquoquantum gravity phenomenol-ogyrdquo which investigates the possibility that Planck-scale type mea-surements might be within reach And thus that perhaps we couldfinally know which of the theoretical hypotheses if any make sense

I now describe the various periods and their main steps in more detail

B21 The Prehistory 1930ndash1957

General relativity is discovered in 1915 quantum mechanics in 1926 Afew years later around 1930 Born Jordan and Dirac are already capableof formalizing the quantum properties of the electromagnetic field Howlong did it take to realize that the gravitational field should presumablybehave quantum mechanically as well Almost no time already in 1916Einstein points out that quantum effects must lead to modifications inthe theory of general relativity [340] In 1927 Oskar Klein suggests thatquantum gravity should ultimately modify the concepts of space and time[341] In the early thirties Rosenfeld [342] writes the first technical pa-pers on quantum gravity applying Paulirsquos method for the quantizationof fields with gauge groups to the linearized Einstein field equations Therelation with a linear spin-2 quantum field is soon unraveled in the worksof Fierz and Pauli [343] and the spin-2 quantum of the gravitational fieldis already a familiar notion in the thirties Its name ldquogravitonrdquo is al-ready in use in 1934 when it appears in a paper by Blokhintsev andGalrsquoperin [344] (published in the ideological magazine Under the Bannerof Marxism) Bohr considers the idea of identifying the neutrino and thegraviton In 1938 Heisenberg [345] points out that the fact that the grav-itational coupling constant is dimensional is likely to cause problems withthe quantum theory of the gravitational field

The history of these early explorations of the quantum properties ofspacetime has recently been reconstructed by John Stachel [346] In par-ticular John describes in his paper the extensive but largely neglectedwork conducted in the middle thirties by a Russian physicist MatveiPetrovich Bronstein Persistent rumors claim that Bronstein was a nephewof Leon Trotsky and that he hid this relation that became dangerous butGennady Gorelik (of the Center for Philosophy and History of Science atBoston University and Institute for the History of Science and Technol-ogy of the Russian Academy of Sciences) assures me that this rumor isfalse Bronstein re-derives the RosenfeldndashPauli quantization of the lin-ear theory but realizes that the unique features of gravitation require aspecial treatment when the full nonlinear theory is taken into account

History 399

He realizes that field quantization techniques must be generalized in sucha way as to be applicable in the absence of a background geometry Inparticular he realizes that the limitation posed by general relativity onthe mass density radically distinguishes the theory from quantum elec-trodynamics and would ultimately lead to the need to ldquoreject riemanniangeometryrdquo and perhaps also to ldquoreject our ordinary concepts of space andtimerdquo [347] The reason Bronstein has remained unknown for so long haspartly to do with the fact that he was executed by the Soviet State Secu-rity Agency (the NKVD) at the age of 32 I am told that in Russia somestill remember Bronstein as ldquosmarter than Landaurdquo (but Gorelik doubtsthis opinion could be shared by a serious physicist) For a discussion ofBronsteinrsquos early work in quantum gravity see [348]

References and many details on these pioneering times are in the fas-cinating paper by John Stachel mentioned above Here I pick up thehistorical evolution after World War II In particular I start from 1949a key year for the history of quantum gravity

1949ndash Peter Bergmann starts his program of phase space quantization of

nonlinear field theories [349] He soon realizes that physical quantumobservables must correspond to coordinate-independent quantities only[350] The search for these gauge-independent observables is started in thegroup that forms around Bergmann at Brooklyn Polytechnic and then inSyracuse For instance Ted Newman develops a perturbation approachfor finding gauge-invariant observables order by order [351] The groupstudies the problems raised by systems with constraints and reaches aremarkable clarity unfortunately often forgotten later on on the problemof what are the observables in general relativity The canonical approachto quantum gravity is born

ndash Bryce DeWitt completes his thesis He applies Schwingerrsquos covariantquantization to the gravitational field

ndash Dirac presents his method for treating constrained hamiltonian sys-tems [113]

1952ndash Following the pioneering works of Rosenfeld Fierz and Pauli Gupta

[352] develops systematically the ldquoflat-space quantizationrdquo of the grav-itational field The idea is simply to introduce a fictitious ldquoflat spacerdquothat is Minkowski metric ημν and quantize the small fluctuations of themetric around Minkowski hμν = gμν minus ημν The covariant approach isfully born The first difficulty appears immediately when searching forthe propagator because of gauge invariance the quadratic term of thelagrangian is singular as for the electromagnetic field Guptarsquos treatmentuses an indefinite norm state space as for the electromagnetic field

400 Appendix B

1957ndash Charles Misner introduces the ldquoFeynman quantization of general rel-

ativityrdquo [353] He quotes John Wheeler for suggesting the expressionint

exp(ih)(Einstein action) d(field histories) (B1)

and studies how to have a well-defined version of this idea Misnerrsquos paper[353] is very remarkable in many respects It explains with complete claritynotions such as why the quantum hamiltonian must be zero why theindividual spacetime points are not defined in the quantum theory andthe need of dealing with gauge invariance in the integral Even moreremarkably the paper opens with a discussion of the possible directionsfor quantizing gravity and lists the three lines of directions ndash covariantcanonical and sum-over-histories ndash describing them almost precisely withthe same words we would today1

At the end of the fifties all the basic ideas and the research programsare clear It is only a matter of implementing them and seeing if theywork The implementation however turns out to be a rather herculeantask that requires the ingenuity of people of the caliber of Feynman andDeWitt on the covariant side and of Dirac and DeWitt on the canonicalside

B22 The Classical Age 1958ndash1969

1958ndash The Bergmann group [129] and Dirac [113 114] work out the gen-

eral hamiltonian theory of a constrained system For a historical recon-struction of this achievement see [354] At the beginning Dirac and theBergmann group work independently The present double classificationinto primary and secondary constraints and into first- and second-classconstraints still reflects this original separation

1959ndash By 1959 Dirac has completely unraveled the canonical structure of

GR [130]

1961ndash Arnowitt Deser and Misner complete what we now call the ADM for-

mulation of GR namely its hamiltonian version in appropriate variables

1To be sure Misner lists a fourth approach as well based on the Schwinger equationsfor the variation of the propagator but notices that ldquothis method has not been appliedindependently to general relativityrdquo a situation that might have changed only veryrecently [145 146]

History 401

which greatly simplify the hamiltonian formulation and make its geomet-ric reading transparent [131]

In relation to the quantization Arnowitt Deser and Misner present aninfluential argument for the finiteness of the self-energy of a point-particlein classical GR and use it to argue that nonperturbative quantum gravityshould be finite

ndash Tullio Regge defines the Regge calculus [285]

1962ndash Feynman attacks the task of computing transition amplitudes in

quantum gravity He shows that tree-amplitudes lead to the physics oneexpects from the classical theory [355]

ndash DeWitt starts developing his background field methods for the com-putation of perturbative transition amplitudes [356]

ndash Bergmann and Komar clarify what one should expect from a Hilbertspace formulation of GR [357]

ndash Following the ADM methods Peres writes the HamiltonndashJacobi for-mulation of GR [358]

G2(qabqcd minus12qacqbd)

δS(q)δqac

δS(q)δqbd

+ det q R[q] = 0 (B2)

which is our fundamental equation (49) written in metric variables andwill soon lead to the WheelerndashDeWitt equation here qab is the ADM3-metric

1963ndash John Wheeler realizes that the quantum fluctuations of the gravita-

tional field must be short-scale fluctuations of the geometry and intro-duces the physical idea of spacetime foam [359] Wheelerrsquos Les Houcheslecture notes are remarkable in many respects and are the source of manyof the ideas still current in the field To mention two others ldquoProblem56rdquo suggests that gravity in 2 + 1 dimensions may not be so trivial afterall and indicates it may be an interesting model to explore ldquoProblem57rdquo suggests studying quantum gravity by means of a Feynman integralover a spacetime lattice

Julian Schwinger introduces the tetrad spin-connection formulation inquantum gravity [80] On the strict relation between this formalism andYangndashMills theories he writes

Weyl the originator of the electromagnetic gauge invariance princi-ple also recognized that the gravitational field can be characterizedby a kind of gauge transformation [79] This is the possibility of alter-ing freely at each point the orientation of a local Lorentz coordinate

402 Appendix B

frame while suitably transforming certain gravitational potentials Ina subsequent development Yang and Mills introduced an arbitrarilyoriented 3d isotopic space at each spacetime point The occasional re-mark that the gravitational field can be viewed as a YangndashMills fieldis thus rather anachronistic

1964ndash Penrose introduces the idea of spin networks and of a discrete struc-

ture of space controlled by SU(2) representation theory The constructionexists only in the form of a handwritten manuscript It gets published onlyin 1971 [181] The idea will surprisingly re-emerge 25 years later whenspin networks will be found to label the states of LQG

ndash Beginning to study loop corrections to GR amplitudes Feynman ob-serves that unitarity is lost for naive diagrammatic rules DeWitt [360]develops the combinatorial means to correct the quantization (requir-ing independence of diagrams from the longitudinal parts of propaga-tors) These correction terms can be put in the form of loops of fictitiousfermionic particles the FaddeevndashPopov ghosts [361] The key role ofDeWitt in this context was emphasized by Veltman in 1974 [362]

Essentially due to this and some deficiencies in his combinatorialmethods Feynman was not able to go beyond one closed loop DeWittin his 1964 Letter and in his subsequent monumental work derivedmost of the things that we know of now That is he considered thequestion of a choice of gauge and the associated ghost particle Indeedhe writes the ghost contribution in the form of a local Lagrangiancontaining a complex scalar field obeying Fermi statistics Somewhatillogically this ghost is now called the FaddeevndashPopov ghost

The designation ldquoFaddeevndashPopov ghostrdquo is far from illogical in compar-ison with the complicated combinatorics of DeWitt the FaddeevndashPopovapproach has the merit of a far greater technical simplicity and a trans-parent geometric interpretation which justifies its popularity It is onlyin the work of Faddeev that the key role played by the gauge orbits asthe true dynamical variables is elucidated [363]

1967ndash Bryce DeWitt publishes the ldquoEinsteinndashSchrodinger equationrdquo [214]

((hG)2(qabqcd minus

12qacqbd)

δ

δqac

δ

δqbdminus det q R[q]

)Ψ(q) = 0 (B3)

which is the main quantum gravity equation (61) in metric variablesBryce will long denote this equation as the ldquoEinsteinndashSchrodinger equa-tionrdquo attributing it to Wheeler ndash while John Wheeler called it the DeWittequation ndash until finally in 1988 at an Osgood Hill conference DeWitt

History 403

gives up and calls it what everybody else had been calling it since thebeginning the ldquoWheelerndashDeWitt equationrdquo

The story of the birth of the WheelerndashDeWitt equation is worth tellingIn 1965 during an air trip John had to stop for a short time at theRaleighndashDurham airport in North Carolina Bryce lived nearby Johnphoned Bryce and proposed to meet at the airport during the wait be-tween two flights Bryce showed up with the HamiltonndashJacobi equationfor GR published by Peres in 1962 and mumbled the idea of doing pre-cisely what Schrodinger did for the hydrogen atom replace the squareof the derivative with a second derivative Surprising Bryce John wasenthusiastic (John is often enthusiastic of course) and declared immedi-ately that the equation of quantum gravity had been found The paperwith the equation the first of Brycersquos celebrated 1967 quantum gravitytrilogy [214 364] was submitted in the spring of 1966 but its publicationwas delayed until 1967 Among the reasons for the delay apparently weredifficulties with publication charges

ndash John Wheeler discusses the idea of the wave function Ψ(q) on thespace of the ldquo3-geometryrdquo q and the notion of superspace the space ofthe 3-geometries in [38]

ndash Roger Penrose starts twistor theory [365]ndash The project of DeWitt and Feynman is concluded A complete and

consistent set of Feynman rules for GR are written down [361 364]

1968ndash Ponzano and Regge define a quantization of 3d euclidean GR [283]

The model will lead to major developments

1969ndash Developing an idea in Brycersquos paper on canonical quantum gravity

Charles Misner starts quantum cosmology the game of truncating theWheelerndashDeWitt equation to a finite number of degrees of freedom [366]The idea is beautiful but it will develop into a long-lasting industry fromwhich after a while little new will be understood

The decade closes with the main lines of the covariant and the canon-ical theory clearly defined It will soon become clear that neither theoryworks

B23 The Middle Ages 1970ndash1983

1970ndash The decade of the seventies opens with a word of caution Reviving a

point made by Pauli a paper by Zumino [367] suggests that the quanti-zation of GR may be problematic and might make sense only by viewing

404 Appendix B

GR as the low-energy limit of a more general theory More than thirtyyears later opinions still diverge on whether this is true

1971ndash Using the technology developed by DeWitt and Feynman for gravity

trsquoHooft and Veltman decide to study the renormalizability of GR Almostas a warm-up exercise they consider the renormalization of YangndashMillstheory and find that the theory is renormalizable ndash a result that wonthem the Nobel prize [368] In a sense one can say that the first physicalresult of the research in quantum gravity is the proof that YangndashMillstheory is renormalizable

ndash David Finkelstein writes his inspiring ldquospacetime coderdquo series of pa-pers [369] (which among other ideas discuss quantum groups)

1973ndash Following the program initiated with Veltman in 1971 trsquoHooft finds

evidence of nonrenormalizable divergences in GR with matter fieldsShortly after trsquoHooft and Veltman as well as Deser and Van Nieuwen-huizen confirm the evidence [370]

1974ndash Hawking announces the derivation of black-hole radiation [248] A

(macroscopic) Schwarzschild black hole of mass M emits thermal radia-tion at the temperature (825) The result comes as a surprise anticipatedonly by the observation by Bekenstein a year earlier that entropy is nat-urally associated with black holes and thus they could be thought of insome obscure sense as ldquohotrdquo [247] and by the BardeenndashCarterndashHawkinganalysis of the analogy between laws of thermodynamics and dynamicalbehavior of black holes [246] Hawkingrsquos result is not directly connectedto quantum gravity ndash it is a skillful application of quantum field theoryin curved spacetime ndash but has a very strong impact on the field It fostersan intense activity in quantum field theory in curved spacetime it opensa new field of research in ldquoblack-hole thermodynamicsrdquo and it opens thequantum-gravitational problems of understanding the statistical origin ofthe black-hole (the BekensteinndashHawking) entropy (827)

An influential clarifying and at the same time intriguing paper is writ-ten two years later by Bill Unruh The paper points out the existence ofa general relation between accelerated observers quantum theory grav-ity and thermodynamics [371] Something deep about Nature should behidden in this tangle of problems but we do not yet know what

1975ndash It becomes generally accepted that GR coupled to matter is not

renormalizable The research program started with Rosenfeld Fierz andPauli is dead

History 405

1976ndash A first attempt to save the covariant program is made by Steven Wein-

berg who explores the idea of asymptotic safety [39] developing earlierideas from Giorgio Parisi [372] Kenneth Wilson and others suggestingthat nonrenormalizable theories could nevertheless be meaningful

ndash To resuscitate the covariant theory even if in modified form thepath has already been indicated find a high-energy modification of GRPreserving general covariance there is not much one can do to modify GRAn idea that attracts much enthusiasm is supergravity [373] it seems thatby simply coupling a spin-32 particle to GR namely with the action (infirst-order form)

S[gΓ ψ] =int

d4xradicminusg

(1

2GRminus i

2εμνρσ ψμγ5γνDρψσ

) (B4)

one can get a theory finite even at two loopsndash Supersymmetric string theory is born [349]

1977ndash Another independent idea is to keep the same kinematics and change

the action The obvious thing to do is to add terms proportional to thedivergences Stelle proves that an action with terms quadratic in the cur-vature

S =int

d4xradicminusg

(αR + βR2 + γRμνRμν

) (B5)

is renormalizable for appropriate values of the coupling constants [375]Unfortunately precisely for these values of the constants the theory is badIt has negative energy modes that make it unstable around the Minkowskivacuum and not unitary in the quantum regime The problem becomesto find a theory renormalizable and unitary at the same time or to cir-cumvent nonunitarity

1978ndash The Hawking radiation is soon re-derived in a number of ways

strongly reinforcing its credibility Several of these derivations point tothermal techniques [376] thus motivating Hawking [40] to revive theWheelerndashMisner ldquoFeynman quantization of general relativityrdquo [353] in theform of a ldquoeuclideanrdquo integral over riemannian 4-geometries g

Z =int

Dg eminusint radic

gR (B6)

Time-ordering and the concept of positive frequency are incorporated intothe ldquoanalytic continuationrdquo to the euclidean sector The hope is double todeal with topology change and that the euclidean functional integral willprove to be a better calculation tool than the WheelerndashDeWitt equation

406 Appendix B

1980ndash Within the canonical approach the discussion focuses on understand-

ing the disappearance of the time coordinate from the WheelerndashDeWitttheory The problem has actually nothing to do with quantum gravitysince the time coordinate disappears in the classical HamiltonndashJacobiform of GR as well and in any case physical observables are coordinateindependent and thus in particular independent from the time coordi-nate in whatever correct formulation of GR But in the quantum contextthere is no single spacetime as there is no trajectory for a quantum par-ticle and the very concepts of space and time become fuzzy This factraises much confusion and a vast interesting discussion (whose many con-tributions I can not possibly summarize here) on the possibility of doingmeaningful fundamental physics in the absence of a fundamental notionof time For early references on the subject see for instance [42 44]

1981ndash Polyakov [377] shows that the cancellation of the conformal anomaly

in the quantization of the string action

S =1

4παprime

intd2σ

radicg gμνpartμX

apartνXbηab (B7)

leads to the critical dimension A new problem is created how to recoverour 4-dimensional world from a string theory which is defined in the crit-ical dimension

1983ndash The hope is still high for supergravity now existing in various ver-

sions as well as for higher-derivative theories whose rescue from nonuni-tarity is explored using a number of ingenious ideas (large-N expansionslarge-d expansions LeendashWick mechanisms ) At the tenth GRG con-ference in Padova in 1983 two physicists of indisputable seriousness GaryHorowitz and Andy Strominger summarize their contributed paper [378]with the words

In sum higher-derivative gravity theories are a viable option forresolving the problem of quantum gravity

At the same conference supergravity is vigorously advertised as the fi-nal solution of the quantum gravity puzzle But very soon it becomesclear that supergravity is nonrenormalizable at higher loops and thathigher-derivatives theories do not lead to viable perturbative expansionsExcitement hope and hype fade away

In its version in 11 dimensions supergravity will find new importancein the late 1990s in connection with string theory High-derivative cor-rections will also re-appear in the low-energy limit of string theory

History 407

ndash Hartle and Hawking [157] introduce the notion of the ldquowave func-tion of the Universerdquo and the ldquono-boundaryrdquo boundary condition for theHawking integral opening up a new intuition on quantum gravity andquantum cosmology But the euclidean integral does not provide a wayof computing genuine field theoretical quantities in quantum gravity anybetter than the WheelerndashDeWitt equation and the atmosphere at themiddle of the eighties is again rather gloomy On the other hand JimHartle [26] develops the idea of a sum-over-histories formulation of GRinto a fully fledged extension of quantum mechanics to the generally co-variant setting The idea will later be developed and formalized by ChrisIsham [379]

ndash Sorkin introduces his poset approach to quantum gravity [380]

B24 The Renaissance 1984ndash1994

1984ndash Green and Schwarz realize that strings might describe ldquoour Universerdquo

[381] Excitement starts to build up around string theory in connectionwith the unexpected anomaly cancellation and the discovery of the het-erotic string [382]

ndash The relation between the ten-dimensional superstrings theory andfour-dimensional low-energy physics is studied in terms of compactifica-tion on CalabyndashYau manifolds [383] and orbifolds The dynamics of thechoice of the vacuum remains unclear but the compactification leads to4d chiral models resembling low-energy physics

ndash Belavin Polyakov and Zamolodchikov publish their analysis of con-formal field theory [384]

1986ndash Goroff and Sagnotti [29] finally compute the two-loop divergences of

pure GR definitely nailing the corpse of pure GR perturbative quantumfield theory into its coffin the divergent term in the effective action is

ΔS =209

737 280π4

1ε

intd4x

radicminusg RμνρσR

ρσεθR

εθμν (B8)

ndash Penrose suggests that the wave function collapse in quantum mechan-ics might be of quantum-gravitational origin [385] The idea is radical andimplies a re-thinking of the basis of mechanics Remarkably the idea maybe testable work is today in progress to study the feasibility of an exper-imental test

ndash String field theory represents a genuine attempt to address the mainproblem of string theory finding a fundamental background-independentdefinition of the theory [386] The string field path however turns out tobe hard

408 Appendix B

ndash The connection formulation of GR is developed by Abhay Ashtekar[132] on the basis of some results by Amitaba Sen [82] At the time thisis denoted the ldquonew variablesrdquo formulation It is a development in clas-sical general relativity but it has long-ranging consequences on quantumgravity as the basis of LQG

1987ndash Fredenhagen and Haag explore the general constraint that general

covariance puts on quantum field theory [387]ndash Green Schwarz and Witten publish their book on superstring theory

In the gauge in which the metric has no superpartner the superstringaction is

S =1

4παprime

intd2σ

radicg

(gμνpartμX

apartνXb minus iψaγμpartμψ

b)ηab (B9)

Interest in the theory grows very rapidly To be sure string theory stillobtains a very small place at the 1991 Marcel Grossmann meeting [388]But the research in supergravity and higher-derivative theories has mergedinto strings and string theory is increasingly viewed as a strong compet-ing candidate for the quantum theory of the gravitational field As aside product many particle physicists begin to study general relativityor at least some bits of it Strings provide a consistent perturbative the-ory The covariant program is fully re-born The problem becomes under-standing why the world described by the theory appears so different fromours

1988ndash Ted Jacobson and Lee Smolin find loop-like solutions to the Wheelerndash

DeWitt equation formulated in the connection formulation [178] openingthe way to LQG

ndash The ldquoloop representation of quantum general relativityrdquo is introducedin [176 177] It is based on the new connection formulation of GR [132]on the JacobsonndashSmolin solutions [178] and on Chris Ishamrsquos ideas onthe need of nongaussian or nonFock representations in quantum gravity[43] Loop quantization had been previously and independently developedby Rodolfo Gambini and his collaborators for YangndashMills theories [179]In the gravitational context the loop representation leads immediately totwo surprising results an infinite family of exact solutions of the WheelerndashDeWitt equation is found and knot theory controls the physical quantumstates of the gravitational field Classical knot theory with its extensionsbecomes a branch of mathematics relevant to describe the diff-invariantstates of quantum spacetime [215] The theory transforms the oldWheelerndashDeWitt theory into a formalism that can be concretely used tocompute physical quantities in quantum gravity The canonical programis fully re-born Nowadays the theory is called ldquoloop quantum gravityrdquo

History 409

ndash Ed Witten introduces the notion of topological quantum field theory(TQFT) [389] In a celebrated paper [390] he uses a TQFT to give afield theoretical representation of the Jones polynomial a knot theoryinvariant The expression used by Witten has an interpretation in LQGit can be seen as the ldquoloop transformrdquo of a quantum state given by theexponential of the ChernndashSimon functional [215]

Formalized by Atiyah [391] the idea of TQFT will have beautiful de-velopments and will strongly influence later developments in quantumgravity General topological theories in any dimensions and in particularBF theory are introduced by Gary Horowitz shortly afterwards [392]

ndash Witten finds an ingenious way of quantizing GR in 2 + 1 spacetimedimensions [393] (thus solving ldquoProblem 56rdquo of the 1963 Wheelerrsquos LesHouches lectures) opening up a big industry of analysis of the theory(for a review see [394]) The quantization method is partially a sumover histories and partially canonical Covariant perturbative quantizationseemed to fail for this theory The theory had been studied a few yearsearlier by Deser Jackiw trsquoHooft Achucarro Townsend and others [395]

1989ndash Amati Ciafaloni and Veneziano find evidence that string theory

implies that distances smaller than the Planck scale cannot be probed[396]

ndash In the string world there is excitement for some nonperturbativemodels of strings ldquoin 0 dimensionrdquo equivalent to 2d quantum gravity[397] The excitement dies fast as is often the case but the models will re-emerge in the nineties [398] and will also inspire the spinfoam formulationof quantum gravity [278]

1992ndash Turaev and Viro [286] define a state sum that on the one hand is a

rigorously defined TQFT and on the other hand can be seen as a reg-ulated and well-defined version of the PonzanondashRegge [283] quantizationof 2+1 gravity Turaev and Ooguri [289] soon find a 4d extension whichwill have a remarkable impact on later developments

ndash The notion of weave is introduced in LQG [189] It is evidence of adiscrete structure of spacetime emerging from LQG The first exampleof a weave which is considered is a 3d mesh of intertwined rings Notsurprisingly the intuition was already in Wheeler (See Figure B1 takenfrom Misner Thorne and Wheeler [399])

1993ndash Gerard rsquot Hooft introduces the idea of holography developed by Lenny

Susskind [375] According to the ldquoholographic principlerdquo the informationon the physical state in the interior of a region can be represented onthe regionrsquos boundary and is limited by the area of this boundary This

410 Appendix B

Fig B1 The weave in Wheelerrsquos vision From Ref [374]

principle can be also interpreted as referring to the information on thesystem accessible from the outside of the region in which case it makesmuch more sense to me

1994ndash Noncommutative geometry often indicated as a tool for describing

certain aspects of Planck-scale geometry finds a strict connection to GR inthe work of Alain Connes Remarkably the ConnesndashChamseddine ldquospec-tral actionrdquo just the trace of a simple function of a suitably defined Dirac-like operator D

S = tr[f(D2(hG))] (B10)

where f is the characteristic function of the [0 1] interval turns out toinclude the standard model action as well as the EinsteinndashHilbert action[401]

B25 Nowadays 1995ndash

1995ndash Nonperturbative aspects of string theory begin to appear branes

[402] dualities [403] the matrix model formulation of M theory[404] (for a review see for instance [405]) The interest in stringsbooms At the plenary conference of a meeting of the American Mathe-matical Society in Baltimore Ed Witten claims that

History 411

The mathematics of the next millennium will be dominated by stringtheory

causing a few eyebrows to raiseThe various dualities appear to relate the different versions of the the-

ory pointing to the existence of a unique fundamental theory The actualconstruction of the fundamental background-independent theory how-ever is still missing and string theory exists so far only in the form of anumber of (related) expansions over assigned backgrounds

ndash Two results in loop gravity appear (i) the overcompleteness of theloop basis is resolved by the discovery of the spin network basis [171] (ii)eigenvalues of area and volume are computed [191] The latter result israpidly extended and derived in a number of alternative ways

The rigorous mathematical framework for LQG starts to be developed[200 201]

ndash Ted Newman and his collaborators introduce the Null Surface For-mulation of GR [406]

1996ndash The BekensteinndashHawking black-hole entropy (822) is computed

within LQG as well as within string theory almost at the same timeThe loop result is obtained by computing the number of (spin network)

states which endow a 2-sphere with a given area [239 249] as well asby loop quantizing the classical theory of the field outside the hole andstudying the boundary states [238] These gravitational surface states[254] can be identified with the states of a ChernndashSimon theory on asurface with punctures [253] The computation is valid for various realisticblack holes The 14 factor in (827) is obtained by fixing the Immirziparameter

In string theory the computation exploits a strong couplingweak cou-pling duality which in certain supersymmetric configurations preservesthe number of states the physical black hole is in a strong coupling situ-ation but the number of its microstates can be computed in a weak-fieldconfiguration that has the same charges at infinity One obtains preciselythe 14 factor of (827) as well as other aspects of the Hawking radiationphenomenology [407] However the calculation method is indirect andworks only for extremal or near-extremal black holes

ndash A rigorously defined finite and anomaly-free hamiltonian constraintoperator is constructed by Thomas Thiemann in LQG [133] Some doubtsare raised on whether the classical limit of this theory is in fact GR (theissue is still open) but the construction defines a consistent generallycovariant quantum field theory in 4d

ndash Intriguing state sum models obtained modifying a TQFT are pro-posed by Barrett and Crane Reisenberger Iwasaki and others as a

412 Appendix B

tentative model for quantum GR All these models appear as sums ofldquospinfoamsrdquo branched surfaces carrying spins

ndash The loop representation is ldquoexponentiatedrdquo a la Feynman givingrise again to a spinfoam model corresponding to canonical LQG Thesedevelopments revive the sum-over-histories approach

1997ndash Intriguing connections between noncommutative geometry and string

theory appear [408]ndash There is a lively discussion on the difficulties of the lattice approaches

in finding a second-order phase transition [409]

1998ndash Juan Maldacena shows [410] that the large-N limit of certain confor-

mal field theories includes a sector describing supergravity on the productof anti-deSitter spacetimes and spheres He conjectures that the com-pactifications of Mstring theory on an anti-deSitter spacetime is dualto a conformal field theory on the spacetime boundary This leads to anew proposal for defining M theory itself in terms of the boundary the-ory an effort to reach background independence (for M theory) usingbackground-dependent methods (for the boundary theory)

A consequence of this ldquoMaldacena conjecturerdquo is an explosion of interestfor rsquot Hooft and Susskind holographic principle (see year 1993)

ndash Two papers in the influential journal Nature [265] raise the hopethat seeing spacetime-foam effects and testing quantum gravity theoriesmight not be as forbidding as usually assumed The idea is that thereare a number of different instances (the neutral kaon system gamma-rayburst phenomenology interferometers ) in which presently operatingmeasurement or observation devices or instruments that are going to besoon constructed involve sensitivity scales comparable to ndash or not toofar from ndash the Planck scale [266] If this direction fails testing quantumgravity might require the investigation of very early cosmology [411]

1999 I stop here because too-recent history is not yet history

B3 The divide

The lines of research that I have summarized in Appendix B2 have foundmany points of contact in the course of their development and have oftenintersected For instance there is a formal way of deriving a sum-over-histories formulation from a canonical theory and vice versa the perturba-tive expansion can also be obtained by expanding the sum-over-historiesstring theory today faces the problem of finding its nonperturbative

History 413

formulation and thus the typical problems of a canonical theory andLQG has mutated into the spinfoam models a sum-over-histories formu-lation using techniques that can be traced to a development of stringtheory of the early nineties However in spite of this continuous cross-fertilization the three main lines of development have kept their essentialseparateness

The three directions of investigation were already clearly identified byCharles Misner in 1959 [353] In the concluding remark of the ConferenceInternationale sur les Theories Relativistes de la Gravitation in 1963Peter Bergmann notes [412]

In view of the great difficulties of this program I consider it a verypositive thing that so many different approaches are being broughtto bear on the problem To be sure the approaches we hope willconverge to one goal

This was 40 years ago The divide is particularly strong between the covariant line of research

more connected to the particle-physics tradition and the canonicalsum-over-histories one more connected to the relativity tradition This dividehas remained through over 70 years of research in quantum gravity Hereis a typical comparison arbitrarily chosen among many On the particle-physics side at the First Marcel Grossmann Meeting Peter van Neuwen-huizen writes [413]

gravitons are treated on exactly the same basis as other parti-cles such as photons and electrons In particular particles (includinggravitons) are always in flat Minkowski space and move as if theyfollowed their geodesics in curved spacetime because of the dynamicsof multiple graviton exchange Pure relativists often become some-what uneasy at this point because of the following two aspects entirelypeculiar to gravitation (1) One must decide before quantizationwhich points are spacelike separated but it is only after quantizationthat the fully quantized metric field can tell us this spacetime struc-ture (2) In a classical curved background one needs positiveand negative frequency solutions but in non-stationary spacetimes itis not clear whether one can define such solutions The strategy of par-ticle physicists has been to ignore these problems for the time beingin the hope that they will ultimately be resolved in the final theoryConsequently we will not discuss them any further

On the relativity side Peter Bergmann comments [414]

The world-point by itself possesses no physical reality It acquires real-ity only to the extent that it becomes the bearer of specific propertiesof the physical fields imposed on the spacetime manifold

The conceptual divide is huge Partially it reflects the different under-standing of the world held by the particle-physics community on the one

414 Appendix B

hand and the relativity community on the other The two communitieshave made repeated and sincere efforts to talk to each other and under-stand each other But the divide remains Both sides have the feeling thatthe other side is incapable of appreciating something basic and essentialthe structure of QFT as it has been understood over half a century of in-vestigation on the particle-physics side the novel physical understandingof space and time that has appeared with GR on the relativity side Bothsides expect that the point of view of the other will turn out at the endof the day to be not very relevant One side because the experience withQFT is on a fixed metric spacetime and thus is irrelevant in a genuinelybackground-independent context The other because GR is only a low-energy limit of a much more complex theory and thus cannot be takentoo seriously as an indication about the deep structure of Nature Hope-fully the recent successes of both lines will force the two sides finallyto face the problems that the other side considers prioritary backgroundindependence on the one hand control of a perturbation expansion onthe other

After 70 years of research there is no consensus no established theoryand no theory that has yet received any direct or indirect experimentalsupport In the course of 70 years many ideas have been explored fashionshave come and gone the discovery of the Holy Grail has been several timesannounced with much later scorn Ars longa vita brevis

However in spite of its age the research in quantum gravity does notseem to have been meandering meaninglessly when seen in its entiretyOn the contrary one sees a logic that has guided the development of theresearch from the early formulation of the problem and the research di-rections in the fifties to nowadays The implementation of the programshas been extremely laborious but has been achieved Difficulties haveappeared and solutions have been proposed which after much difficultyhave led to the realization at least partial of the initial hopes It wassuggested in the early seventies that GR could perhaps be seen as thelow-energy theory of a theory without uncontrollable divergences today30 years later such a theory ndash string theory ndash is known In 1957 CharlesMisner indicated that in the canonical framework one should be able tocompute eigenvalues and in 1995 37 years later eigenvalues were com-puted ndash within loop quantum gravity The road is not yet at the endmuch remains to be understood some of the current developments mightlead nowhere But looking at the entire development of the subject it isdifficult to deny that there has been progress

Appendix C

On method and truth

I collect in this appendix some simple reflections on scientific methodology and onthe content of scientific theories relevant for quantum gravity In particular I try tomake more explicit the methodological assumptions at the root of some of the researchdescribed in this book and to give it some justification

I am no professional philosopher and what follows has no ambition in that senseI am convinced however of the utility of a dialog between physics and philosophyThis dialog has played a major role during the other periods in which science has facedfundamental problems I think that most physicists underestimate the effect of theirown epistemological prejudices on their research And many philosophers underestimatethe effect ndash positive or negative ndash they have on fundamental research On the one handa more acute philosophical awareness would greatly help the physicists engaged infundamental research As I have argued in Chapter 1 during the second half of thetwentieth century fundamentals were clear in theoretical physics and the problems weretechnical but today foundational problems are back on the table as they were at thetime of Newton Faraday Heisenberg and Einstein These physicists couldnrsquot certainlyhave done what they have done if they werenrsquot nurtured by (good or bad) philosophyOn the other hand I wish contemporary philosophers concerned with science would bemore interested in the ardent lava of the fundamental problems science is facing todayIt is here I believe that stimulating and vital issues lie

C1 The cumulative aspects of scientific knowledge

Part of the reflection about the science of the last decades has emphasizedthe ldquononcumulativerdquo aspect in the development of scientific knowledgethe evolution of scientific theories is marked by large or small break-ing points in which to put it crudely empirical facts are reorganizedwithin new theories which are to some extent ldquoincommensurablerdquo withrespect to their antecedent These ideas ndash correctly understood or misun-derstood ndash have had a strong influence on the physicists

The approach to quantum gravity described in this book assumes a dif-ferent reading of the evolution of scientific knowledge Indeed I have basedthe discussion on quantum gravity on the expectation that the central

415

416 Appendix C

physical tenets of QM and GR represent our best guide for accessingeven the extreme and unexplored territories of the quantum-gravitationalregime In my opinion the emphasis on the incommensurability betweentheories has clarified an important aspect of science but risks to obscuresomething of the internal logic according to which historically physicshas extended knowledge There is a subtle but very definite cumulativeaspect in the progress of physics which goes far beyond the growth in thevalidity and precision of the empirical content of the theories In movingfrom a theory to the theory that supersedes it we do not save only theverified empirical content of the old theory but more This ldquomorerdquo is acentral concern for good physics It is the source I think of the spectac-ular and undeniable predictive power of theoretical physics I think thatby playing it down one risks misleading theoretical research into a lesseffective methodology

Let me illustrate this point with a historical case There was a problembetween Maxwell equations and Galileo transformations There were twoobvious ways out To consider Maxwell theory as a theory of limitedvalidity a phenomenological theory of some yet-to-be-discovered aetherrsquosdynamics Or to consider that galilean equivalence of inertial systemshad limited validity thus accepting the idea that inertial systems arenot equivalent in electromagnetic phenomena Both directions are logicaland were pursued at the end of the nineteenth century Both are soundapplications of the idea that a scientific revolution changes in depth whatold theories teach us about the world Which of the two ways did Einsteinsuccessfully take

Neither For Einstein Maxwell theory was a source of awe He rhap-sodizes about his admiration for the theory For him Maxwell had openeda new window on the world Given the astonishing success of Maxwelltheory empirical (electromagnetic waves) technological (radio) as wellas conceptual (understanding the nature of light) Einsteinrsquos admirationis comprehensible But Einstein had a tremendous respect for Galileorsquosinsight as well Young Einstein was amazed by a book with Huygensrsquoderivation of collision theory virtually out of galilean invariance aloneEinstein understood that Galileorsquos great intuition ndash that the notion ofvelocity is only relative ndash could not be wrong I am convinced that in thisfaith of Einstein in the core of the great galilean discovery there is verymuch to learn for the philosophers of science as well as for the con-temporary theoretical physicists So Einstein believed the two theoriesMaxwell and Galileo and assumed that their tenets would hold far beyondthe regime in which they had been tested He assumed that Galileo hadgrasped something about the physical world which was simply correctAnd so had Maxwell Of course details had to be adjusted The coreof Galileorsquos insight was that all inertial systems are equivalent and that

On method and truth 417

velocity is relative not the details of the galilean transformations Einsteinknew the Lorentz transformations (found by Poincare) and was able tosee that they do not contradict Galileorsquos insight If there was contradictionin putting the two together the problem was ours we were surreptitiouslysneaking some incorrect assumption into our deductions He found the in-correct assumption which of course was that simultaneity could be welldefined It was Einsteinrsquos faith in the essential physical correctness of theold theories that guided him to his spectacular discovery

There are very many similar examples in the history of physics thatcould equally well illustrate this point Einstein found GR ldquoout of purethoughtrdquo having Newton theory on the one hand and special relativ-ity ndash the understanding that any interaction is mediated by a field ndash onthe other Dirac found quantum field theory from Maxwell equations andquantum mechanics Newton combined Galileorsquos insight that accelerationgoverns dynamics with Keplerrsquos insight that the source of the force thatgoverns the motion of the planets is the Sun The list could be long Inall these cases confidence in the insight that came with some theory orldquotaking a theory seriouslyrdquo led to major advances that greatly extendedthe original theory itself Far be it from me to suggest that there is any-thing simple or automatic in figuring out where the true insights areand in finding the way of making them work together But what I amsaying is that figuring out where the true insights are and finding theway of making them work together is the work of fundamental physicsThis work is grounded on confidence in the old theories not on a randomsearch for new ones

One of the central concerns of the modern philosophy of science is toface the apparent paradox that scientific theories change but are never-theless credible Modern philosophy of science is to some extent an af-tershock reaction to the fall of newtonian mechanics A tormented recog-nition that an extremely successful scientific theory can nevertheless beuntrue But I think that a notion of truth which is challenged by the eventof a successful physical theory being superseded by a more successful oneis a narrow-minded notion of truth

A physical theory is a conceptual structure that we develop and use inorder to organize read and understand the world and make predictionsabout it A successful physical theory is a theory that does so effectivelyand consistently In the light of our experience there is no reason not toexpect that a more effective conceptual structure might always exist Aneffective theory may always show its limits and be replaced by a betterone However a novel conceptualization cannot but rely on what theprevious one has already understood Thought is in constant evolutionand in constant reorganization It is not a static entity Science is itselfthe process of the evolution of thinking

418 Appendix C

When we move to a new city we are at first confused about its geog-raphy Then we find some reference points and we make a first roughmental map of the city in terms of these points Perhaps there is partof the city on the hills and part on the plain As time goes on the mapimproves There are moments in which we suddenly realize that we hadit wrong Perhaps there were two areas with hills and we were previ-ously confusing the two Or we had mistaken a large square called Earthsquare for the downtown while downtown was farther away around asquare called Sun square So we update the mental map Sometime laterwe have learned names and features of neighborhoods and streets andthe hills as references fade away The neighborhood structure of knowl-edge is more effective than the hillplain one The structure changesbut the knowledge increases And Earth square now we know it is notdowntown and we know it forever

There are discoveries that are forever That the Earth is not the cen-ter of the Universe that simultaneity is relative that absolute velocity ismeaningless That we do not get rain by dancing These are steps human-ity takes and does not take back Some of these discoveries amount simplyto clearing from our thinking wrong encrusted or provisional credencesBut also discovering classical mechanics or discovering electromagnetismor quantum mechanics are discoveries forever Not because the details ofthese theories cannot change but because we have discovered that a largeportion of the world admits to being understood in certain terms andthis is a fact that we will have to keep facing forever

One of the main theses of this book is that general relativity is theexpression of one of these insights which will stay with us ldquoforeverrdquo Theinsight is that the physical world does not have a stage that localiza-tion and motion are relational only that this background independenceis required for any fundamental description of our world

How can a theory be effective even outside the domain for which it wasfound How could Maxwell predict radio waves Dirac predict antimatterand GR predict black holes How can theoretical thinking be so magicallypowerful

It has been suggested that these successes are due to chance and seemgrand only because of the historically deformed perspective A sort ofdarwinian natural selection for theories has been suggested there arehundreds of theories proposed most of them die the ones that surviveare the ones remembered There is alway somebody who wins the lotterybut this is not a sign that humans can magically predict the outcome ofthe lottery My opinion is that such an interpretation of the developmentof science is unjust and worse misleading It may explain something butthere is more in science There are tens of thousands of persons playingthe lottery there were only two relativistic theories of gravity in 1916


when Einstein predicted that light would be deflected by the Sun preciselyby an angle of 175 seconds of arc Familiarity with the history of physicsI feel confident to claim rules out the lottery picture

I think that the answer is simpler Say somebody predicts that theSun will rise tomorrow and the Sun rises This successful prediction isnot a matter of chance there arenrsquot hundreds of people making randompredictions about all sorts of strange objects appearing at the horizonThe prediction that tomorrow the Sun will rise is sound However itcannot be taken for granted A neutron star could rush in close to thespeed of light and sweep the Sun away Who or what grants the rightof induction Why should I be confident that the Sun will rise just be-cause it has been rising so many times in the past I do not know theanswer to this question But what I know is that the predictive power ofa theory beyond its own domain is precisely of the same sort Simply welearn something about Nature and what we learn is effective in guidingus to predict Naturersquos behavior Thus the spectacular predictive powerof theoretical physics is nothing less and nothing more than commoninduction it follows from the successful assumption that there are reg-ularities in Nature at all levels The spectacular success of science inmaking predictions about territories not yet explored is as comprehensi-ble (or as incomprehensible) as my ability to predict that the Sun willrise tomorrow Simply Nature around us happens to be full of regular-ities that we recognize whether or not we understand why regularitiesexist at all These regularities give us strong confidence ndash although notcertainty ndash that the Sun will rise tomorrow as well as that the basic factsabout the world found with QM and GR will be confirmed not violatedin the quantum-gravitational regimes that we have not yet empiricallyprobed

This view is not dominant in theoretical physics nowadays Other at-titudes dominate The ldquopessimisticrdquo scientist has little faith in the pos-sibilities of theoretical physics because he worries that all possibilitiesare open and anything might happen between here and the Planck scaleThe ldquowildrdquo scientist observes that great scientists had the courage tobreak with ldquoold and respected assumptionsrdquo and to explore some novelldquostrangerdquo hypotheses From this observation the ldquowildrdquo scientist con-cludes that to do great science one has to explore all sorts of strangehypotheses and to violate respected ideas The wilder the hypothesis thebetter I think wildness in physics is sterile The greatest revolutionaries inscience were extremely almost obsessively conservative So was certainlythe greatest revolutionary Copernicus and so was Planck Copernicuswas pushed to the great jump from his pedantic labor on the minute tech-nicalities of the ptolemaic system (fixing the equant) Kepler was forcedto abandon the circles by his extremely technical work on the details

420 Appendix C

of Marsrsquo orbit He was using ellipses as approximations to the epicycle-deferent system when he began to realize that the approximation wasfitting the data better than the supposedly exact curve And Einsteinand Dirac were also extremely conservative Their vertigo-inducing leapsahead were not pulled out of the blue sky They did not come from thethrill of violating respected ideas or trying a new pretty idea They wereforced out of respect towards previous physical insights Today insteadwe have plenty of seminars on ldquoa new pretty ideardquo regularly soon forgot-ten and superseded by a new fad In physics novelty has always emergedfrom new data or from a humble devoted interrogation of the old theo-ries From turning these theories over and over in onersquos mind immersingoneself in them making them clash merge talk until through them themissing gear can be seen

Finally the ldquopragmaticrdquo scientist ignores conceptual questions andphysical insights and only cares about developing a theory This is anattitude successful in the sixties in arriving at the standard model But inthe sixties empirical data were flowing in daily to keep research on trackToday theoreticians have no new data The ldquopragmaticrdquo theoretician doesnot care He does not trust the insight of the old theories He focuses onlyon the development of the novel theory and cannot care less if the worldpredicted by the theory resembles less and less the world we see He iseven excited that the theory looks so different from the world thinkingthat this is evidence of how much ahead he has advanced in knowledgewhich is a complete nonsense Theoretical physics becomes a mental gameclosed in on itself and the connection with reality is lost

In my opinion precious research energies are today wasted in these at-titudes A philosophy of science that downplays the component of factualknowledge in physical theories might have part of the responsibility

C2 On realism

A scientific theory is a conceptual structure that we use to read organizeand understand the world at some level of our knowledge It is one stepalong a process since knowledge increases In my view scientific thinkingis not much different from common-sense thinking In fact it is only a bet-ter instance of the same activity thinking about the world and updatingour mental schemes Science is the organized enterprise of continuouslyexploring the possible ways of thinking about the world and constantlyselecting the ones that work best

If this is correct there cannot be any qualitative difference between thetheoretical notions introduced in science and the terms in our everydaylanguage A fundamental intuition of classical empiricism is that nothinggrants us that the concepts that we use to organize our perceptions refer


to ldquorealrdquo entities Some modern philosophy of science has emphasized theapplication of this intuition to the concepts introduced by science Thuswe are warned to doubt the ldquorealityrdquo of the theoretical objects (electronsfields black holes ) I find these warnings incomprehensible Not be-cause they are ill founded but because they are not applied consistentlyThe fathers of empiricism consistently applied this intuition to any phys-ical object Who grants me the reality of a chair Why should a chairbe more than a theoretical concept organizing certain regularities in myperceptions I will not venture here in disputing nor in agreeing with thisdoctrine What I find incomprehensible is the position of those who grantthe solid status of reality to a chair but not to an electron The argu-ments against the reality of the electron apply to the chair as well Thearguments in favor of the reality of the chair apply to the electron as wellA chair as well as an electron is a concept that we use to read organizeand understand the world They are equally real They are equally volatileand uncertain

Perhaps this curious schizophrenic attitude of being antirealist withelectrons and iron-realist with chairs is the result of a tortuous historicalevolution initiated by the rebellion against ldquometaphysicsrdquo and with itthe granting of confidence to science alone From this point of view meta-physical questioning on the reality of chairs is sterile ndash true knowledge is inscience Thus it is to scientific knowledge that we apply empiricist rigorBut understanding science in empiricistsrsquo terms required making sense ofthe raw empirical data on which science is based With time the idea ofraw empirical data showed more and more its limits The common-senseview of the world was reconsidered as a player in our picture of knowl-edge This common-sense view should give us a language and a foundationfrom which to start ndash the old antimetaphysical prejudice still preventingus however from applying empiricist rigor to this common-sense view ofthe world But if one is not interested in questioning the reality of chairsfor the very same reason why should one be interested in questioning theldquoreality of the electronsrdquo

Again I think this point is important for science itself The factual con-tent of a theory is our best tool The faith in this factual content does notprevent us from being ready to question the theory itself if sufficientlycompelled to do so by novel empirical evidence or by putting the theoryin relation to other things we know about the world or we learn about itScientific antirealism in my opinion is not only a shortsighted applicationof a deep classical empiricist insight it is also a negative influence overthe development of science H Stein (private communication) has recentlybeautifully illustrated a case in which a great scientist Poincare wasblocked from getting to a major discovery (special relativity) by a philos-ophy that restrained him from ldquotaking seriouslyrdquo his own findings

422 Appendix C

Science teaches us that our naive view of the world is imprecise inap-propriate biased It constructs better views of the world (Better for someuse worse for others of course which is why it is silly to think of ourgirlfriend as a collection of electrons) Electrons if anything are ldquomorerealrdquo than chairs not ldquoless realrdquo in the sense that they underpin a way ofconceptualizing the world which is in many respects more powerful Onthe other hand the process of scientific discovery and the experience ofthe twentieth century in particular has made us painfully aware of theprovisional character of any form of knowledge Our mental and mathe-matical pictures of the world are only mental and mathematical picturesBetween our images of reality and our experience of reality there is alwaysan hiatus This is true for abstract scientific theories as well as for theimage we have of our dining room (not to even mention the image we haveof our girlfriend) Nevertheless the pictures are effective and we canrsquot doany better

C3 On truth

So is there anything we can say with confidence about the ldquoreal physicalworldrdquo A large part of the recent reflection on science has taught us thatraw data do not exist that any information about the world is alreadydeeply filtered and interpreted by the theory and that theories are alllikely to be superseded It has been useful and refreshing to learn thisFar more radically the European reflection and part of the American aswell has emphasized the fact that truth is always internal to the theorythat we can never leave language that we can never exit the circle ofdiscourse within which we are speaking As a scientist I appreciate andshare these ideas

But the fact that the only notion of truth is internal to our discoursedoes not imply that we should lose confidence in it If truth is internal toour discourse then this internal truth is what we mean by truth Indeedthere may be no valid notion of truth outside our own discourse but it isprecisely ldquofrom withinrdquo this discourse not from without it that we canand do assert the truth of the reality of the world and the truth of whatwe have learned about it More significantly still it is structural to ourlanguage to be a language about the world and to our thinking to be athinking of the world1

Therefore there is no sense in denying the truth of what we have learnedabout the world precisely because there is no notion of truth except the

1The rational investigation of the world started with the pre-Socratic λoγoς (logos)which is the principle (that we seek) governing the cosmos as well as human reasoningand speaking about the cosmos it is at the same time the truth and our reasoningabout it


one within our own discourse If there is no place we can go which isoutside our language in which place are they standing those who questionthe truth we find It can only be a pleasant short dreamy place wherewe are happy to stay for a short while smiling as if we were wise andthen come back to reality The world is real solid and understandableprecisely because the language our only home states so The best we cansay about the physical world and about what is physically real out thereis what good physics says about it2

At the same time there is no reason that our perceiving understand-ing and conceptualizing the world should not be in continuous evolutionScience is the form of this evolution At every stage the best we can sayabout the reality of the world is precisely what we are saying The factwe will understand it better later on does not make our present under-standing less valuable or less credible When we walk in the mountainswe do not dismiss our map just because there may exist a better mapwhich we donrsquot have Searching for a fixed point on which to rest ourrestlessness is in my opinion naive useless and counterproductive forthe development of knowledge It is only by believing our insights andat the same time questioning our mental habits that we can go ahead Ibelieve that this process of cautious faith and self-confident doubt is thecore of scientific thinking Science is the human adventure that consists inexploring possible ways of thinking of the world Being ready to subvertif required anything we have been thinking so far

I think this is among the best of human adventures Research in quan-tum gravity in its effort to conceptualize quantum spacetime and thusmodify in depth the notions of space and time is a step in this adventure

2I certainly do not wish to suggest that the physical description of the world exhaustsit It would be like saying that if I understand the physics of a brick I immediatelyknew why a cathedral stands or why it is splendid

References

Preface and terminology and notation

[1] C Rovelli Loop space representation In New Perspectives in CanonicalGravity ed A Ashtekar et al (Napoli Bibliopolis 1988)

[2] C Rovelli Ashtekar formulation of general relativity and loop space non-perturbative Quantum Gravity a report Class and Quantum Grav 8(1991) 1613ndash1675

[3] A Ashtekar Non-perturbative Canonical Gravity (Singapore World Sci-entific 1991)

[4] L Smolin Time measurement and information loss in quantum cos-mology In Brill Feschrift Proceedings ed B Hu and T Jacobson(Cambridge Cambridge University Press 1993) Recent developments innonperturbative quantum gravity In Quantum Gravity and Cosmology edJ Perez-Mercader J Sola and E Verdaguer (Singapore World Scientific1993)

[5] J Baez Knots and Quantum Gravity (Oxford Oxford University Press1994)

[6] B Brugmann Loop representations In Canonical Gravity from Classicalto Quantum ed J Ehlers and H Friedrich (Berlin Springer-Verlag 1994)

[7] R Gambini and J Pullin Loops Knots Gauge Theories and QuantumGravity (Cambridge Cambridge University Press 1996)

[8] J Kowalski-Glikman Towards quantum gravity Lecture Notes in Physics541 (2000) (Berlin Springer)

[9] A Ashtekar Background independent quantum gravity A Status reportClass Quant Grav 21 (2004) R53

[10] L Smolin An invitation to loop quantum gravity hep-th0408048[11] T Thiemann Lectures on loop quantum gravity Lecture Notes in Physics

631 (2003) 41ndash135 gr-qc0210094[12] C Rovelli Loop quantum gravity Living Reviews in Relativity electronic

journal http wwwlivingreviewsorgArticlesVolume11998-1rovelli

424

References 425

[13] A Ashtekar Quantum geometry and gravity recent advances to appear inthe Proc 16th Int Conf on General Relativity and Gravitation DurbanS Africa July 2001 gr-qc9901023

[14] M Gaul and C Rovelli Loop quantum gravity and the meaning of diffeo-morphism invariance Lecture Notes in Physics 541 (2000) 277ndash324 (BerlinSpringer) gr-qc9910079

[15] C Rovelli and P Upadhya Loop quantum gravity and quanta of space aprimer gr-qc9806079

[16] J Baez and J Muniain Gauge Fields Knots and Gravity (SingaporeWorld Scientific 1994)

[17] JC Baez An introduction to spin foam models of BF theory and quan-tum gravity In Geometry and Quantum Physics ed H Gausterer andH Grosse Lecture Notes in Physics 543 (1999) 25ndash94 (Berlin Springer-Verlag) gr-qc9905087

[18] D Oriti Spacetime geometry from algebra spin foam models fornon-perturbative quantum gravity Rept Prog Phys 64 (2001) 1489gr-qc0106091

[19] A Perez Spin foam models for quantum gravity Class and QuantumGrav 20 (2002) gr-qc0301113

[20] T Thiemann Modern Canonical Quantum General Relativity (CambridgeCambridge University Press 2004 in press) a preliminary version is in gr-qc0110034

[21] J Ambjorn B Durhuus and T Jonsson Quantum Geometry (CambridgeCambridge University Press 1997)

Chapter 1 General ideas and heuristic picture

[22] M Gell-Mann Strange Beauty (London Vintage 2000) pp 303ndash304[23] L Smolin Towards a background independent approach to M theory

hep-th9808192 The cubic matrix model and duality between strings andloops hep-th0006137 A candidate for a background independent formu-lation of M theory Phys Rev D62 (2000) 086001 hep-th9903166

[24] L Smolin Strings as perturbations of evolving spin networks Nucl PhysProc Suppl 88 (2000) 103ndash113 hep-th9801022

[25] D Amati M Ciafaloni and G Veneziano Can spacetime be probed belowthe string size Phys Lett B216 (1989) 41

[26] J Hartle Spacetime quantum mechanics and the quantum mechanicsof spacetime In Proceedings 1992 Les Houches School Gravitation andQuantisation ed B Julia and J Zinn-Justin (Paris Elsevier Science1995) p 285

[27] SA Fulling Aspects of Quantum Field Theory in Curved Spacetime(Cambridge Cambridge University Press 1989)

[28] RM Wald Quantum Field Theory on Curved Spacetime and Black HoleThermodynamics (Chicago University of Chicago Press 1994)

426 References

[29] MH Goroff and A Sagnotti Quantum gravity at two loops Phys LettB160 (1985) 81 The ultraviolet behaviour of Einstein gravity Nucl PhysB266 (1986) 709

[30] G Horowitz Quantum gravity at the turn of the millenium plenary talkat the Marcell Grossmann Conf Rome 2000 gr-qc0011089

[31] S Carlip Quantum gravity a progress report Rept Prog Phys 64 (2001)885 gr-qc0108040

[32] CJ Isham Conceptual and geometrical problems in quantum gravity InRecent Aspects of Quantum Fields ed H Mitter and H Gausterer (BerlinSpringer-Verlag 1991) p 123

[33] C Rovelli Strings loops and the others a critical survey on the present ap-proaches to quantum gravity In Gravitation and Relativity At the turn ofthe Millenium ed N Dadhich and J Narlikar (Pune Inter-University Cen-tre for Astronomy and Astrophysics 1998) pp 281ndash331 gr-qc9803024

[34] C Callender and N Hugget eds Physics Meets Philosophy at the PlanckScale (Cambridge Cambridge University Press 2001)

[35] C Rovelli Halfway through the woods In The Cosmos of Scienceed J Earman and JD Norton (University of Pittsburgh Press andUniversitats Verlag-Konstanz 1997)

[36] C Rovelli Quantum spacetime what do we know In Physics MeetsPhilosophy at the Planck Length ed C Callender and N Hugget(Cambridge Cambridge University Press 1999) gr-qc9903045

[37] C Rovelli The century of the incomplete revolution searching for generalrelativistic quantum field theory J Math Phys Special Issue 2000 41(2000) 3776 hep-th9910131

[38] JA Wheeler Superspace and the nature of quantum geometrodynamicsIn Batelle Rencontres 1967 ed C DeWitt and JW Wheeler Lectures inMathematics and Physics 242 (New York Benjamin 1968)

[39] S Weinberg Ultraviolet divergences in quantum theories of gravitationIn General Relativity An Einstein Centenary Survey ed SW Hawkingand W Israel (Cambridge Cambridge University Press 1979)

[40] SW Hawking The path-integral approach to quantum gravity In GeneralRelativity An Einstein Centenary Survey ed SW Hawking and W Israel(Cambridge Cambridge University Press 1979)

[41] SW Hawking Quantum cosmology In Relativity Groups and TopologyLes Houches Session XL ed B DeWitt and R Stora (Amsterdam NorthHolland 1984)

[42] K Kuchar Canonical methods of quantization In Oxford 1980 Proceed-ings Quantum Gravity 2 (Oxford Oxford University Press 1984)

[43] CJ Isham Topological and global aspects of quantum theory In RelativityGroups and Topology Les Houches 1983 ed BS DeWitt and R Stora(Amsterdam North Holland 1984)

[44] CJ Isham Quantum gravity an overview In Oxford 1980 ProceedingsQuantum Gravity 2 (Oxford Oxford University Press 1984)

References 427

[45] C J Isham Structural problems facing quantum gravity theory In Proc14th Int Conf on General Relativity and Gravitation ed M FrancavigliaG Longhi L Lusanna and E Sorace (Singapore World Scientific 1997)pp 167ndash209

[46] MB Green J Schwarz and E Witten Superstring Theory (CambridgeCambridge University Press 1987) J Polchinski String Theory(Cambridge Cambridge University Press 1998)

[47] C Rovelli A dialog on quantum gravity Int J Mod Phys 12 (2003) 1hep-th0310077

[48] L Smolin How far are we from the quantum theory of gravity hep-th0303185

[49] A Connes Non Commutative Geometry (New York Academic Press1994)

[50] L Smolin Three Roads to Quantum Gravity (Oxford Oxford UniversityPress 2000)

[51] KS Robinson Blue Mars (New York Bantam 1996)[52] G Egan Schild Ladder (London Gollancz 2001)[53] E Palandri Anna prende il volo (Milano Feltrinelli 2000)

Chapter 2 General relativity

[54] S Holst Barberorsquos Hamiltonian derived from a generalized Hilbert-Palatini action Phys Rev D53 (1996) 5966ndash5969

[55] L Russo La rivoluzione dimenticata (Milano Feltrinelli 1997)[56] JP Bourguignon and P Gauduchon Spineurs operateurs de Dirac et

variations de metriques Comm Math Phys 144 (1992) 581[57] T Schucker Forces from Connesrsquo geometry hep-th0111236 Lectures at

the Autumn School Topology and Geometry in Physics Rot an der Rot2001 ed E Bick and F Steffen (Lecture Notes in Physics Springer 2004)

[58] L Russo Flussi e riflussi (Feltrinelli Milano 2003)[59] M Faraday Experimental Researches in Electricity (London Bernard

Quaritch 1855) pp 436ndash437[60] R Descartes Principia Philosophiae (1644) Translated by VR Miller and

RP Miller (Dordrecht Reidel 1983)[61] I Newton De Gravitatione et Aequipondio Fluidorum translation in

Unpublished Papers of Isaac Newton ed AR Hall and MB Hall(Cambridge Cambridge University Press 1962)

[62] I Newton Principia Mathematica Philosophia Naturalis 1687 Englishtranslation The Principia Mathematical Principles of Natural Philosophy(City University of California Press 1999)

[63] A Einstein and M Grossmann Entwurf einer verallgemeinerten Rela-tivitatstheorie und einer Theorie der Gravitation Z fur Mathematik undPhysik 62 (1914) 225

428 References

[64] A Einstein Grundlage der allgemeinen Relativitatstheorie Ann der Phys49 (1916) 769ndash822

[65] M Pauri and M Vallisneri Ephemeral point-events is there a last rem-nant of physical objectivity DIALOGOS 79 (2002) 263ndash303 L Lusannaand M Pauri General covariance and the objectivity of space-time point-events the physical role of gravitational and gauge degrees of freedomhttpphilsci-archivepitteduarchive00000959 (2002)

[66] P Hajicek Lecture Notes in Quantum Cosmology (Bern University ofBern 1990)

[67] AS Eddington The Nature of the Physical World (New York MacMillan1930) pp 99ndash102

[68] SJ Earman A Primer on Determinism (Dordrecht D Reidel 1986)[69] B DeWitt in Gravitation An Introduction to Current Research ed

L Witten (New York Wiley 1962)[70] JD Brown and D Marolf On relativistic material reference systems Phys

Rev D53 (1996) 1835[71] C Rovelli What is observable in classical and quantum gravity Class

and Quantum Grav 8 (1991) 297 Quantum reference systems Class andQuantum Grav 8 (1991) 317

[72] PG Bergmann Phys Rev 112 (1958) 287 Observables in general covari-ant theories Rev Mod Phys 33 (1961) 510

[73] BW Parkinson and JJ Spilker eds Global Positioning System The-ory and Applications Prog in Astronautics and Aeronautics Nos 163ndash164 (Amer Inst Aero Astro Washington 1996) ED Kaplan Under-standing GPS Principles and Applications Mobile Communications Se-ries (Boston Artech House 1996) B Hofmann-Wellenhof H Lichteneg-ger and J Collins Global Positioning System Theory and Practice (NewYork Springer-Verlag 1993)

[74] B Guinot Application of general relativity to metrology Metrologia 34(1997) 261 F de Felice MG Lattanzi A Vecchiato and PL BernaccaGeneral relativistic satellite astrometry I A non-perturbative approachto data reduction Astron Astrophy 332 (1998) 1133 TB BahderFermi Coordinates of an Observer Moving in a Circle in MinkowskiSpace Apparent Behavior of Clocks Army Research Laboratory AdelphiMaryland USA Technical Report ARL-TR-2211 May 2000 ARThompson JM Moran and GW Swenson Interferometry and Synthe-sis in Radio Astronomy (Malabar Florida Krieger Pub Co 1994) pp138ndash139 PNAM Visser Gravity field determination with GOCE andGRACE Adv Space Res 23 (1999) 771

[75] S Weinberg Gravitation and Cosmology (New York Wiley 1972)[76] RM Wald General Relativity (Chicago The University of Chicago Press

1989)[77] Y Choquet-Bruhat C DeWitt-Morette and M Dillard-Bleick Analysis

Manifolds and Physics (Amsterdam North Holland 1982)

References 429

[78] I Ciufolini and J Wheeler Gravitation and Inertia (Princeton PrincetonUniversity Press 1996)

[79] H Weyl Electron and gravitation Z Physik 56 (1929) 330[80] J Schwinger Quantized gravitational field Phys Rev 130 (1963) 1253[81] JF Plebanski On the separation of Einsteinian substructures J Math

Phys 18 (1977) 2511[82] A Sen Gravity as a spin system Phys Lett 119B (1982) 89[83] J Samuel A lagrangian basis for Ashtekarrsquos reformulation of canonical

gravity Pramana J Phys 28 (1987) L429 T Jacobson and L SmolinCovariant action for Ashtekarrsquos form of canonical gravity Class andQuantum Grav 5 (1988) 583

[84] R Capovilla J Dell and T Jacobson General relativity without the met-ric Phys Rev Lett 63 (1991) 2325 R Capovilla J Dell T Jacobsonand L Mason Selfndashdual 2ndashforms and gravity Class and Quantum Grav8 (1991) 41

[85] JD Norton How Einstein found his field equations 1912ndash1915 HistoricalStudies in the Physical Sciences 14 (1984) 253ndash315 Reprinted in Einsteinand the History of General Relativity Einstein Studies ed D Howard andJ Stachel Vol I (Boston Birkhauser 1989) pp 101ndash159

[86] J Stachel Einsteinrsquos search for general covariance 1912ndash1915 In Einsteinand the History of General Relativity Einstein Studies ed D Howard andJ Stachel Vol 1 (Boston Birkhauser 1989) pp 63ndash100

[87] E Kretschmann Uber den physikalischen Sinn der RelativitatpostulateAnn Phys Leipzig 53 (1917) 575

[88] JL Anderson Principles of Relativity Physics (New York AcademicPress 1967)

[89] J Barbour Absolute or Relative Motion (Cambridge Cambridge Univer-sity Press 1989)

[90] J Earman and J Norton What price spacetime substantivalism The holestory Brit J Phil Sci 38 (1987) 515ndash525

[91] J Earman World Enough and Space-time Absolute Versus RelationalTheories of Spacetime (Cambridge MIT Press 1989)

[92] G Belot Why general relativity does need an interpretation Phil Sci 63(1998) S80ndashS88

[93] J Earman and G Belot Pre-Socratic quantum gravity In Physics MeetsPhilosphy at the Planck Scale ed C Callander (Cambridge CambridgeUniversity Press 2001)

[94] C Rovelli Analysis of the different meaning of the concept of time indifferent physical theories Il Nuovo Cimento 110B (1995) 81

[95] JT Fraser Of Time Passion and Knowledge (Princeton PrincetonUniversity Press 1990)

[96] H Reichenbach The Direction of Time (Berkeley University of CaliforniaPress 1956) PCW Davies The Physics of Time Asymmetry (England

430 References

Surrey University Press 1974) R Penrose in General Relativity AnEinstein Centenary Survey ed SW Hawking and W Israel (CambridgeCambridge University Press 1979) HD Zee The Physical Basis ofthe Direction of Time (Berlin Springer 1989) J Halliwel and JAPerez-Mercader eds Proceedings of the International Workshop PhysicalOrigins of Time Asymmetry Huelva Spain September 1991 (CambridgeCambridge University Press 1992)

[97] CJ Isham Canonical quantum gravity and the problem of timeLectures presented at the NATO Advanced Institute Recent Problems inMathematical Physics Salamanca June 15 1992 K Kuchar Time andinterpretations of quantum gravity In Proc 4th Canadian Conference onGeneral Relativity and Relativistic Astrophysics ed G Kunstatter D Vin-cent and J Williams (Singapore World Scientific 1992) A Ashtekar andJ Stachel eds Proc Osgood Hill Conference Conceptual Problems inQuantum Gravity Boston 1988 (Boston Birkhauser 1993)

[98] C Rovelli Time in quantum gravity an hypothesis Phys Rev D43 (1991)442

[99] J Hartle Classical physics and hamiltonian quantum mechanics as relicsof the big bang Physica Scripta T36 (1991) 228

[100] A Grunbaum Philosophical Problems of Space and Time (New YorkKnopf 1963) T Gold and DL Shumacher eds The Nature of Time(Ithaca Cornell University Press 1967) P Kroes Time its Structure andRole in Physical Theories (Dordrecht D Reidel 1985)

[101] C Rovelli GPS observables in general relativity Phys Rev D65 (2002)044017 gr-qc0110003

[102] TB Bahder Navigation in curved space-time Amer J Phys 69 (2001)315ndash321

[103] M Blagojevic J Garecki FW Hehl and Yu N Obukhov Real nullcoframes in general relativity and GPS type coordinates gr-qc0110078

Chapter 3 Mechanics

[104] VI Arnold Matematiceskie Metody Klassiceskoj Mechaniki (MoskowMir 1979) See in particular Chapter IX Section C

[105] JM Souriau Structure des Systemes Dynamiques (Paris Dunod 1969)[106] JL Lagrange Memoires de la Premiere Classe des Sciences Mathema-

tiques et Physiques (Paris Institute de France 1808)[107] WR Hamilton On the application to dynamics of a general mathematical

method previously applied to optics British Association Report (1834)513ndash518

[108] C Crnkovic and E Witten Covariant description of canonical formal-ism in geometrical theories In Newtonrsquos Tercentenary Volume ed SWHawking and W Israel (Cambridge Cambridge University Press 1987)

References 431

A Ashtekar L Bombelli and O Reula In Mechanics Analysis and Geom-etry 200 Years after Lagrange ed M Francaviglia (Amsterdam Elsevier1991)

[109] MJ Gotay J Isenberg and JE Marsden (with the collaboration of RMontgomery J Sniatycki and PB Yasskin) Momentum maps and classi-cal relativistic fields Part 1 covariant field theory physics9801019

[110] T DeDonder Theorie Invariantive du Calcul des Variationes (ParisGauthier-Villars 1935)

[111] Hesiod Theogony translated by HG Evelyn-White (London HarvardUniversity Press 1914) pp 125ndash130 [Instigated by mother Γαια Kρoνoςthen slaughters and castrates father O

vρανoς]

[112] C Rovelli The statistical state of the universe Class and Quantum Grav10 (1993) 1567

[113] PAM Dirac Generalized Hamiltonian dynamics Can J Math Phys 2(1950) 129ndash148

[114] PAM Dirac Lectures on Quantum Mechanics (New York Belfer Gradu-ate School of Science Yeshiva University 1964)

[115] A Hanson T Regge and C Teitelboim Constrained Hamiltonian Systems(Roma Accademia nazionale dei Lincei 1976) M Henneaux and C Teit-elboim Quantization of Gauge Systems (Princeton Princeton UniversityPress 1972)

[116] C Rovelli Partial observables Phy Rev D65 (2002) 124013 gr-qc0110035

[117] C Rovelli A note on the foundation of relativistic mechanics I Relativis-tic observables and relativistic states In Proc 15th SIGRAV Conferenceon General Relativity and Gravitational Physics 2002 (Bristol IOP Pub-lishing 2004) in press gr-qc0111037

[118] C Rovelli Covariant hamiltonian formalism for field theory symplecticstructure and HamiltonndashJacobi equation on the space G In Decoherenceand Entropy in Complex Systems Selected Lectures from DICE 2002Lecture Notes in Physics 633 ed HT Elze (Berlin SpringerndashVerlag2003) gr-qc0207043

[119] M Montesinos C Rovelli and T Thiemann SL(2 R) model with twoHamiltonian constraints Phys Rev D60 (1999) 044009

[120] H Weil Geodesic fields in the calculus of variations Ann Math 36 (1935)607ndash629

[121] J Kijowski A finite dimensional canonical formalism in the classical fieldtheory Comm Math Phys 30 (1973) 99ndash128 M Ferraris and M Fran-caviglia The Lagrangian approach to conserved quantities in general rel-ativity In Mechanics Analysis and Geometry 200 Years after Lagrangeed M Francaviglia (Amsterdam Elsevier Sci Publ 1991) pp 451ndash488IV Kanatchikov Canonical structure of classical field theory in the poly-momentum phase space Rep Math Phys 41 (1998) 49 F Helein andJ Kouneiher Finite dimensional Hamiltonian formalism for gauge and

432 References

field theories math-ph0010036 H Rund The HamiltonndashJacobi Theory inthe Calculus of Variations (New York Krieger 1973) H Kastrup Canon-ical theories of Lagrangian dynamical systems in physics Phys Rep 101(1983) 1

[122] C Rovelli Statistical mechanics of gravity and thermodynamical origin oftime Class and Quantum Grav 10 (1993) 1549

[123] A Connes and C Rovelli Von Neumann algebra automorphisms and timeversus thermodynamics relation in general covariant quantum theoriesClass and Quantum Grav 11 (1994) 2899

[124] P Martinetti and C Rovelli Diamondsrsquo temperature Unruh effect forbounded trajectories and thermal time hypothesis Class and QuantumGrav 20 (2003) 4919ndash4932 gr-qc0212074

[125] M Montesinos and C Rovelli Statistical mechanics of generally covariantquantum theories a Boltzmann-like approach Class and Quantum Grav18 (2001) 555ndash569

Chapter 4 Hamiltonian general relativity

[126] D Giulini Ashtekar variables in Classical General Relativity In CanonicalGravity From Classical to Quantum ed J Ehlers and H Friedrich (BerlinSpringer-Verlag 1994) p 81

[127] S Alexandrov E Buffenoir P Roche Plebanski theory and covariantcanonical formulation gr-qc0612071

[128] A Perez C Rovelli Physical effects of the Immirzi parameter Phys RevD73 (2006) 044013

[129] P Bergmann Phys Rev 112 (1958) 287 Rev Mod Phys 33 (1961)P Bergmann and A Komar The phase space formulation of general rel-ativity and approaches towards quantization Gen Rel Grav 1 (1981)pp 227ndash254 In General Relativity and Gravitation ed A Held (1981) pp227ndash254 A Komar General relativistic observables via HamiltonndashJacobifunctionals Phys Rev D4 (1971) 923ndash927

[130] PAM Dirac The theory of gravitation in Hamiltonian form Proc RoyalSoc London A246 (1958) 333 Phys Rev 114 (1959) 924

[131] R Arnowitt S Deser and CW Misner The dynamics of general relativityIn Gravitation An Introduction to Current Research ed L Witten (NewYork Wiley 1962) p 227

[132] A Ashtekar New variables for classical and quantum gravity Phys RevLett 57 (1986) 2244 New Hamiltonian formulation of general relativityPhys Rev D36 (1987) 1587

[133] T Thiemann Anomaly-free formulation of nonperturbative 4-dimensionalLorentzian quantum gravity Phys Lett B380 (1996) 257

[134] F Barbero Real Ashtekar variables for Lorentzian signature spacetimesPhys Rev D51 (1995) 5507 gr-qc9410014 Phys Rev D51 (1995) 5498

References 433

[135] G Immirzi Quantum gravity and Regge calculus Nucl Phys Proc Suppl57 (1997) 65 Real and complex connections for canonical gravity Classand Quantum Grav 14 (1997) L177ndashL181

[136] L Fatibene M Francaviglia C Rovelli On a Covariant Formulation ofthe Barbero-Immirzi Connection gr-qc0702134

[137] C Rovelli and T Thiemann The Immirzi parameter in quantum generalrelativity Phys Rev D57 (1998) 1009ndash1014 gr-qc9705059

[138] G Esposito G Gionti and C Stornaiolo Space-time covariant form ofAshtekarrsquos constraints Nuovo Cimento 110B (1995) 1137ndash1152

[139] C Rovelli A note on the foundation of relativistic mechanics II Covarianthamiltonian general relativity gr-qc0202079

[140] M Ferraris and M Francaviglia The Lagrangian approach to conservedquantities in General Relativity In Mechanics Analysis and Geometry200 Years after Lagrange ed M Francaviglia (Amsterdam Elsevier SciPubl 1991) pp 451ndash488 W Szczyrba A symplectic structure of the setof Einstein metrics a canonical formalism for general relativity CommMath Phys 51 (1976) 163ndash182 J Sniatcki On the canonical formulationof general relativity In Proc Journees Relativistes (Caen Faculte des Sci-ences 1970) J Novotny On the geometric foundations of the Lagrangeformulation of general relativity In Differential Geometry ed G Soos andJ Szenthe (Amsterdam North-Holland 1982)

[141] A Peres Nuovo Cimento 26 (1962) 53 U Gerlach Phys Rev 177 (1969)1929 K Kuchar J Math Phys 13 (1972) 758 P Horava On a covariantHamiltonndashJacobi framework for the EinsteinndashMaxwell theory Class andQuantum Grav 8 (1991) 2069 ET Newman and C Rovelli Generalizedlines of force as the gauge invariant degrees of freedom for general relativityand YangndashMills theory Phys Rev Lett 69 (1992) 1300 J Kijowski and GMagli Unconstrained Hamiltonian formulation of General Relativity withthermo-elastic sources Class and Quantum Grav 15 (1998) 3891ndash3916

Chapter 5 Quantum mechanics

[142] E Schrodinger Quantisierung als Eigenwertproblem Ann der Phys 79(1926) 489 Part 2 English translation in Collected Papers on QuantumMechanics (Chelsea Publications 1982)

[143] E Schrodinger Quantisierung als Eigenwertproblem Ann der Phys 79(1926) 361 Part 1 English translation op cit

[144] M Reisenberger and C Rovelli Spacetime states and covariant quan-tum theory Phys Rev D65 (2002) 124013 gr-qc0111016 D Marolf andC Rovelli Relativistic quantum measurement Phys Rev D66 (2002)023510 gr-qc0203056

[145] F Conrady L Doplicher R Oeckl C Rovelli and M Testa Minkowskivacuum from background independent quantum gravity Phys Rev D164(2004) 064019 gr-qc0307118

434 References

[146] F Conrady and C Rovelli Generalized Schrodinger equation in Euclideanquantum field theory Int J Mod Phys in press hep-th0310246

[147] M Montesinos The double role of Einsteinrsquos equations as equations ofmotion and as vanishing energy-momentum tensor gr-qc0311001

[148] PAM Dirac Principles of Quantum Mechanics 1st edition (OxfordOxford University Press 1930)

[149] C Rovelli Is there incompatibility between the ways time is treated ingeneral relativity and in standard quantum mechanics In ConceptualProblems of Quantum Gravity ed A Ashtekar and J Stachel (New YorkBirkhauser 1991)

[150] J Halliwell The WheelerndashdeWitt equation and the path integral in mini-superspace quantum cosmology In Conceptual Problems of QuantumGravity A Ashtekar and J Stachel (New York Birkhauser 1991)

[151] C Rovelli Quantum mechanics without time a model Phys Rev D42(1991) 2638

[152] C Rovelli Quantum evolving constants Phys Rev D44 (1991) 1339[153] R Oeckl A lsquogeneral boundaryrsquo formulation for quantum mechanics and

quantum gravity hep-th0306025 Schroedingerrsquos cat and the clock lessonsfor quantum gravity gr-qc0306007

[154] L Doplicher Generalized TomonagandashSchrodinger equation from theHadamard formula gr-qc0405006

[155] S Tomonaga Prog Theor Phys 1 (1946) 27 J Schwinger Quantumelectrodynamics I A covariant formulation Phys Rev 74 (1948) 1439

[156] NC Tsamis and RP Woodard Physical Greenrsquos functions in quantumgravity Annals of Phys 215 (1992) 96

[157] JB Hartle and SW Hawking Wave function of the Universe Phys RevD28 (1983) 2960

[158] D Marolf Group averaging and refined algebraic quantization where arewe In Proceedings of the IXth Marcel Grossmann Conference Rome ItalyJuly 2ndash9 2000 ed RT Jantzen GM Keiser and R Ruffini (World Sci-entific 1996) gr-qc0011112

[159] C Rovelli Relational quantum mechanics Int J Theor Phys 35 (1996)1637ndash1678

[160] C Rovelli Incerto tempore incertisque loci Can we compute the exacttime at which a quantum measurement happens Foundations of Physics28 (1998) 1031ndash1043

[161] F Laudisa The EPR argument in a relational interpretation of quantummechanics Foundations of Physics Letters 14 (2) (2001) 119ndash132

[162] A Grinbaum Elements of information theoretic derivation of the formal-ism of quantum theory Int J Quant Information 1 (2003) 1

[163] F Laudisa and C Rovelli Relational quantum mechanics In The StanfordEncyclopedia of Philosophy (Spring 2002 Edition) ed Edward N ZaltaURL httpplatostanfordeduarchivesspr2002entriesqm-relational

References 435

[164] M Bitbol Relations et correlations en Physique Quantique In Un Sieclede Quanta ed M Crozon and Y Sacquin (Paris EDP Sciences 2000)

[165] J Wheeler Information physics quantum the search for the links Proc3rd Int Symp Foundations of Quantum Mechanics Tokyo 1989 p 354

[166] J Wheeler It from Bit In Sakharov Memorial Lectures on Physics Vol 2ed L Keldysh and V Feinberg (New York Nova Science 1992)

[167] CU Fuchs Quantum foundations in the light of quantum information InProc NATO Advanced Research Workshop on Decoherence and its Impli-cations in Quantum Computation and Information Transfer ed A Gonis(New York Plenum 2001) quant-ph0106166quant-ph0205039

[168] D Finkelstein Quantum Relativity (Berlin Springer 1996)

Chapter 6 Quantum space

[169] L Freidel and ER Livine Spin networks for non-compact groups J MathPhys 44 (2003) 1322ndash1356

[170] C Rovelli Loop representation in quantum gravity In Conceptual Prob-lems of Quantum Gravity ed A Ashtekar and J Stachel (New YorkBirkhauser 1991)

[171] C Rovelli and L Smolin Spin networks and quantum gravity Phys RevD52 (1995) 5743ndash5759 gr-qc9505006

[172] J Lewandowski ET Newman and C Rovelli Variations of the parallelpropagator and holonomy operator and the Gauss law constraint J MathPhys 34 (1993) 4646

[173] A Ashtekar and J Lewandowski Quantum theory of geometry I Areaoperators Class and Quantum Grav 14 (1997) A55 II Volume operatorsAdv Theor Math Phys 1 (1997) 388ndash429

[174] LH Kauffman and SL Lins Temperley-Lieb Recoupling Theory and In-variant of 3-Manifolds (Princeton Princeton University Press 1994)

[175] R De Pietri and C Rovelli Geometry eigenvalues and scalar product fromrecoupling theory in loop quantum gravity Phys Rev D54 (1996) 2664gr-qc9602023 T Thiemann Closed formula for the matrix elements of thevolume operator in canonical quantum gravity J Math Phys 39 (1998)3347ndash3371 gr-qc9606091

[176] C Rovelli and L Smolin Knot theory and quantum gravity Phys RevLett 61 (1988) 1155

[177] C Rovelli and L Smolin Loop space representation for quantum generalrelativity Nucl Phys B331 (1990) 80

[178] T Jacobson and L Smolin Nonperturbative quantum geometries NuclPhys B299 (1988) 295

[179] R Gambini and A Trias Phys Rev D22 (1980) 1380 On the geometricalorigin of gauge theories Phys Rev D23 (1981) 553 Nucl Phys B278(1986) 436 C di Bartolo F Nori R Gambini and A Trias Loop spaceformulation of free electromagnetism Nuovo Cimento Lett 38 (1983) 497

436 References

[180] B Brugmann R Gambini and J Pullin Knot invariants as nondegeneratequantum geometries Phys Rev Lett 68 (1992) 431 Jones polynomials forintersecting knots as physical states of quantum gravity Nucl Phys B385(1992) 587 Gen Rel Grav 25 (1993) 1 J Pullin in Proc 5th MexicanSchool of Particles and Fields ed J Lucio (Singapore World Scientific1993)

[181] R Penrose Theory of quantized directions unpublished manuscript An-gular momentum an approach to combinatorial spacetime In QuantumTheory and Beyond ed T Bastin (Cambridge Cambridge UniversityPress 1971) pp 151ndash180

[182] L Smolin The future of spin networks gr-qc9702030[183] JC Baez Spin networks in gauge theory Adv Math 117 (1996) 253 JC

Baez Spin networks in nonperturbative quantum gravity In Interface ofKnot Theory and Physics ed L Kauffman (Providence Rhode IslandAmerican Mathematical Society 1996) gr-qc9504036

[184] N Grott and C Rovelli Moduli spaces structure of knots with intersec-tions J Math Phys 37 (1996) 3014

[185] W Fairbairn and C Rovelli Separable Hilbert space in loop quantumgravity J Math Phys to appear gr-qc0403047

[186] J Zapata A combinatorial approach to diffeomorphism invariant quantumgauge theories J Math Phys 38 (1997) 5663ndash5681 A combinatorial spacefor loop quantum gravity Gen Rel Grav 30 (1998) 1229

[187] J Lewandowski A Okolow H Sahlmann T Thiemann Comm MathPhys 267 (2006) 703ndash733

[188] C Fleischhack Representations of the Weyl Algebra in Quantum Geome-try math-ph0407006

[189] A Ashtekar C Rovelli and L Smolin Weaving a classical metric withquantum threads Phys Rev Lett 69 (1992) 237 hep-th9203079

[190] C Rovelli A generally covariant quantum field theory and a prediction onquantum measurements of geometry Nucl Phys B405 (1993) 797

[191] C Rovelli and L Smolin Discreteness of area and volume in quantumgravity Nucl Phys B442 (1995) 593 Erratum Nucl Phys B456 (1995)734

[192] S Frittelli L Lehner and C Rovelli The complete spectrum of the areafrom recoupling theory in loop quantum gravity Class and Quantum Grav13 (1996) 2921

[193] R Loll The volume operator in discretized quantum gravity Phys RevLett 75 (1995) 3048 Spectrum of the volume operator in quantum gravityNucl Phys B460 (1996) 143ndash154 gr-qc9511030

[194] J Lewandowski Volume and quantizations Class and Quantum Grav 14(1997) 71ndash76

[195] T Tsushima The expectation value of the Gaussian weave state in loopquantum gravity gr-qc0212117

References 437

[196] T Thiemann A length operator for canonical quantum gravity J MathPhys 39 (1998) 3372ndash3392 gr-qc9606092

[197] A Ashtekar A Corichi and J Zapata Quantum theory of geometry IIINoncommutativity of Riemannian structures Class and Quantum Grav15 (1998) 2955

[198] S Major Operators for quantized directions Class Quant Grav 16(1999) 3859ndash3877

[199] S Major A Spin Network Primer Am J Phys 67 (1999) 972ndash980[200] A Ashtekar and CJ Isham Representations of the holonomy algebra

of gravity and non-abelian gauge theories Class and Quantum Grav 9(1992) 1433 hep-th9202053

[201] A Ashtekar and J Lewandowski Representation theory of analytic holon-omy Clowast-algebras In Knots and Quantum Gravity ed J Baez (OxfordOxford University Press 1994) Differential geometry on the space of con-nections via graphs and projective limits J Geom and Phys 17 (1995)191

[202] A Ashtekar Mathematical problems of non-perturbative quantum generalrelativity In Les Houches Summer School on Gravitation and Quantiza-tions Les Houches France Jul 5ndashAug 1 1992 ed J Zinn-Justin andB Julia (Amsterdam North-Holland 1995) gr-qc9302024

[203] J Baez Generalized measures in gauge theory Lett Math Phys 31 (1994)213ndash223

[204] A Ashtekar J Lewandowski D Marolf J Mourao and T ThiemannQuantization of diffeomorphism invariant theories of connections with localdegrees of freedom J Math Phys 36 (1995) 6456 gr-qc9504018

[205] R De Pietri On the relation between the connection and the loop rep-resentation of quantum gravity Class and Quantum Grav 14 (1997) 53gr-qc9605064

[206] A Ashtekar C Rovelli and L Smolin Gravitons and loops Phys RevD44 (1991) 1740ndash1755 hep-th9202054 J Iwasaki and C Rovelli Gravi-tons as embroidery on the weave Int J Mod Phys D1 (1993) 533 Gravi-tons from loops non-perturbative loop-space quantum gravity containsthe graviton-physics approximation Class and Quantum Grav 11 (1994)1653 M Varadarajan Gravitons from a loop representation of linearizedgravity Phys Rev D66 (2002) 024017 gr-qc0204067

[207] A Corichi and JM Reyes A Gaussian weave for kinematical loop quan-tum gravity Int J Mod Phys D10 (2001) 325 gr-qc0006067

Chapter 7 Dynamics and matter

[208] R Borissov R De Pietri and C Rovelli Matrix elements of Thiemannrsquoshamiltonian constraint in loop quantum gravity Class and QuantumGrav 14 (1997) 2793 gr-qc9703090

438 References

[209] T Thiemann The phoenix project master constraint programme for loopquantum gravity gr-qc0305080

[210] M Gaul and C Rovelli A generalized hamiltonian constraint operator inloop quantum gravity and its simplest euclidean matrix elements Classand Quantum Grav 18 (2001) 1593ndash1624 gr-qc0011106

[211] T Thiemann QSD V Quantum gravity as the natural regulator of thehamiltonian constraint of matter quantum field theories Class and Quan-tum Grav 15 (1998) 1281ndash1314 gr-qc9705019

[212] S Alexandrov SO(4C)-covariant AshtekarndashBarbero gravity and theImmirzi parameter Class and Quantum Grav 17 (2000) 4255ndash4268S Alexandrov and ER Livine SU(2) loop quantum gravity seen fromcovariant theory Phys Rev D67 (2003) 044009 gr-qc0209105

[213] S Alexandrov and DV Vassilevich Area spectrum in Lorentz covariantloop gravity gr-qc0103105

[214] BS DeWitt Quantum theory of gravity I the canonical theory PhysRev 160 (1967) 1113

[215] B Brugmann R Gambini and J Pullin Jones polynomials for intersectingknots as physical states of quantum gravity Nucl Phys B385 (1992) 587Knot invariants as nondegenerate quantum geometries Phys Rev Lett 68(1992) 431 C Di Bartolo R Gambini J Griego and J Pullin Consistentcanonical quantization of general relativity in the space of Vassiliev knotinvariants Phys Rev Lett 84 (2000) 2314

[216] K Ezawa Nonperturbative solutions for canonical quantum gravity anoverview Phys Repts 286 (1997) 271ndash348 gr-qc9601050

[217] C Rovelli and L Smolin The physical hamiltonian in nonperturbativequantum gravity Phys Rev Lett 72 (1994) 44

[218] C Rovelli Outline of a general covariant quantum field theory and a quan-tum theory of gravity J Math Phys 36 (1995) 6529

[219] T Thiemann Quantum spin dynamics (QSD) Class and Quantum Grav15 (1998) 839ndash73 gr-qc9606089 QSD II The kernel of the WheelerndashDeWitt constraint operator Class and Quantum Grav 15 (1998) 875ndash905 gr-qc9606090 QSD III Quantum constraint algebra and physicalscalar product in quantum general relativity Class and Quantum Grav15 (1998) 1207ndash1247 gr-qc9705017 QSD IV 2 + 1 euclidean quantumgravity as a model to test 3 + 1 lorentzian quantum gravity Class andQuantum Grav 15 (1998) 1249ndash1280 gr-qc9705018 QSD VI QuantumPoincare algebra and a quantum positivity of energy theorem for canon-ical quantum gravity Class and Quantum Grav 15 (1998) 1463ndash1485gr-qc9705020 QSD VII Symplectic structures and continuum lattice for-mulations of gauge field theories Class and Quantum Grav 18 (2001)3293ndash3338 hep-th0005232

[220] L Smolin Quantum gravity with a positive cosmological constant hep-th0209079 L Freidel and L Smolin The linearization of the Kodamastate hep-th0310224

[221] T Thiemann Loop Quantum Gravity An Inside View hep-th0608210

References 439

[222] C Rovelli and H Morales-Tecotl Fermions in quantum gravity Phys RevLett 72 (1995) 3642 Loop space representation of quantum fermions andgravity Nucl Phys B451 (1995) 325

[223] J Baez and K Krasnov Quantization of diffeomorphism-invariant theorieswith fermions J Math Phys 39 (1998) 1251ndash1271 hep-th9703112

[224] T Thiemann Kinematical Hilbert spaces for fermionic and Higgs quan-tum field theories Class and Quantum Grav 15 (1998) 1487ndash1512 gr-qc9705021

[225] M Montesinos and C Rovelli The fermionic contribution to the spec-trum of the area operator in nonperturbative quantum gravity Class andQuantum Grav 15 (1998) 3795ndash3801

[226] S O Bilson-Thompson F Markopoulou L Smolin hep-th0603022[227] A Alekseev AP Polychronakos and M Smedback On area and entropy

of a black hole Phys Lett B574 (2003) 296 AP Polychronakos Areaspectrum and quasinormal modes of black holes hep-th0304135

[228] S Major and L Smolin Quantum deformations of quantum gravity NuclPhys B473 (1996) 267ndash290 gr-qc9512020 R Borissov S Major andL Smolin The geometry of quantum spin networks Class and QuantumGrav 13 (1996) 3183ndash3196

[229] LH Kauffman Map coloring q-deformed spin networks and TuraevndashViroinvariants for three manifolds Int J Mod Phys B6 (1992) 1765ndash1794Erratum B6 (1992) 3249

[230] N Reshetikhin and V Turaev Ribbon graphs and their invariants derivedfrom quantum groups Comm Math Phys 127 (1990) 1ndash26

[231] K Noui and Ph Roche Cosmological deformation of Lorentzian spin foammodels Class and Quantum Grav 20 (2003) 3175ndash3214 gr-qc0211109

Chapter 8 Applications

[232] M Bojowald and HA Morales-Tecotl Cosmological applications of loopquantum gravity to appear in Proc 5th Mexican School (DGFM) TheEarly Universe and Observational Cosmology gr-qc0306008

[233] M Domagala L Lewandowski Black-hole entropy from quantum geome-try Class Quant Grav 21 (2004) 52335243

[234] K A Meissner Black-hole entropy in loop quantum gravity Class QuantGrav 21 (2004) 52455252

[235] A Ashtekar An Introduction to Loop Quantum Gravity Through Cos-mology gr-qc0702030

[236] L Modesto Disappearance of Black Hole Singularity in Quantum GravityPhys Rev D70 (2004) 124009

[237] A Ashtekar and M Bojowald Quantum geometry and Schwarzschild sin-gularity Class Quant Grav 23 (2006) 391ndash411

440 References

[238] A Ashtekar J Baez A Corichi and K Krasnov Quantum geometryand black hole entropy Phys Rev Lett 80 (1998) 904 gr-qc9710007A Ashtekar JC Baez and K Krasnov Quantum geometry of isolatedhorizons and black hole entropy Adv Theor Math Phys 4 (2001) 1ndash94gr-qc0005126

[239] C Rovelli Black hole entropy from loop quantum gravity Phys Rev Lett14 (1996) 3288 Loop quantum gravity and black hole physics Helv PhysActa 69 (1996) 583

[240] GH Hardy and S Ramanujan Proc London Math Soc 2 (1918) 75[241] G Amelino-Camelia Are we at dawn with quantum gravity phenomenol-

ogy Lectures given at 35th Winter School of Theoretical Physics FromCosmology to Quantum Gravity Polanica Poland 2ndash12 Feb 1999 Lec-ture Notes in Physics 541 (2000) 1ndash49 gr-qc9910089

[242] C Rovelli and S Speziale Reconcile Planck-scale discreteness and theLorentzndashFitzgerald contraction Phys Rev D67 (2003) 064019

[243] M Bojowald Absence of singularity in loop quantum cosmology PhysRev Lett 86 (2001) 5227ndash5230 gr-qc0102069

[244] M Bojowald Inflation from quantum geometry Phys Rev Lett 89 (2002)261301 gr-qc0206054

[245] SW Hawking Black holes in general relativity Comm Math Phys 25(1972) 152

[246] JM Bardeen B Carter and SW Hawking The four laws of black holemechanics Comm Math Phys 31 (1973) 161

[247] JD Bekenstein Black holes and the second law Nuovo Cimento Lett 4(1972) 737ndash740 Black holes and entropy Phys Rev D7 (1973) 2333ndash2346Generalized second law for thermodynamics in black hole physics PhysRev D9 (1974) 3292ndash3300

[248] SW Hawking Black hole explosions Nature 248 (1974) 30 Particle cre-ation by black holes Comm Math Phys 43 (1975) 199

[249] K Krasnov Geometrical entropy from loop quantum gravity Phys RevD55 (1997) 3505 On statistical mechanics of gravitational systems GenRel Grav 30 (1998) 53ndash68 gr-qc9605047 On statistical mechanics of aSchwarzschild black hole Gen Rel Grav 30 (1998) 53

[250] JW York Dynamical origin of black hole radiance Phys Rev D28 (1983)2929

[251] G rsquot Hooft Horizon operator approach to black hole quantization gr-qc9402037 L Susskind Some speculations about black hole entropy in stringtheory hep-th9309145 L Susskind L Thorlacius and J Uglum PhysRev D48 (1993) 3743 C Teitelboim Statistical thermodynamics of ablack hole in terms of surface fields Phys Rev D53 (1996) 2870ndash2873 ABuonanno M Gattobigio M Maggiore L Pilo and C Ungarelli EffectiveLagrangian for quantum black holes Nucl Phys B451 (1995) 677

[252] A Ashtekar C Beetle O Dreyer et al Isolated horizons and their appli-cations Phys Rev Lett 85 (2000) 3564ndash3567 gr-qc0006006 A Ashtekar

References 441

Classical and quantum physics of isolated horizons Lect Notes Phys 541(2000) 50ndash70

[253] L Smolin Linking topological quantum field theory and nonperturbativequantum gravity J Math Phys 36 (1995) 6417ndash6455

[254] AP Balachandran L Chandar and A Momen Edge states in gravity andblack hole physics Nucl Phys B461 (1996) 581ndash596 A Momen Edgedynamics for BF theories and gravity Phys Lett 394 (1997) 269ndash274 SCarlip Black hole entropy from conformal field theory in any dimensionPhys Rev Lett 82 (1999) 2828ndash2831

[255] S Hod Bohrrsquos correspondence principle and the area spectrum of quantumblack holes Phys Rev Lett 81 (1998) 4293 Gen Rel Grav 31 (1999)1639 Kerr black hole quasinormal frequencies Phys Rev D67 (2003)081501

[256] O Dreyer Quasinormal modes the area spectrum and black hole entropyPhys Rev Lett 90 (2003) 081301

[257] A Corichi On quasinormal modes black hole entropy and quantum ge-ometry Phys Rev D67 (2003) 087502 gr-qc0212126

[258] JD Bekenstein and VF Mukhanov Spectroscopy of the quantum blackhole Phys Lett B360 (1995) 7ndash12

[259] L Smolin Macroscopic deviations from Hawking radiation In Matters ofGravity 7 gr-qc9602001

[260] M Barreira M Carfora and C Rovelli Physics with loop quantum gravityradiation from quantum black hole Gen Rel Grav 28 (1996) 1293

[261] R Gambini and J Pullin Nonstandard optics from quantum spacetimePhys Rev D59 (1999) 124021 gr-qc9809038 Quantum gravity experi-mental physics Gen Rel Grav 31 (1999) 1631ndash1637

[262] J Alfaro HA Morales-Tecotl and LF Urrutia Quantum gravity cor-rections to neutrino propagation Phys Rev Lett 84 (2000) 2318ndash2321gr-qc9909079 Loop quantum gravity and light propagation Phys RevD65 (2002) 103509 hep-th0108061

[263] C Kozameh and F Parisi Lorentz invariance and the semiclassical ap-proximation of loop quantum gravity gr-qc0310014

[264] J Collins A Perez D Sudarsky L Urrutia H Vucetich Lorentz in-variance An Additional fine tuning problem Phys Rev Lett 93 (2004)191301

[265] G Amelino-Camelia J Ellis NE Mavromatos DV Nanopoulos andS Sarkar Potential sensitivity of gamma-ray buster observations to wavedisperion in vacuo Nature 393 (1998) 763 astro-ph9712103 G Amelino-Camelia An interferometric gravitational wave detector as a quantumgravity apparatus Nature 398 (1999) 216

[266] J Ellis J Hagelin D Nanopoulos and M Srednicki Search for vio-lations of quantum mechanics Nucl Phys B241 (1984) 381 J EllisNE Mavromatos and DV Nanopoulos Testing quantum mechanics inthe neutral kaon system Phys Lett B293 (1992) 142 IC Percival

442 References

and WT Strunz Detection of space-time fluctuations by a model mat-ter interferometer quant-ph9607011 G Amelino-Camelia J Ellis NEMavromatos and DV Nanopoulos Distance measurement and wave dis-persion in a Liouville string approach to quantum gravity Int J ModPhys A12 (1997) 607

Chapter 9 Spinfoams

[267] J Barrett Quantum gravity as topological quantum field theory J MathPhys 36 (1995) 6161ndash6179 gr-qc9506070

[268] ME Peskin and DV Schroeder An Introduction to Quantum Field The-ory (Reading Addison Wesley 1995)

[269] JW Barrett and L Crane Relativistic spin networks and quantum grav-ity J Math Phys 39 (1998) 3296ndash3302

[270] M Bojowald and A Perez Spin foam quantization and anomalies gr-qc0303026

[271] JC Baez JD Christensen TR Halford and DC Tsang Spin foammodels of riemannian quantum gravity Class and Quantum Grav 19(2002) 4627ndash4648 gr-qc0202017

[272] G Roepstorff Path Integral Approach to Quantum Physics An Introduc-tion (Berlin Springer-Verlag 1994)

[273] AS Wightman Quantum field theory in terms of vacuum expectationvalues Phys Rev 101 (1956) 860 RF Streater and AS WightmanPCT Spin and Statistics and All That Mathematical Physics MonographSeries (Reading MA Benjamin-Cummings 1964) K Osterwalder and RSchrader Axioms for euclidean Greenrsquos functions Comm Math Phys 31(1973) 83 Axioms for euclidean Greenrsquos functions 2 42 (1975) 281

[274] A Perez and C Rovelli Observables in quantum gravity gr-qc0104034[275] A Ashtekar D Marolf J Mourao and T Thiemann Constructing Hamil-

tonian quantum theories from path integrals in a diffeomorphism invari-ant context Class and Quantum Grav 17 (2000) 4919ndash4940 quant-ph9904094

[276] J Baez Strings loops knots and gauge fields In Knots and QuantumGravity ed J Baez (Oxford Oxford University Press 1994)

[277] J Iwasaki A reformulation of the PonzanondashRegge quantum gravity modelin terms of surfaces gr-qc9410010 A definition of the PonzanondashReggequantum gravity model in terms of surfaces J Math Phys 36 (1995)6288

[278] M Reisenberger Worldsheet formulations of gauge theories and gravitytalk given at the 7th Marcel Grossmann Meeting Stanford July 1994 gr-qc9412035 A lattice worldsheet sum for 4-d Euclidean general relativitygr-qc9711052

[279] M Reisenberger and C Rovelli Sum over surfaces form of loop quantumgravity Phys Rev D56 (1997) 3490ndash3508 gr-qc9612035

References 443

[280] C Rovelli Quantum gravity as a lsquosum over surfacesrsquo Nucl Phys (ProcSuppl) B57 (1997) 28ndash43

[281] C Rovelli The projector on physical states in loop quantum gravity PhysRev D59 (1999) 104015 gr-qc9806121

[282] F Markopoulou Dual formulation of spin network evolution gr-qc9704013

[283] G Ponzano and T Regge Semiclassical limit of Racah coefficients InSpectroscopy and Group Theoretical Methods in Physics ed F Bloch(Amsterdam North-Holland 1968)

[284] L Crane and D Yetter On the classical limit of the balanced state sumgr-qc9712087 JW Barrett The classical evaluation of relativistic spinnetworks Adv Theor Math Phys 2 (1998) 593ndash600 JW Barrett andRM Williams The asymptotics of an amplitude for the 4-simplex AdvTheor Math Phys 3 (1999) 209ndash214 L Freidel and K Krasnov Simplespin networks as Feynman graphs J Math Phys 41 (2000) 1681ndash1690

[285] T Regge General relativity without coordinates Nuovo Cimento 19(1961) 558ndash571

[286] VG Turaev and OY Viro State sum invariants of 3-manifolds and quan-tum 6j symbols Topology 31 (1992) 865 VG Turaev Quantum Invari-ants of Knots and 3-Manifolds (New York de Gruyter 1994)

[287] D V Boulatov A model of three-dimensional lattice gravity Mod PhysLett A7 (1992) 1629ndash1648

[288] E Brezin C Itzykson G Parisi and J Zuber Comm Math Phys 59(1978) 35 F David Nucl Phys B257 (1985) 45 J Ambjorn B Durhuusand J Frolich Nucl Phys B257 (1985) 433 VA Kazakov IK Kos-tov and AA Migdal Phys Lett 157 (1985) 295 DV Boulatov VAKazakov IK Kostov and AA Migdal Nucl Phys B275 (1986) 641 MDouglas and S Shenker Nucl Phys B335 (1990) 635 D Gross and AAMigdal Phys Rev Lett 64 (1990) 635 E Brezin and VA Kazakov PhysLett B236 (1990) 144 O Alvarez E Marinari and P Windey RandomSurfaces and Quantum Gravity (New York Plenum Press 1991)

[289] H Ooguri Topological lattice models in four dimensions Mod Phys LettA7 (1992) 2799

[290] VG Turaev Quantum Invariants of Knots and 3-manifolds (New Yorkde Gruyter 1994)

[291] AS Schwartz The partition function of degenerate quadratic function-als and RayndashSinger invariants Lett Math Phys 2 (1978) 247ndash252G Horowitz Exactly soluble diffeomorphism-invariant theories CommMath Phys 125 (1989) 417ndash437 D Birmingham M Blau M Rakowskiand G Thompson Topological field theories Phys Rep 209 (1991) 129ndash340

[292] L Crane and D Yetter A categorical construction of 4-D topological quan-tum field theories In Quantum Topology ed L Kauffman and R Baadhio(Singapore World Scientific 1993) pp 120ndash130 hep-th9301062 L

444 References

Crane L Kauffman and D Yetter State-sum invariants of 4-manifoldsI J Knot Theor Ramifications 6 (1997) 177ndash234 hep-th9409167 JDRoberts Skein theory and the TuraevndashViro invariants Topology 34 (1995)771ndash787

[293] C Rovelli Basis of the PonzanondashReggendashTuraevndashVirondashOoguri quantumgravity model is the loop representation basis Phys Rev D48 (1993)2702

[294] R De Pietri and L Freidel SO(4) Plebanski action and relativisticstate sum models Class and Quantum Grav 16 (1999) 2187ndash2196 gr-qc9804071

[295] M Reisenberger Classical Euclidean GR from left-handed area = right-handed area Class and Quantum Grav 16 (1999) 1357ndash1371 gr-qc9804061 On relativistic spin network vertices J Math Phys 40 (1999)2046ndash2054 gr-qc9711052

[296] A Perez Spin foam quantization of Plebanskirsquos action Adv Theor MathPhys 5 (2002) 947ndash968 gr-qc0203058

[297] A Barbieri Quantum tetrahedron and spin networks Nucl Phys B518(1998) 714ndash728

[298] J Baez and J Barrett The quantum tetrahedron in 3 and 4d Adv TheorMath Phys 3 (1999) 815 gr-qc9903060

[299] J Baez Spin foam models Class and Quantum Grav 15 (1998) 1827ndash1858 gr-qc9709052

[300] R De Pietri L Freidel K Krasnov and C Rovelli BarrettndashCrane modelfrom a BoulatovndashOoguri field theory over a homogeneous space NuclPhys B574 (2000) 785ndash806 hep-th9907154

[301] A Perez and C Rovelli A spinfoam model without bubble divergencesNucl Phys B599 (2001) 255ndash282

[302] D Oriti and RM Williams Gluing 4-simplices a derivation of theBarrettndashCrane spinfoam model for Euclidean quantum gravity Phys RevD63 (2001) 024022

[303] M Reisenberger and C Rovelli Spinfoam models as Feynman diagramsgr-qc0002083 Spacetime as a Feynman diagram the connection formu-lation Class and Quantum Grav 18 (2001) 121ndash140 gr-qc0002095

[304] A Mikovic Quantum field theory of spin networks gr-qc0102110[305] A Perez Finiteness of a spinfoam model for euclidean GR Nucl Phys

B599 (2001) 427ndash434[306] L Freidel Group field theory An Overview Int J Theor Phys 44 (2005)

1769ndash1783[307] D Oriti The group field theory approach to quantum gravity gr-

qc0607032[308] L Freidel ER Levine 3d Quantum Gravity and Effective Non-

Commutative Quantum Field Theory Phys Rev Lett 96 (2006) 221301[309] L Freidel and D Louapre Nonperturbative summation over 3d discrete

topologies hep-th0211026

References 445

[310] L Freidel A PonzanondashRegge model of lorentzian 3-dimensional gravityNucl Phys Proc Suppl 88 (2000) 237ndash240 gr-qc0102098

[311] JW Barrett and L Crane A lorentzian signature model for quantumgeneral relativity Class and Quantum Grav 17 (2000) 3101ndash3118 gr-qc9904025

[312] A Perez and C Rovelli Spin foam model for lorentzian general relativityPhys Rev D63 (2001) 041501 gr-qc0009021

[313] L Crane A Perez and C Rovelli Perturbative finiteness in spin foamquantum gravity Phys Rev Lett 87 (2001) 181301 A finiteness proof forthe lorentzian state sum spin foam model for quantum general relativitygr-qc0104057

[314] A Perez and C Rovelli 3 + 1 spin foam model of quantum gravity withspacelike and timelike components Phys Rev D64 (2001) 064002 gr-qc0011037

[315] L Freidel ER Livine and C Rovelli Spectra of length and area in 2 + 1lorentzian loop quantum gravity Class and Quantum Grav 20 (2003)1463ndash1478 gr-qc0212077

[316] L Freidel and K Krasnov Spin foam models and the classical actionprinciple Adv Theor Phys 2 (1998) 1221ndash1285 hep-th9807092

[317] J Ambjorn J Jurkiewicz and R Loll Lorentzian and euclidean quan-tum gravity analytical and numerical results hep-th0001124 A non-perturbative lorentzian path integral for gravity Phys Rev Lett 85 (2000)924 hep-th0002050 J Ambjorn A Dasgupta J Jurkiewicz and R LollA lorentzian cure for euclidean troubles Nucl Phys Proc Suppl 106(2002) 977ndash979 hep-th0201104 R Loll and A Dasgupta A proper timecure for the conformal sickness in quantum gravity Nucl Phys B606(2001) 357ndash379 hep-th0103186

[318] ER Livine and D Oriti Implementing causality in the spin foam quantumgeometry Nucl Phys B663 (2003) 231ndash279 gr-qc0210064

[319] M Arnsdorf Relating covariant and canonical approaches to triangulatedmodels of quantum gravity Class and Quantum Grav 19 (2002) 1065ndash1092 gr-qc0110026

[320] ER Livine Projected spin networks for lorentz connection linking spinfoams and loop gravity Class and Quantum Grav 19 (2002) 5525ndash5542gr-qc0207084

[321] K Noui A Perez Three-dimensional loop quantum gravity Physicalscalar product and spin foam models Class Quant Grav 22 (2005) 1739ndash1762

[322] R Capovilla M Montesinos VA Prieto and E Rojas BF gravity andthe Immirzi parameter Class and Quantum Grav 18 (2001) L49ndashL52

[323] T Thiemann and O Winkler Coherent states for canonical quantum gen-eral relativity and the infinite tensor product extension Nucl Phys B606(2001) 401ndash440 gr-qc0102038

[324] F Markopoulou An insiderrsquos guide to quantum causal histories NuclPhys Proc Suppl 88 (2000) 308ndash313 hep-th9912137

446 References

[325] A Mikovic Spin foam models of matter coupled to gravity Class andQuantum Grav 19 (2002) 2335 Spinfoam models of YangndashMills theorycoupled to gravity Class and Quantum Grav 20 (2003) 239ndash246

[326] D Oriti and H Pfeiffer A spin foam model for pure gauge theory coupledto quantum gravity Phys Rev D66 (2002) 124010

[327] NJ Vilenkin Special Functions and the Theory of Group Representations(Providence Rhode Island American Mathematical Society 1968)

[328] W Ruhl The Lorentz Group and Harmonic Analysis (New YorkWA Benjamin Inc 1970)

[329] L Modesto C Rovelli Particle scattering in loop quantum gravity PhysRev Lett 95 (2005) 191301

[330] C Rovelli Graviton propagator from background-independent quantumgravity Phys Rev Lett 97 151301 (2006) E Bianchi L ModestoC Rovelli and S Speziale Graviton propagator in loop quantum grav-ity Class Quant Grav 23 6989 (2006)

[331] S Speziale Towards the graviton from spinfoams The 3-D toy modelJHEP (2006) 0605039 ER Livine S Speziale JL Willis Towards thegraviton from spinfoams Higher order corrections in the 3-D toy modelPhys Rev D75 (2007) 024038 ER Livine S Speziale Group IntegralTechniques for the Spinfoam Graviton Propagator gr-qc0608131

Chapter 10 Conclusion and Appendices

[332] G Galilei Dialogo dei massimi system (Firenze 1632)[333] DM Brink and GR Satchler Angular Momentum (Oxford Clarendon

Press 1968)[334] R Penrose In Combinatorial Mathematics and its Application

ed D Welsh (New York Academic Press 1971)[335] LH Kauffman Int J Mod Phys A5 (1990) 93[336] LH Kauffman In Knots Topology and Quantum Field Theories

ed L Lusanna (Singapore World Scientific 1991)[337] JP Moussoris in Advances in Twistor Theory Research Notes in Mathe-

matics ed JP Huston and RS Ward (Boston Pitman 1979) pp 308ndash312

[338] I Levinson Liet TSR Mokslu Acad Darbai B Ser 2 (1956) 17[339] AP Yutsin JB Levinson and VV Vanagas Mathematical Apparatus of

the Theory of Angular Momentum (Jerusalem Israel Program for ScientificTranslation 1962)

[340] A Einstein Naeherungsweise Integration der Feldgleichungen derGravitation Preussische Akademie der Wissenschaften (Berlin) Sitzungs-berichte (1916) p 688

[341] O Klein Zur Funfdimensionalen Darstellung der RelativitaetstheorieZ fur Physik 46 (1927) 188

References 447

[342] L Rosenfeld Zur Quantelung der Wellenfelder Ann der Physik 5 (1930)113 Uber die Gravitationswirkungen des Lichtes Z fur Physik 65 (1930)589

[343] M Fierz Hel Physica Acta 12 (1939) 3 W Pauli and M Fierz On rela-tivistic field equations of particles with arbitrary spin in an electromagneticfield Hel Physica Acta 12 (1939) 297

[344] DI Blokhintsev and FM Galrsquoperin Pod Znamenem Marxisma 6 (1934)147

[345] W Heisenberg Z fur Physik 110 (1938) 251[346] J Stachel Early history of quantum gravity (1916ndash1940) Presented at

the HGR5 Notre Dame July 1999 Early history of quantum gravity InBlack Holes Gravitational Radiation and the Universe ed BR Iyer andB Bhawal (Netherlands Kluwer Academic Publishers 1999)

[347] MP Bronstein Quantentheories schwacher Gravitationsfelder Physika-lische Z der Sowietunion 9 (1936) 140

[348] GE Gorelik First steps of quantum gravity and the planck values InStudies in the History of General Relativity [Einstein Studies Vol 3]ed J Eisenstaedt and AJ Kox (Boston Birkhauser 1992) pp 364ndash379GE Gorelik and VY Frenkel Matvei Petrovic Bronstein and the SovietTheoretical Physics in the Thirties (Boston Birkhauser-Verlag 1994)

[349] PG Bergmann Non-linear field theories Phys Rev 75 (1949) 680 Non-linear field theories II canonical equations and quantization Rev ModPhys 21 (1949)

[350] PG Bergmann Nuovo Cimento 3 (1956) 1177[351] ET Newman and PG Bergmann Observables in singular theories by

systematic approximation Rev Mod Phys 29 (1957) 443[352] S Gupta Proc Phys Soc A65 (1952) 608[353] C Misner Feynman quantization of general relativity Rev Mod Phys 29

(1957) 497[354] P Bergmann The canonical formulation of general relativistic theories the

early years 1930ndash1959 In Einstein and the History of General Relativityed D Howard and J Stachel (Boston Birkhauser 1989)

[355] R Feynman Quantum theory of gravitation Acta Physica Polonica 24(1963) 697

[356] B DeWitt In Conference Internationale sur les Theories Relativistes dela Gravitation ed Gauthier-Villars (Warsaw Editions Scientifiques dePologne 1964)

[357] PG Bergmann and A Komar The coordinate group symmetries of gen-eral relativity Int J Theor Phys 5 (1972) 15

[358] A Peres Nuovo Cimento 26 (1962) 53[359] JA Wheeler Geometrodynamics and the issue of the final state In Rel-

ativity Groups and Topology ed C DeWitt and BS DeWitt (New Yorkand London Gordon and Breach 1964) p 316

448 References

[360] BS DeWitt Phys Rev Lett 12 (1964) 742 In Dynamical Theory ofGroups and Fields (New York Wiley 1965)

[361] LD Faddeev and VN Popov Feynman diagrams for the YangndashMills fieldPhys Lett 25B (1967) 30

[362] M Veltman in Proc 6th Int Symp Electron and Photon Interactions atHigh Energies ed H Rollnik and W Pfeil (Amsterdam North Holland1975)

[363] LD Faddeev and VN Popov Perturbation theory for gauge invariantfields Kiev Inst Theor Phys Acad Sci 67-036 (Fermilab Publication72-057-T)

[364] BS DeWitt Quantum theory of gravity II The manifestly covariant the-ory Phys Rev 162 (1967) 1195 Quantum theory of gravity III Applica-tions of the covariant theory Phys Rev 162 (1967) 1239

[365] R Penrose Twistor theory J Math Phys 8 (1967) 345[366] C Misner Quantum cosmology Phys Rev 186 (1969) 1319[367] B Zumino Effective lagrangians and broken symmetries In Brandeis Uni-

versity Lectures On Elementary Particles And Quantum Field Theory Vol2 ed S Deser (MIT Press Cambridge MA 1971) pp 437ndash500

[368] G rsquot Hooft Renormalizable lagrangians for massive YangndashMills fieldsNucl Phys B35 (1971) 167 G rsquot Hooft and M Veltman Regularizationand renormalization of gauge fields Nucl Phys B44 (1972) 189

[369] D Finkelstein Space-time code Phys Rev 184 (1969) 1261ndash1279[370] G trsquoHooft An algorithm for the poles at dimension four in the dimensional

regularization Nucl Phys B62 (1973) 444 G trsquoHooft and M VeltmanOne-loop divergencies in the theory of gravitation Ann Inst Poincare 20(1974) 69 S Deser and P Van Nieuwenhuizen One loop divergences ofthe quantized EinsteinndashMaxwell fields Phys Rev D10 (1974) 401 Non-renormalizability of the quantized DiracndashEinstein system Phys Rev D10(1974) 411

[371] WG Unruh Notes on black hole evaporation Phys Rev D14 (1976) 870[372] G Parisi The theory of non-renormalizable interactions 1 the large-N

expansion Nucl Phys B100 (1975) 368[373] S Ferrara P van Nieuwenhuizen and DZ Freedman Progress toward

a theory of supergravity Phys Rev D13 (1976) 3214 S Deser and PZumino Consistent supergravity Phys Lett B62 (1976) 335 For a reviewsee P van Nieuwenhuizen Supergravity Physics Reports 68 (1981) 189

[374] L Brink P Di Vecchia and P Howe A locally supersymmetric andreparameterization-invariant action for the spinning string Phys LettB65 (1976) 471ndash474 S Deser and B Zumino A complete action for thespinning string Phys Lett B65 (1976) 369

[375] KS Stelle Renormalization of higher derivatives quantum gravity PhysRev D16 (1977) 953

[376] JB Hartle and SW Hawking Path integral derivation of the black holeradiance Phys Rev D13 (1976) 2188

References 449

[377] AM Polyakov Quantum geometry of the bosonic string Phys Lett 103B(1981) 207 Quantum geometry of the fermionic string Phys Lett 103B(1981) 211

[378] G Horowitz and A Strominger in 10th Int Conf on General Relativityand Gravitation ndash Contributed Papers ed B Bertotti F de Felice and APascolini (Padova Universita di Padova 1983)

[379] CJ Isham Quantum logic and the histories approach to quantum theoryJ Math Phys 35 (1994) 2157 gr-qc9308006

[380] RD Sorkin Posets as lattice topologies In General Relativity and Grav-itation Proceedings of the GR10 Conference Volume I ed B BertottiF de Felice and A Pascolini (Rome Consiglio Nazionale Delle Ricerche1983) p 635

[381] MB Green and JH Schwarz Anomaly cancellation in supersymmetricd = 10 gauge theory requires SO(32) Phys Lett 149B (1984) 117

[382] DJ Gross JA Harvey E Martinec and R Rohm The heterotic stringPhys Rev Lett 54 (1985) 502ndash505

[383] P Candelas GT Horowitz A Strominger and E Witten Vacuum con-figurations for superstrings Nucl Phys B258 (1985) 46

[384] AA Belavin AM Polyakov and AB Zamolodchikov Infinite conformalsymmetry in two-dimensional quantum field theory Nucl Phys B241(1984) 333

[385] R Penrose Gravity and state vector reduction In Quantum Concepts inSpace and Time ed R Penrose and CJ Isham (Oxford Clarendon Press1986) p 129

[386] GT Horowitz J Lykken R Rohm and A Strominger A purely cubicaction for string field theory Phys Rev Lett 57 (1986) 283

[387] K Fredenhagen and R Haag Generally covariant quantum field theoryand scaling limits Comm Math Phys 108 (1987) 91

[388] H Sato and T Nakamura eds Marcel Grossmann Meeting on GeneralRelativity (Singapore World Scientific 1992)

[389] E Witten Topological quantum field theory Comm Math Phys 117(1988) 353

[390] E Witten Quantum field theory and the Jones polynomial Comm MathPhys 121 (1989) 351

[391] MF Atiyah Topological quantum field theories Publ Math Inst HautesEtudes Sci Paris 68 (1989) 175 The Geometry and Physics of Knots edAccademia Nazionale dei Lincei (Cambridge Cambridge University Press1990)

[392] GT Horowitz Exactly soluble diffeomorphism invariant theories CommMath Phys 125 (1989) 417

[393] E Witten (2 + 1)-dimensional gravity as an exactly soluble system NuclPhys B311 (1988) 46

[394] S Carlip Lectures on (2 + 1)-dimensional gravity (lecture given at theFirst Seoul Workshop on Gravity and Cosmology February 24ndash25 1995)

450 References

[395] S Deser and R Jackiw Three-dimensional cosmological gravity dynam-ics of constant curvature Ann Phys 153 (1984) 405 S Deser R Jackiwand G rsquot Hooft Three-dimensional Einstein gravity dynamics of flatspace Ann Phys 152 (1984) 220 A Achucarro and PK Townsend AChernndashSimon action for three-dimensional antidesitter supergravity theo-ries Phys Lett B180 (1986) 89


[397] D Gross and A Migdal Nonperturbative two-dimensional quantum grav-ity Phys Rev Lett 64 (1990) 635 M Douglas and S Shenker Nucl PhysB335 (1990) 635 E Brezin and VA Kazakov Phys Lett B236 (1990)144 Random Surfaces and Quantum Gravity ed O Alvarez E Marinariand P Windey (New York Plenum Press 1991)

[398] CG Callan BS Giddings JA Harvey and A Strominger Evanescentblack holes Phys Rev D45 (1992) 1005

[399] CW Misner KS Thorne and JA Wheeler Gravitation (San FranciscoFreeman 1973)

[400] G rsquotHooft Dimensional reduction in quantum gravity Utrecht PreprintTHU-9326 gr-qc9310026 L Susskind The world as a hologram JMath Phys 36 (1995) 6377

[401] AH Chamseddine and A Connes Universal formula for noncommutativegeometry actions unification of gravity and the standard model PhysRev Lett 24 (1996) 4868 The spectral action principle Comm MathPhys 186 (1997) 731

[402] J Polchinski Dirichlet branes and RamonndashRamon charges Phys RevLett 75 (1995) 4724

[403] CM Hull and PK Townsend Unity of superstring dualities Nucl PhysB438 (1995) 109

[404] T Banks W Fischler SH Shenker and L Susskind M-theory as a matrixmodel a conjecture Phys Rev D55 (1997) 5112

[405] MJ Duff M-Theory (the theory formerly known as strings) Int J ModPhys A11 (1996) 5623

[406] S Frittelli C Kozameh and ET Newman GR via characteristic surfacesJ Math Phys 5 (1995) 4984 5005 6397 T Newman in On EinsteinrsquosPath ed A Harvey (New York Berlin Heidelberg Springer-Verlag 1999)

[407] A Strominger and G Vafa Microscopic origin of the BekensteinndashHawkingentropy Phys Lett B379 (1996) 99 G Horowitz and A Strominger Blackstrings and p-branes Nucl Phys B360 (1991) 197 J Maldacena and AStrominger Black hole grey body factor and D-brane spectroscopy PhysRev D55 (1997) 861 G Horowitz Quantum states of black holes In ProcSymp Black Holes and Relativistic Stars in Honor of S ChandrasekharDecember 1996 gr-qc9704072

[408] A Connes MR Douglas and A Schwarz Noncommutative geometry andmatrix theory compactification on tori JHEP 9802 (1998) 003

References 451

[409] J Ambjorn M Carfora and A Marzuoli The Geometry of DynamicalTriangulations Lecture Notes in Physics (Berlin Springer-Verlag 1997)

[410] JM Maldacena The large-N limit of superconformal field theories and su-pergravity Adv Theor Math Phys 2 (1998) 231 Int J Theor Phys 38(1999) 1113 E Witten Anti-deSitter space and holography Adv TheorMath Phys 2 (1998) 253

[411] M Gasperini and G Veneziano Pre-Big Bang in string cosmology As-tropart Phys 1 (1993) 317

[412] P Bergmann in Conference Internationale sur les Theories Relativistesde la Gravitation ed Gauthier-Villars (Warsaw Scientifiques de Pologne1964)

[413] P van Nieuwenhuizen in Proc First Marcel Grossmann Meeting on Gen-eral Relativity ed R Ruffini (Amsterdam North Holland 1977)

[414] P Bergmann in Cosmology and Gravitation ed P Bergmann and V DeSabbata (New York Plenum Press 1980)

Index

n-j symbols 377SO(2 1) 359SO(3) 36 146 331 360SO(3 1) 34 359SO(4) 341 343 390

Casimirs 349 355SO(n) 390SU(2) 227 377

local gauge invariance 146 234so(3) basis xxiiso(3 1C) 36su(2) basis xxii 380

4-simplex 332 343

Achucarro 409action-at-a-distance 49 50ADM formalism 400Alexandrov 295Alfaro 315Amati 409ambient isotopic 384Anderson 96arc 282area

definition 43 150in spinfoams 350operator 249 different orderings

293spectrum degenerate sector 259

main sequence 249Aristotle 53 77Arnold 98

Arnowit 163 400Ashtekar xxv 163 275 318 408AshtekarndashLewandowski measure 231Atiyah 409

background 12background independence 7ndash31

263ndash265 281 371in strings 8 412

background-independent QFT 13Baez 274 364Barbero connection 156 163 276Barbieri 364Barbour 96Bardeen 301 318Barrett 349 364 411BarrettndashCrane model 330 348Bekenstein 301 312 404BekensteinndashMukhanov effect 293

312Belavin 407Bergmann 162 399ndash401 413BF theory 340 350binor calculus 383bivector 354black hole

entropy 301 308 372 404 411extremal 411ringing modes 312

Bohr 28 398Bojowald 318 353Boltzmann 29

452

Index 453

boosted geometrical operators 318Boulatov 364boundary data space 123 134BrinkndashSatcheler diagrams 390Bronstein 398bubble 338 342

CalabyndashYau manifolds 407Cartan 60 96

structure equation 35 46 331 333Carter 301 318ChernndashSimon

functional 409theory 411

ChoquetndashBruhat 96Christoffel connection 46chromatic evaluation 386Ciafaloni 409Ciufolini 96classical limit 103 179 268 373ClebschndashGordan condition 379color 252coloring 234 252 326configuration space

nonrelativistic 98relativistic 101 105 107

conformal field theory 407Connes 10 32 144 206 410contiguity 77 220Copernicus 419correlation 105 175 218cosmic-ray energy thresholds 316cosmological constant 35 293 330

343 371covariant derivative 34 37 38 46Crane 349 355 411CranendashYetter model 330 342curvature 34curve xxicycle 345cylindrical functions 227

DeDonder 144hamiltonian 132

degenerate sector 253De Pietri 275 364 365Descartes 28 48 53 77

Deser 163 400 404 409DeWitt 4 17 399 401 402DeWittndashMorette 96diffeomorphisms 41 146 229 238

active and passive 62extended 232 266

DillardndashBleick 96Dirac 4 144 162 211 221 400 417divergences 359

infrared 293 339 342ultraviolet 277 282 291 339 407

Earman 96edge 325 329 332Egan 32Einstein 10 28 33 47 48 50 51

55 65 66 68 71 74 209 415416

equations 35 36 38 39 47EinsteinndashSchrodinger equation 226

402energy 203energy-momentum 39event 105

space 105evolution equation 106extended loop representation 294eyeglasses graph 238

face 325 329 332Faddeev 402FaddeevndashPopov ghosts 402Faraday 17 28 49 415

lines 16 49Fermi theory 7 8fermion 36 38 287Feynman 4 320 321 401 402

expansion 343 348Feynman rules for GR 403Fierz 399finiteness 280 289 359Finkelstein 5 222 404flat 35Fleischhack 274Fock space 190 272four-simplex 329 341Fraser 96

454 Index

Fredenhagen 408Freidel 364ndash365Friedmann

equation 57 297models 296

functional representation 186

Galileo 28 29 374 416Gambini 274 294 315 408gamma-ray bursts 316 412Gelfand spectrum 231Gelfand triple 167GelfandndashNeimarkndashSiegal

construction 362GFT (group field theory) 330 343

356lorentzian 359

Goedelrsquos solution 76Gorelik 398Goroff 31 407GPS coordinates 88grains of space 18graph 227

4-strand 345null 231subspaces 230 proper 231

grasp 244 245 253 282gravitational electric field 148 243graviton 272 398Green 32 407 408Gupta 399

Haag 408Hamilton 104

equations 99 111function 103 120 in field theory

135 of GR 151HamiltonndashJacobi

formalism 102 in field theory 137in GR 146

function characteristic 103principal 103

relativistic formalism 113hamiltonian

nonrelativistic 99relativistic 108 177

hand 250harmonic oscillator 104

Hartle 164 407Hawking 4 32 301 302 318 320

404 405 407Heisenberg 28 398 415higher-derivative theories 7 406Hodge star xxihole argument 68 80holographic principle 412holonomy 44 227 242Holst 36 156Horowitz 406 409Huygens 416

Immirzi parameter 19 156 163 250303 365

Inertia 57Inertial frame 57ndash61inflation 297 301inflaton 297 301information 218initial singularity 298 372intertwiners 199 237 381Isham 32 275 408Iwasaki 411

Jackiw 409Jacobson 274 408JacobsonndashSmolin solutions 408Jones polynomial 409

Kauffman brackets 384KauffmanndashLins diagrams 389Kepler 419Klein 398KMS 205knot 241Komar 401Krasnov 318Kretschmann 78 96Kuchar 32

Lagrange 98 129lagrangian 98Landau 33Lapse function 111lattice YangndashMills theory 16 198

227 228 229 230 269 271Leibnitz 55

Index 455

lengthdefinition 43operator 275

LensendashThirring effect 76Lewandowski 274 275linear connection 46link 234 329

virtual 257 258 267 342 352 381Livine 365local Schrodinger equation 194Loll 274loop

operator 250state 15 228 236transform 228 409

Lorentzlocal transformations 41

Lorentz invariance 316lorentzian theory 292

M-theory 412Mach 55

principle 75Major 275Maldacena 412Mandelstam 16 17Markopoulou 364 365Maxwell 28 416measurement 210

quantum gravitational 368metric 46metric geometry 61Minkoswski solution 40Misner 163 400 403 409 413Modesto 318Morales-Tecotl 315motion 99 105 107motion absolute or relative 53moving frame 60Mukhanov 312multiloop 228multiplicity 234

new variable 408Newman 399 411Newton 28 30 48 54 415 417

bucket 54 76constant 35

NewtonndashWigner operator 190node 234 329Norton 96Noui 363 365null surface formulation 411

observablecomplete 178gravitational 367nonrelativistic 105partial 105 172 177 in field

theory 130relativistic 105

observablespartial 107

observer 210Oeckl 221Okolow 274Ooguri 364 409

model 330orbit 101Oriti 364

Palandri 32Palatini 96Parisi 405particle

global 196in GR 39local 196scattering 363

path xxiPauli 399Pauli matrices 380pendulum 104 105

timeless double 109 112 113 122181

Penrose 4 5 274 383 402 403407

evaluation 386Peres 401Perez 363ndash365PeterndashWeyl theorem 230phase space

nonrelativistic 99of GR 150relativistic 98 102 106 107

Planck 419

456 Index

Plebanskiconstraints 37 159 353 355two-form 36 96 147 160

Poincare one-form 100Polchinksi 32Polyakov 16 17 406 407PonzanondashRegge

ansatz 336model 330 334 364 403

presymplectic mechanics 100110

projective family 230projector 170 178 226propagator 167 185 321Pullin 274 294 315pulsar velocities 316

quadritangent 131quanta of area 19 249quantum cosmology 296 403

loop 299quantum event 211quantum gravity phenomenology

412quantum group 293 340 343quantum tetrahedron 355

reality conditions 128 145 208recoupling

theorem 257 388theory 251 383

reduction formulae 388Regge 5

calculus 335 401Reisenberger 365 411relationalism 77 220relativistic particle 118 122Ricci

scalar 35tensor 35 47

Riemannconnection 46geometry 47tensor 47

rigged Hilbert space 167Robinson 32Rosenfeld 398 399

s-knot 241 263Sagnotti 31 407Sahlmann 271scale factor 297Schrodinger 211

equation 165 168 370local equation 194picture 370representation 191

Schwarz 32 407 408Schwarzschild solution 269 302 304Schwinger

equations 400function 192

selfdualconnection 36projector 36

Sen 96 408seperating 205simple representation 349 355 390simplicity relation 390Smolin 32 274 365 408soldering form 60Sorkin 5 407Souriau 98space entity or relation 52spacetime coincidences 70 74 95

220spacetime foam 28spectral action 410spherical harmonics 390spherical vector 391spike 339spin 230spin connection 34spin network 16 17 200 234 274

347 372 402abstract 19embedded 21lorentzian 294state 200 234 236

spinfoam 26 325 412GFT duality 343

spinor 377spinor calculus 383Stachel 398

Index 457

standard model 5 38 276 286 291state 210n-particle 189boundary 174 176 178empty 203 273equilibrium 141 204

Gibbs 204instantaneous 102 116kinematical 169ndash178 193 229one-particle 188particle 195relativistic 106spacetime 168statistical 141 204

state-sum 320Stelle 405string field theory 407string theory 7 406Strominger 406substantivalism 77sum-over-geometries 28 320 322sum-over-surfaces 320 338supergravity 7 405 406superstrings 408supersymmetry 7 13Susskind 412symplectic mechanics 99

rsquot Hooft 4 5 404 409 412tangles 384target space 130tetrad 34 60tetrahedral net 387thermal clock 143thermal fluctuations of the geometry

305thermal time hypothesis 143 206Thiemann 163 274 284 295 310

365 411Thorne 409time

clock 84coordinate 83cosmological 31 86newtonian 85parameter 86

proper 84thermal 142 205thermodynamical 85

TOCY model 330 344Tomita flow 205topological field theory 332 340 409torsion 34 46

free spin connection 34Townsend 409transition amplitudes 178 200 323

346triangulation 328 332

dual 332independence 340

Turaev 409TuraevndashViro model 330 340twistor theory 403two-complex 326two-skeleton 332

unitarity 369Unruh 404Urrita 315

vacuum 202covariant 23 176 178 197dynamical 174 175 193Minkowski 175 188 192 271 273

363 374Van Nieuwenhuizen 404variational principle 102 108 131Veltman 402 404Veneziano 409vertex 325 329 332

amplitude 328Viro 409volume

definition 44 150operator 260 different

regularizations 294quanta of 263

Wald 96wave function of the Universe 407weave 268Weinberg 4 32 96 405Weyl 96 144

458 Index

whale 9 75Wheeler 4 17 28 32 96 310 400

401 403 409WheelerndashDeWitt equation 169 178

185 226 276 295 403Wilson 16 17 405

loop 15

Winkler 365Witten 32 408ndash410

YangndashMills field 37 286

Zamolodchikov 407Zumino 403

Cover

Front matter

Title13

Copyright13

Contents13

Foreword13

Preface13


Acknowledgements


Part I Relativistic foundations








121 Why loops







21 Formalism







221 Einsteinrsquos first problem a field theory for the newtonian interaction


223 The key idea



23 Interpretation



24 Complements

241 Mach principles


243 Has general covariance any physical content




3 Mechanics

31 Nonrelativistic mechanics





324 Mechanics is about relations between observables




33 Field theory











421 Euclidean GR



43 Hamiltonian GR






5 Quantum mechanics





514 Boundary state space K and covariant vacuum |0gt13


52 Relativistic QM









535 Boundary state space K and covariant vacuum |0gt


54 Quantum gravity



55 Complements







563 Information


Part II Loop quantum gravity

6 Quantum space


62 The kinematical state space K

621 Structures in K



63 Internal gauge invariance The space Ko



64 Diffeomorphism invariance The space K subscript diff



65 Operators



66 Operators on K subscript 0


662 Quanta of area




67 Quantum geometry




711 Finiteness

712 Matrix elements

713 Variants


721 YangndashMills

722 Fermions

723 Scalars




741 Variants

8 Applications


811 Inflation










921 Boundaries

93 Models


932 BF theory


934 BC models





10 Conclusion


1011 GR and QM





Part III Appendices

Appendix A Groups and recoupling theory





A23 Normalizations


Appendix B History


B2 Five periods






B3 The divide

Appendix C On method and truth


C2 On realism

C3 On truth

References

Index

CAMBRIDGE MONOGRAPHS ONMATHEMATICAL PHYSICS

General editors P V Landshoff D R Nelson S Weinberg

S J Aarseth Gravitational N-Body Simulations

J Ambjorn B Durhuus and T Jonsson Quantum Geometry A Statistical Field Theory

Approach

A M Anile Relativistic Fluids and Magneto-Fluids

J A de Azcarrage and J M Izquierdo Lie Groups Lie Algebras Cohomology and Some

Applications in Physicsdagger

O Babelon D Bernard and M Talon Introduction to Classical Integrable Systems

V Belinkski and E Verdaguer Gravitational Solitons

J Bernstein Kinetic Theory in the Expanding Universe

G F Bertsch and R A Broglia Oscillations in Finite Quantum Systems

N D Birrell and PCW Davies Quantum Fields in Curved spacedagger

M Burgess Classical Covariant Fields

S Carlip Quantum Gravity in 2+1 Dimensions

J C Collins Renormalizationdagger

M Creutz Quarks Gluons and Latticesdagger

P D DrsquoEath Supersymmetric Quantum Cosmology

F de Felice and C J S Clarke Relativity on Curved Manifoldsdagger

B S DeWitt Supermanifolds 2nd editiondagger

P G O Freund Introduction to Supersymmetrydagger

J Fuchs Affine Lie Algebras and Quantum Groupsdagger

J Fuchs and C Schweigert Symmetries Lie Algebras and Representations A Graduate Course

for Physicistsdagger

Y Fujii and K Maeda The ScalarndashTensor Theory of Gravitation

A S Galperin E A Ivanov V I Orievetsky and E S Sokatchev Harmonic Superspace

R Gambini and J Pullin Loops Knots Gauge Theories and Quantum Gravitydagger

M Gockeler and T Schucker Differential Geometry Gauge Theories and Gravitydagger

C Gomez M Ruiz Altaba and G Sierra Quantum Groups in Two-dimensional Physics

M B Green J H Schwarz and E Witten Superstring Theory volume 1 Introductiondagger

M B Green J H Schwarz and E Witten Superstring Theory volume 2 Loop Amplitudes

Anomalies and Phenomenologydagger

V N Gribov The Theory of Complex Angular Momenta

S W Hawking and G F R Ellis The Large-Scale Structure of Space-Timedagger

F Iachello and A Arima The Interacting Boson Model

F Iachello and P van Isacker The Interacting BosonndashFermion Model

C Itzykson and J-M Drouffe Statistical Field Theory volume 1 From Brownian Motion to

Renormalization and Lattice Gauge Theorydagger

C Itzykson and J-M Drouffe Statistical Field Theory volume 2 Strong Coupling Monte

Carlo Methods Conformal Field Theory and Random Systemsdagger

C Johnson D-Branes

J I Kapusta Finite-Temperature Field Theorydagger

V E Korepin A G Izergin and N M Boguliubov The Quantum Inverse Scattering Method

and Correlation Functionsdagger

M Le Bellac Thermal Field Theorydagger

Y Makeenko Methods of Contemporary Gauge Theory

N Manton and P Sutcliffe Topological Solitons

N H March Liquid Metals Concepts and Theory

I M Montvay and G Munster Quantum Fields on a Latticedagger

L Orsquo Raifeartaigh Group Structure of Gauge Theoriesdagger

T Ortın Gravity and Strings

A Ozorio de Almeida Hamiltonian Systems Chaos and Quantizationdagger

R Penrose and W Rindler Spinors and Space-Time volume 1 Two-Spinor Calculus and

Relativistic Fieldsdagger

R Penrose and W Rindler Spinors and Space-Time volume 2 Spinor and Twistor Methods in

Space-Time Geometrydagger

S Pokorski Gauge Field Theories 2nd edition

J Polchinski String Theory volume 1 An Introduction to the Bosonic String

J Polchinski String Theory volume 2 Superstring Theory and Beyond

V N Popov Functional Integrals and Collective Excitationsdagger

R J Rivers Path Integral Methods in Quantum Field Theorydagger

R G Roberts The Structure of the Protondagger

C Rovelli Quantum Gravity

W C Saslaw Gravitational Physics of Stellar and Galactic Systemsdagger

H Stephani D Kramer M A H MacCallum C Hoenselaers and E Herlt Exact Solutions

of Einsteinrsquos Field Equations 2nd edition

J M Stewart Advanced General Relativitydagger

A Vilenkin and E P S Shellard Cosmic Strings and Other Topological Defectsdagger

R S Ward and R O Wells Jr Twistor Geometry and Field Theoriesdagger

J R Wilson and G J Mathews Relativistic Numerical Hydrodynamics

daggerIssued as a paperback

Quantum Gravity

CARLO ROVELLICentre de Physique Theorique de LuminyUniversite de la Mediterranee Marseille

cambridge university press Cambridge New York Melbourne Madrid Cape Town

Singapore Satildeo Paulo Delhi Tokyo Mexico City

Cambridge University Press The Edinburgh Building Cambridge CB2 8RU UK

Published in the United States of America by Cambridge University Press New York

wwwcambridgeorg Information on this title wwwcambridgeorg9780521715966

copy Cambridge University Press 2004

This publication is in copyright Subject to statutory exception and to the provisions of relevant collective licensing agreements no reproduction of any part may take place without the written

permission of Cambridge University Press

First published 2004Reprinted 2005

First paperback edition published with correction 2008 Reprinted 2010

A catalogue record for this publication is available from the British Library

Library of Congress Cataloguing in Publication data

isbn 978-0-521-83733-0 Hardback isbn 978-0-521-71596-6 Paperback

Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in

this publication and does not guarantee that any content on such websites is or will remain accurate or appropriate Information regarding prices travel

timetables and other factual information given in this work is correct at the time of first printing but Cambridge University Press does not guarantee

the accuracy of such information thereafter

Contents

Foreword by James Bjorken page xi

Preface xiii

Preface to the paperback edition xvii

Acknowledgements xix

Terminology and notation xxi

Part 1 Relativistic foundations 1

1 General ideas and heuristic picture 311 The problem of quantum gravity 3

111 Unfinished revolution 3112 How to search for quantum gravity 4113 The physical meaning of general relativity 9114 Background-independent quantum field theory 10

12 Loop quantum gravity 13121 Why loops 14122 Quantum space spin networks 17123 Dynamics in background-independent QFT 22124 Quantum spacetime spinfoam 26

13 Conceptual issues 28131 Physics without time 29

v

vi Contents

2 General Relativity 3321 Formalism 33

211 Gravitational field 33212 ldquoMatterrdquo 37213 Gauge invariance 40214 Physical geometry 42215 Holonomy and metric 44

22 The conceptual path to the theory 48221 Einsteinrsquos first problem a field theory for the

newtonian interaction 48222 Einsteinrsquos second problem relativity of motion 52223 The key idea 56224 Active and passive diffeomorphisms 62225 General covariance 65

23 Interpretation 71231 Observables predictions and coordinates 71232 The disappearance of spacetime 73

24 Complements 75241 Mach principles 75242 Relationalism versus substantivalism 76243 Has general covariance any physical content 78244 Meanings of time 82245 Nonrelativistic coordinates 87246 Physical coordinates and GPS observables 88

3 Mechanics 9831 Nonrelativistic mechanics 9832 Relativistic mechanics 105

321 Structure of relativistic systems partialobservablesrelativistic states 105

322 Hamiltonian mechanics 107323 Nonrelativistic systems as a special case 114324 Mechanics is about relations between observables 118325 Space of boundary data G and Hamilton function

S 120326 Evolution parameters 126327 Complex variables and reality conditions 128

33 Field theory 129331 Partial observables in field theory 129332 Relativistic hamiltonian mechanics 130333 The space of boundary data G and the Hamilton

function S 133

Contents vii

334 HamiltonndashJacobi 13734 Thermal time hypothesis 140

4 Hamiltonian general relativity 14541 EinsteinndashHamiltonndashJacobi 145

411 3d fields ldquoThe length of the electric field is theareardquo 147

412 Hamilton function of GR and its physical meaning 15142 Euclidean GR and real connection 153

421 Euclidean GR 153422 Lorentzian GR with a real connection 155423 Barbero connection and Immirzi parameter 156

43 Hamiltonian GR 157431 Version 1 real SO(3 1) connection 157432 Version 2 complex SO(3) connection 157433 Configuration space and hamiltonian 158434 Derivation of the HamiltonndashJacobi formalism 159435 Reality conditions 162

5 Quantum mechanics 16451 Nonrelativistic QM 164

511 Propagator and spacetime states 166512 Kinematical state space K and ldquoprojectorrdquo P 169513 Partial observables and probabilities 172514 Boundary state space K and covariant vacuum |0〉 174515 Evolving constants of motion 176

52 Relativistic QM 177521 General structure 177522 Quantization and classical limit 179523 Examples pendulum and timeless double

pendulum 18053 Quantum field theory 184

531 Functional representation 186532 Field propagator between parallel boundary

surfaces 190533 Arbitrary boundary surfaces 193534 What is a particle 195535 Boundary state space K and covariant vacuum |0〉 197536 Lattice scalar product intertwiners and spin

network states 19854 Quantum gravity 200

541 Transition amplitudes in quantum gravity 200542 Much ado about nothing the vacuum 202

viii Contents

55 Complements 204551 Thermal time hypothesis and Tomita flow 204552 The ldquochoicerdquo of the physical scalar product 206553 Reality conditions and scalar product 208

56 Relational interpretation of quantum theory 209561 The observer observed 210562 Facts are interactions 215563 Information 218564 Spacetime relationalism versus quantum

relationalism 220

Part II Loop quantum gravity 223

6 Quantum space 22561 Structure of quantum gravity 22562 The kinematical state space K 226

621 Structures in K 230622 Invariances of the scalar product 231623 Gauge-invariant and diffeomorphism-invariant

states 23363 Internal gauge invariance The space K0 234

631 Spin network states 234632 Details about spin networks 236

64 Diffeomorphism invariance The space Kdiff 238641 Knots and s-knot states 240642 The Hilbert space Kdiff is separable 241

65 Operators 242651 The connection A 242652 The conjugate momentum E 243

66 Operators on K0 246661 The operator A(S) 246662 Quanta of area 249663 n-hand operators and recoupling theory 250664 Degenerate sector 253665 Quanta of volume 259

67 Quantum geometry 262671 The texture of space weaves 268

7 Dynamics and matter 27671 Hamiltonian operator 277

711 Finiteness 280712 Matrix elements 282

Contents ix

713 Variants 28472 Matter kinematics 285

721 YangndashMills 286722 Fermions 287723 Scalars 288724 The quantum states of space and matter 289

73 Matter dynamics and finiteness 28974 Loop quantum gravity 291

741 Variants 293

8 Applications 29681 Loop quantum cosmology 296

811 Inflation 30182 Black-hole thermodynamics 301

821 The statistical ensemble 303822 Derivation of the BekensteinndashHawking entropy 308823 Ringing modes frequencies 311824 The BekensteinndashMukhanov effect 312


9 Quantum spacetime spinfoams 32091 From loops to spinfoams 32192 Spinfoam formalism 327

921 Boundaries 32893 Models 329

931 3d quantum gravity 331932 BF theory 340933 The spinfoamGFT duality 343934 BC models 348935 Group field theory 356936 Lorentzian models 359

94 Physics from spinfoams 361941 Particlesrsquo scattering and Minkowski vacuum 363

10 Conclusion 366101 The physical picture of loop gravity 366

1011 GR and QM 3661012 Observables and predictions 3671013 Space time and unitarity 3681014 Quantum gravity and other open problems 370


x Contents

Part III Appendices 375

Appendix A Groups and recoupling theory 377A1 SU(2) spinors intertwiners n-j symbols 377A2 Recoupling theory 383

A21 Penrose binor calculus 383A22 KL recoupling theory 385A23 Normalizations 388

A3 SO(n) and simple representations 390

Appendix B History 393B1 Three main directions 393B2 Five periods 395

B21 The Prehistory 1930ndash1957 398B22 The Classical Age 1958ndash1969 400B23 The Middle Ages 1970ndash1983 403B24 The Renaissance 1984ndash1994 407B25 Nowadays 1995ndash 410

B3 The divide 412

Appendix C On method and truth 415C1 The cumulative aspects of scientific knowledge 415C2 On realism 420C3 On truth 422

References 424

Index 452

Foreword

The problem of what happens to classical general relativity at the extremeshort-distance Planck scale of 10minus33 cm is clearly one of the most pressingin all of physics It seems abundantly clear that profound modifications ofexisting theoretical structures will be mandatory by the time one reachesthat distance scale There exist several serious responses to this chal-lenge These include effective field theory string theory loop quantumgravity thermogravity holography and emergent gravity Effective fieldtheory is to gravitation as chiral perturbation theory is to quantum chro-modynamics ndash appropriate at large distances and impotent at short Itsprimary contribution is the recognition that the EinsteinndashHilbert action isno doubt only the first term in an infinite series constructed out of higherpowers of the curvature tensor String theory emphasizes the possibleroles of supersymmetry extra dimensions and the standard-model inter-nal symmetries in shaping the form of the microscopic theory Loop grav-ity most directly attacks the fundamental quantum issues and featuresthe construction of candidate wave-functionals which are background in-dependent Thermogravity explores the apparent deep connection of semi-classical gravity to thermodynamic concepts such as temperature and en-tropy The closely related holographic ideas connect theories defined inbulk spacetimes to complementary descriptions residing on the bound-aries Finally emergent gravity suggests that the time-tested symbioticrelationship between condensed matter theory and elementary particletheory should be extended to the gravitational and cosmological contextsas well with more lessons yet to be learned

In each of the approaches difficult problems stand in the way of at-taining a fully satisfactory solution to the basic issues Each has its bandof enthusiasts the largest by far being the string community Most of theapproaches come with rather strong ideologies especially apparent whenthey are popularized The presence of these ideologies tends to isolate the

xi

xii Foreword

communities from each other In my opinion this is extremely unfortu-nate because it is probable that all these ideologies including my own(which is distinct from the above listing) are dead wrong The evidence ishistory from the Greeks to Kepler to Newton to Einstein there has beenno shortage of grand ideas regarding the Basic Questions In the presenceof new data available to us and not them only fragments of those grandvisions remain viable The clutter of thirty-odd standard model param-eters and the descriptive nature of modern cosmology suggests that wetoo have quite a way to go before ultimate simplicity is attained Thisdoes not mean abandoning ideologies ndash they are absolutely essential indriving us all to work hard on the problems But it does mean that anattitude of humility and of high sensitivity toward alternative approachesis essential

This book is about only one approach to the subject ndash loop quantumgravity It is a subject of considerable technical difficulty and the litera-ture devoted to it is a formidable one This feature alone has hindered thecross-fertilization which is as delineated above so essential for progressHowever within these pages one will find a much more accessible de-scription of the subject put forward by one of its leading architects anddeepest thinkers The existence of such a fine book will allow this im-portant subject quite likely to contribute significantly to the unknownultimate theory to be assimilated by a much larger community of the-orists If this does indeed come to pass its publication will become oneof the most important developments in this very active subfield since itsonset

James Bjorken

Preface

A dream I have long held was to write a ldquotreatiserdquo on quantum gravityonce the theory had been finally found and experimentally confirmed Weare not yet there There is neither experimental support nor sufficienttheoretical consensus Still a large amount of work has been developedover the last twenty years towards a quantum theory of spacetime Manyissues have been clarified and a definite approach has crystallized Theapproach variously denoted1 is mostly known as ldquoloop quantum gravityrdquo

The problem of quantum gravity has many aspects Ideas and resultsare scattered in the literature In this book I have attempted to collect themain results and to present an overall perspective on quantum gravity asdeveloped during this twenty-year period The point of view is personaland the choice of subjects is determined by my own interests I apologizeto friends and colleagues for what is missing the reason so much is missingis due to my own limitations for which I am the first to be sorry

It is difficult to over-estimate the vastitude of the problem of quantumgravity The physics of the early twentieth century has modified the veryfoundation our understanding of the physical world changing the meaningof the basic concepts we use to grasp it matter causality space and timeWe are not yet able to paint a consistent picture of the world in whichthese modifications taken together make sense The problem of quantumgravity is nothing less than the problem of finding this novel consistentpicture finally bringing the twentieth century scientific revolution to anend

Solving a problem of this sort is not just a matter of mathematical skillAs was the case with the birth of quantum mechanics relativity electro-magnetism and newtonian mechanics there are conceptual and founda-tional problems to be addressed We have to understand which (possibly

1See the notation section

xiv Preface

new) notions make sense and which old notions must be discarded inorder to describe spacetime in the quantum relativistic regime What weneed is not just a technique for computing say gravitonndashgraviton scatter-ing amplitudes (although we certainly want to be able to do so eventu-ally) We need to re-think the world in the light of what we have learnedabout it with quantum theory and general relativity

General relativity in particular has modified our understanding of thespatio-temporal structure of reality in a way whose consequences havenot yet been fully explored A significant part of the research in quantumgravity explores foundational issues and Part I of this book (ldquoRelativisticfoundationsrdquo) is devoted to these basic issues It is an exploration of howto rethink basic physics from scratch after the general-relativistic con-ceptual revolution Without this we risk asking any tentative quantumtheory of gravity the wrong kind of questions

Part II of the book (ldquoLoop quantum gravityrdquo) focuses on the loopapproach The loop theory described in Part II can be studied by itselfbut its reason and interpretation are only clear in the light of the generalframework studied in Part I Although several aspects of this theory arestill incomplete the subject is mature enough to justify a book A theorybegins to be credible only when its original predictions are reasonablyunique and are confirmed by new experiments Loop quantum gravity isnot yet credible in this sense Nor is any other current tentative theory ofquantum gravity The interest of the loop theory in my opinion is thatat present it is the only approach to quantum gravity leading to well-defined physical predictions (falsifiable at least in principle) and moreimportantly it is the most determined effort for a genuine merging ofquantum field theory with the world view that we have discovered withgeneral relativity The future will tell us more

There are several other introductions to loop quantum gravity Clas-sic reports on the subject [1ndash10 in chronological order] illustrate variousstages of the development of the theory For a rapid orientation and toappreciate different points of view see the review papers [11ndash15] Muchuseful material can be found in [16] Good introductions to spinfoam the-ory are to be found in [1117ndash19] This book is self-contained but I havetried to avoid excessive duplications referring to other books and reviewpapers for nonessential topics well developed elsewhere This book focuseson physical and conceptual aspects of loop quantum gravity ThomasThiemannrsquos book [20] which is going to be completed soon focuses onthe mathematical foundation of the same theory The two books are com-plementary this book can almost be read as Volume 1 (ldquoIntroduction andconceptual frameworkrdquo) and Thiemannrsquos book as Volume 2 (ldquoCompletemathematical frameworkrdquo) of a general presentation of loop quantumgravity

Preface xv

The book assumes that the reader has a basic knowledge of general rel-ativity quantum mechanics and quantum field theory In particular theaim of the chapters on general relativity (Chapter 2) classical mechan-ics (Chapter 3) hamiltonian general relativity (Chapter 4) and quantumtheory (Chapter 5) is to offer the fresh perspective on these topics whichis needed for quantum gravity to a reader already familiar with the con-ventional formulation of these theories

Sections with comments and examples are printed in smaller fonts (seeSection 131 for first such example) Sections that contain side or morecomplex topics and that can be skipped in a first reading without com-promising the understanding of what follows are marked with a star (lowast)in the title References in the text appear only when strictly needed forcomprehension Each chapter ends with a short bibliographical sectionpointing out essential references for the reader who wants to go into moredetail or to trace original works on a subject I have given up the immensetask of collecting a full bibliography on loop quantum gravity On manytopics I refer to specific review articles where ample bibliographic infor-mation can be found An extensive bibliography on loop quantum gravityis given in [9] and [20]

I have written this book thinking of a researcher interested in workingin quantum gravity but also of a good PhD student or an open-mindedscholar curious about this extraordinary open problem I have found thejourney towards general relativistic quantum physics towards quantumspacetime a fascinating adventure I hope the reader will see the beautyI see and that he or she will be capable of completing the journey Thelandscape is magic the trip is far from being over


Three years have lapsed since the first edition of this book During thesethree years the research in loop gravity has been developing briskly and inseveral directions Remarkable new results are for instance the proof thatspinfoam and hamiltonian loop theory are equivalent in 3d the proof ofthe unicity of the loop representation (the ldquoLOSTrdquo theorem) the resolu-tion of the r = 0 black hole singularity major advances in loop cosmologythe result that in 3d loop quantum gravity plus matter yields an effectivenon-commutative quantum field theory the ldquomaster constraintrdquo programfor the definition of the quantum dynamics the idea of deriving parti-cles from linking the recalculation of the Immirzi parameter from blackhole thermodynamics and last but not least the first steps toward calcu-lating scattering amplitudes from the background independent quantumtheory I am certainly neglecting something that will soon turn out to beimportant

I have added notes and pointers to recent literature or recent reviewpapers where the interested reader can find updates on specific topicsIn spite of these rapid developments however it is too early for a full-fledged second edition of this book it seems to me that the book as itis still provides a comprehensive introduction to the field In fact severalof these developments reinforce the point of view of this book namelythat the lines of research considered form a coherent picture and definea common language in which a consistent quantum field theory without(background) space and time can be defined

When I feel pessimistic I see the divergence between research linesand the impressive number of problems that are still open When I feeloptimistic I see their remarkable coherence and I dream we might bewith respect to quantum gravity as Einstein was in 1914 with all themachinery ready trying a number of similar field equations Then itseems to me that a quantum theory of gravity (certainly not the final

xvii

xviii Preface to the paperback edition

theory of everything) is truly at hand maybe we have it maybe what weneed is just the right combination of techniques a few more details orone last missing key idea

Once again my wish is that among the readers of this paperback editionthere is she or he who will give us this last missing idea

Acknowledgements

I am indebted to the many people that have sent suggestions and cor-rections to the draft of this book posted online and to its first editionAmong them are M Carling Alexandru Mustatea Daniele Oriti JohnBaez Rafael Kaufmann Nedal Colin Hayhurst Jurgen Ehlers ChrisGauthier Gianluca Calcagni Tomas Liko Chang Chi-Ming YoungsubYoon Martin Bojowald and Gen Zhang Special thanks in particular toJustin Malecki Jacob Bourjaily and Leonard Cottrell

My great gratitude goes to the friends with whom I have had the priv-ilege of sharing this adventure

To Lee Smolin companion of adventures and friend His unique creativ-ity and intelligence intellectual freedom and total honesty are among thevery best things I have found in life

To Abhay Ashtekar whose tireless analytical rigor synthesis capacityand leadership have been a most precious guide Abhay has solidified ourideas and transformed our intuitions into theorems This book is a resultof Leersquos and Abhayrsquos ideas and work as much as my own

To Laura Scodellari and Chris Isham my first teachers who guided meinto mathematics and quantum gravity

To Ted Newman who with Sally parented the little boy just arrivedfrom the Empirersquos far provinces I have shared with Ted a decade ofintellectual joy His humanity generosity honesty passion and love forthinking are the example against which I judge myself

I would like to thank one by one all the friends working in this fieldwho have developed the ideas and results described in this book butthey are too many I can only mention my direct collaborators and afew friends outside this field Luisa Doplicher Simone Speziale ThomasSchucker Florian Conrady Daniele Colosi Etera Livine Daniele OritiFlorian Girelli Roberto DePietri Robert Oeckl Merced MontesinosKirill Krasnov Carlos Kozameh Michael Reisenberger Don Marolf

xx Acknowledgements

Berndt Brugmann Junichi Iwasaki Gianni Landi Mauro Carfora JormaLouko Marcus Gaul Hugo Morales-Tecotl Laurent Freidel Renate LollAlejandro Perez Giorgio Immirzi Philippe Roche Federico LaudisaJorge Pullin Thomas Thiemann Louis Crane Jerzy Lewandowski JohnBaez Ted Jacobson Marco Toller Jeremy Butterfield John Norton JohnBarrett Jonathan Halliwell Massimo Testa David Finkelstein GaryHorowitz John Earman Julian Barbour John Stachel Massimo PauriJim Hartle Roger Penrose John Wheeler and Alain Connes

With all these friends I have had the joy of talking about physics ina way far from problem-solving from outsmarting each other or frommaking weapons to make ldquousrdquo stronger than ldquothemrdquo I think that physicsis about escaping the prison of the received thoughts and searching fornovel ways of thinking the world about trying to clear a bit the mistylake of our insubstantial dreams which reflect reality like the lake reflectsthe mountainsForemost thanks to Bonnie ndash she knows why


bull In this book ldquorelativisticrdquo means ldquogeneral relativisticrdquo unless other-wise specified When referring to special relativity I say so explicitlySimilarly ldquononrelativisticrdquo and ldquoprerelativisticrdquo mean ldquonon-general -relativisticrdquo and ldquopre-general -relativisticrdquo The choice is a bit unusual(special relativity in this language is ldquononrelativisticrdquo) One reason forit is simply to make language smoother the book is about general rela-tivistic physics and repeating ldquogeneralrdquo every other line sounds too muchlike a Frenchman talking about de Gaulle But there is a more substantialreason the complete revolution in spacetime physics which truly deservesthe name of relativity is general relativity not special relativity This opin-ion is not always shared today but it was Einsteinrsquos opinion Einstein hasbeen criticized on this but in my opinion the criticisms miss the full reachof Einsteinrsquos discovery about spacetime One of the aims of this book isto defend in modern language Einsteinrsquos intuition that his gravitationaltheory is the full implementation of relativity in physics This point isdiscussed at length in Chapter 2

bull I often indulge in the physicistsrsquo bad habit of mixing up function fand function values f(x) Care is used when relevant Similarly I followstandard physicistsrsquo abuse of language in denoting a field such as theMaxwell potential as Aμ(x) A(x) or A where the three notations aretreated as equivalent manners of denoting the field Again care is usedwhere relevant

bull All fields are assumed to be smooth unless otherwise specified All state-ments about manifolds and functions are local unless otherwise specifiedthat is they hold within a single coordinate patch In general I do notspecify the domain of definition of functions clearly equations hold wherefunctions are defined

xxi

xxii Terminology and notation

bull Index notation follows the most common choice in the field Greek in-dices from the middle of the alphabet μ ν = 0 1 2 3 are 4d spacetimetangent indices Upper case Latin indices from the middle of the alphabetI J = 0 1 2 3 are 4d Lorentz tangent indices (In the special relativis-tic context the two are used without distinction) Lower case Latin indicesfrom the beginning of the alphabet a b = 1 2 3 are 3d tangent indicesLower case Latin indices from the middle of the alphabet i j = 1 2 3are 3d indices in R3 Coordinates of a 4d manifold are usually indicated asx y while 3d manifold coordinates are usually indicated as x y (alsoas τ) Thus the components of a spacetime coordinate x are

xμ = (t x) = (x0 xa)

while the components of a Lorentz vector e are

eI = (e0 ei)

bull ηIJ is the Minkowski metric with signature [minus+++] The indicesI J are raised and lowered with ηIJ δij is the Kronecker delta or theR3 metric The indices i j are raised and lowered with δij

bull For reasons explained at the beginning of Chapter 2 I call ldquogravita-tional fieldrdquo the tetrad field eIμ(x) instead of the metric tensor gμν(x) =ηIJ eIμ(x)eJν (x)

bull εIJKL or εμνρσ is the completely antisymmetric object with ε0123 = 1Similarly for εabc or εijk in 3d The Hodge star is defined by

F lowastIJ =

12εIJKL FKL

in flat space and by the same equation where FIJeIμe

Jν = Fμν and

F lowastIJe

Iμe

Jν = F lowast

μν in the presence of gravity Equivalently

F lowastμν =

radicminusdet g

12εμνρσ F ρσ = | det e| 1

2εμνρσ F ρσ

bull Symmetrization and antisymmetrization of indices is defined with a halfA(ab) = 1

2(Aab + Aba) and A[ab] = 12(Aab minusAba)

bull I call ldquocurverdquo on a manifold M a map

γ I rarr M

s rarr γa(s)

where I is an interval of the real line R (possibly the entire R) I callldquopathrdquo an oriented unparametrized curve namely an equivalence class of

Terminology and notation xxiii

curves under change of parametrization γa(s) rarr γprimea(s) = γa(sprime(s)) withdsprimeds gt 0

bull An orthonormal basis in the Lie algebras su(2) and so(3) is chosen onceand for all and these algebras are identified with R3 For so(3) the basisvectors (vi)jk can be taken proportional to εi

jk for su(2) the basis vec-

tors (vi)AB can be taken proportional to the Pauli matrices see AppendixA1 Thus an algebra element ω in su(2) sim so(3) has components ωi

bull For any antisymmetric quantity vij with two 3d indices i j I use alsothe one-index notation

vi = 12εijk vjk vij = εijk vk

the one-index and the two-indices notation are considered as definingthe same object For instance the SO(3) connections ωij and Aij areequivalently denoted ωi and Ai

Symbols Here is a list of symbols with their name and the equationchapter or section where they are introduced or defined

A area Section 214A YangndashMills connection Equation (230)AAi

μ(x) selfdual 4d gravitational connection Equation (219)AAi

a(x) selfdual or real 3d gravitational connection Sections 41142

C relativistic configuration space Section 321Dμ covariant derivative Equation (231)Diff lowast extended diffeomorphism group Section 622eIμ(x) gravitational field Equation (21)e determinant of eIμe edge (of spinfoam) Section 91EEa

i (x) gravitational electric field Section 411f face (of spinfoam) Section 91F curvature two-form Section 211g or U group elementG Newton constantG space of boundary data Sections 325ndash

333hγ U(A γ) Section 71H relativistic hamiltonian Section 32H0 nonrelativistic (conventional) hamiltonian Section 32H quantum state space Chapter 5H0 nonrelativistic quantum state space Chapter 5

xxiv Terminology and notation

in intertwiner on spin network node n Section 63ie intertwiner on spinfoam edge e Chapter 9j irreducible representation (for SU(2) spin)jl spin associated to spin network link l Section 621jf representation associated to spinfoam face f Chapter 9K kinematical quantum state space Section 52K0 SU(2) invariant quantum state space Section 623Kdiff diff-invariant quantum state space Section 623K boundary quantum space Sections 514

535l link (of spin network) Section 91lP Planck length

radicGcminus3

L length Section 214M spacetime manifoldn node (of spin network) Section 91pa relativistic momenta (including pt) Section 32pt momentum conjugate to t Section 32P the ldquoprojectorrdquo operator Section 52PG group G projector Equation

(9117)PH subgroup H projector Equation

(9119)P transition probability Chapter 5P path ordered Equation

(281)qa partial observables Section 32RI

J μν(x) curvature Equation(28)

R(j)αβ(g) matrix of group element g in representation j

R 3d region Section 214s s-knot abstract spin network Equation

(641)|s〉 s-knot state Equation

(641)SBH black-hole entropy Section 82S embedded spin network Section 63|S〉 spin network state Section 631S 2d surface Section 214S space of fast decrease functions Chapter 5S0 space of tempered distributions Chapter 5S[γ] action functional Section 32S(qa) HamiltonndashJacobi function Section 322S(qa qa0) Hamilton function Section 325

Terminology and notation xxv

tρ thermal time Sections 34551

T target space of a field theory Section 331U or g group elementU(A γ) holonomy Section 215v vertex (of spinfoam) Section 91V volume Section 214W (qa qprimea) propagator Chapter 5W transition amplitudes propagator Section 52x 4d spacetime coordinatesx 3d coordinatesZ partition function Chapter 9α loop closed pathβ inverse temperature Section 34γ pathγ motion (in C) Section 321γ Immirzi parameter Section 423γ motion in Ω Section 32Γ relativistic phase space Section 321Γ graph Section 62Γ two-complex Chapter 9θ PoincarendashCartan form on Σ Section 322θ Poincare form on Ω Equation (39)ηIJ ημν Minkowski metric = diag[minus1111]λ cosmological constant Equation (211)λ gauge parameter Section 213ρ statistical state Sections 34 551Σ constraint surface H = 0 Section 322σΣ 3d boundary surface Chapter 4σ spinfoam Chapter 9φ(x) scalar field Equation (232)ψ(x) fermion field Equation (235)ω presymplectic form on Σ Section 322ωIμJ(x) spin connection Equation (22)

ω symplectic form on Ω Section 322Ω space of observables and momenta Sections 32ndash3326j Wigner 6j symbol Equation (933)10j Wigner 10j symbol Equation (9103)15j Wigner 15j symbol Equation (956)|0〉 covariant vacuum in K Sections 514 535|0t〉 dynamical vacuum in Kt Sections 514 532|0M〉 Minkowski vacuum in H Sections 514 531

xxvi Terminology and notation

bull The name of the theory Finally a word about the name of the quantumtheory of gravity described in this book The theory is known as ldquoloopquantum gravityrdquo (LQG) or sometimes ldquoloop gravityrdquo for short Howeverthe theory is also designated in the literature using a variety of othernames I list here these other names and the variations of their use forthe benefit of the disoriented reader

ndash ldquoQuantum spin dynamicsrdquo (QSD) is used as a synonym of LQG WithinLQG it is sometimes used to designate in particular the dynamical as-pects of the hamiltonian theory

ndash ldquoQuantum geometryrdquo is also sometimes used as a synonym of LQGWithin LQG it is used to designate in particular the kinematical as-pects of the theory The expression ldquoquantum geometryrdquo is generic it isalso widely used in other approaches to quantum spacetime in particulardynamical triangulations [21] and noncommutative geometry

ndash ldquoNonperturbative quantum gravityrdquo ldquocanonical quantum gravityrdquo andldquoquantum general relativityrdquo (QGR) are often used to designate LQGalthough their proper meaning is wider

ndash The expression ldquoAshtekar approachrdquo is still used sometimes to desig-nate LQG it comes from the fact that a key ingredient of LQG is thereformulation of classical GR as a theory of connections developed byAbhay Ashtekar

ndash In the past LQG was also called ldquothe loop representation of quantumgeneral relativityrdquo Today ldquoloop representationrdquo and ldquoconnection repre-sentationrdquo are used within LQG to designate the representations of thestates of LQG as functionals of loops (or spin networks) and as functionalsof the connection respectively The two are related in the same manneras the energy (ψn = 〈n|ψ〉) and position (ψ(x) = 〈x|ψ〉) representationsof the harmonic oscillator states

Part I

Relativistic foundations

I know that I am mortal and the creature of aday but when I search out the massed wheeling circlesof the stars my feet no longer touch the earthside by side with Zeus himself I drink my fill ofambrosia food of the gods

Claudius Ptolemy Mathematical Syntaxis

1General ideas and heuristic picture

The aim of this chapter is to introduce the general ideas on which this book is based andto present the picture of quantum spacetime that emerges from loop quantum gravityin a heuristic and intuitive manner The style of the chapter is therefore conversationalwith little regard for precision and completeness In the course of the book the ideasand notions introduced here will be made precise and the claims will be justified andformally derived



Quantum mechanics (QM) and general relativity (GR) have greatlywidened our understanding of the physical world A large part of thephysics of the last century has been a triumphant march of exploration ofnew worlds opened up by these two theories QM led to atomic physics nu-clear physics particle physics condensed matter physics semiconductorslasers computers quantum optics GR led to relativistic astrophysicscosmology GPS technology and is today leading us hopefully towardsgravitational wave astronomy

But QM and GR have destroyed the coherent picture of the worldprovided by prerelativistic classical physics each was formulated in termsof assumptions contradicted by the other theory QM was formulatedusing an external time variable (the t of the Schrodinger equation) ora fixed nondynamical background spacetime (the spacetime on whichquantum field theory is defined) But this external time variable and thisfixed background spacetime are incompatible with GR In turn GR wasformulated in terms of riemannian geometry assuming that the metric isa smooth and deterministic dynamical field But QM requires that anydynamical field be quantized at small scales it manifests itself in discretequanta and is governed by probabilistic laws

3


We have learned from GR that spacetime is dynamical and we havelearned from QM that any dynamical entity is made up of quanta andcan be in probabilistic superposition states Therefore at small scales thereshould be quanta of space and quanta of time and quantum superpositionof spaces But what does this mean We live in a spacetime with quantumproperties a quantum spacetime What is quantum spacetime How canwe describe it

Classical prerelativistic physics provided a coherent picture of the phys-ical world This was based on clear notions such as time space matterparticle wave force measurement deterministic law This picture haspartially evolved (in particular with the advent of field theory and spe-cial relativity) but it has remained consistent and quite stable for threecenturies GR and QM have modified these basic notions in depth GRhas modified the notions of space and time QM the notions of causalitymatter and measurement The novel modified notions do not fit togethereasily The new coherent picture is not yet available With all their im-mense empirical success GR and QM have left us with an understandingof the physical world which is unclear and badly fragmented At the foun-dations of physics there is today confusion and incoherence

We want to combine what we have learnt about our world from the twotheories and to find a new synthesis This is a major challenge ndash perhapsthe major challenge ndash in todayrsquos fundamental physics GR and QM haveopened a revolution The revolution is not yet complete

With notable exceptions (Dirac Feynman Weinberg DeWitt WheelerPenrose Hawking rsquot Hooft among others) most of the physicists of thesecond half of the last century have ignored this challenge The urgencywas to apply the two theories to larger and larger domains The develop-ments were momentous and the dominant attitude was pragmatic Apply-ing the new theories was more important than understanding them Butan overly pragmatic attitude is not productive in the long run Towardsthe end of the twentieth century the attention of theoretical physics hasbeen increasingly focusing on the challenge of merging the conceptualnovelties of QM and GR

This book is the account of an effort to do so


How to search for this new synthesis Conventional field quantizationmethods are based on the weak-field perturbation expansion Their appli-cation to GR fails because it yields a nonrenormalizable theory Perhapsthis is not surprising GR has changed the notions of space and time tooradically to docilely agree with flat space quantum field theory Somethingelse is needed


In science there are no secure recipes for discovery and it is important toexplore different directions at the same time Currently a quantum theoryof gravity is sought along various paths The two most developed are loopquantum gravity described in this book and string theory Other researchdirections include dynamical triangulations noncommutative geometryHartlersquos quantum mechanics of spacetime (this is not really a specificquantum theory of gravity but rather a general theoretical frameworkfor general-relativistic quantum theory) Hawkingrsquos euclidean sum overgeometries quantum Regge calculus Penrosersquos twistor theory Sorkinrsquoscausal sets rsquot Hooftrsquos deterministic approach and Finkelsteinrsquos theoryThe reader can find ample references in the general introductions to quan-tum gravity mentioned in the note at the end of this chapter Here I sketchonly the general ideas that motivate the approach described in this bookplus a brief comment on string theory which is currently the most popularalternative to loop gravity

Our present knowledge of the basic structure of the physical universe issummarized by GR quantum theory and quantum field theory (QFT) to-gether with the particle-physics standard model This set of fundamentaltheories is inconsistent But it is characterized by an extraordinary em-pirical success nearly unique in the history of science Indeed currentlythere is no evidence of any observed phenomenon that clearly escapesquestions or contradicts this set of theories (or a minor modification ofthe same to account say for a neutrino mass or a cosmological constant)This set of theories becomes meaningless in certain physical regimes Inthese regimes we expect the predictions of quantum gravity to becomerelevant and to differ from the predictions of GR and the standard modelThese regimes are outside our experimental or observational reach at leastso far Therefore we have no direct empirical guidance for searching forquantum gravity ndash as say atomic spectra guided the discovery of quan-tum theory

Since quantum gravity is a theory expected to describe regimes that areso far inaccessible one might worry that anything could happen in theseregimes at scales far removed from our experience Maybe the search isimpossible because the range of the possible theories is too large Thisworry is unjustified If this was the problem we would have plenty ofcomplete predictive and coherent theories of quantum gravity Insteadthe situation is precisely the opposite we havenrsquot any The fact is that wedo have plenty of information about quantum gravity because we haveQM and we have GR Consistency with QM and GR is an extremely strictconstraint

A view is sometime expressed that some totally new radical and wildhypothesis is needed for quantum gravity I do not think that this isthe case Wild ideas pulled out of the blue sky have never made science


advance The radical hypotheses that physics has successfully adoptedhave always been reluctantly adopted because they were forced upon usby new empirical data ndash Keplerrsquos ellipses Bohrrsquos quantization ndash or bystringent theoretical deductions ndash Maxwellrsquos inductive current Einsteinrsquosrelativity (see Appendix C) Generally arbitrary novel hypotheses leadnowhere

In fact today we are precisely in one of the typical situations in whichtheoretical physics has worked at its best in the past Many of the moststriking advances in theoretical physics have derived from the effort offinding a common theoretical framework for two basic and apparently con-flicting discoveries For instance the aim of combining the keplerian or-bits with galilean physics led to newtonian mechanics combining Maxwelltheory with galilean relativity led to special relativity combining specialrelativity and nonrelativistic quantum theory led to the theoretical discov-ery of antiparticles combining special relativity with newtonian gravityled to general relativity and so on In all these cases major advances havebeen obtained by ldquotaking seriouslyrdquo1 apparently conflicting theories andexploring the implications of holding the key tenets of both theories fortrue Today we are precisely in one of these characteristic situations Wehave learned two new very general ldquofactsrdquo about Nature expressed byQM and GR we have ldquojustrdquo to figure out what they imply taken to-gether Therefore the question we have to ask is what have we reallylearned about the world from QM and from GR Can we combine theseinsights into a coherent picture What we need is a conceptual scheme inwhich the insights obtained with GR and QM fit together

This view is not the majority view in theoretical physics at presentThere is consensus that QM has been a conceptual revolution but manydo not view GR in the same way According to many the discovery of GRhas been just the writing of one more field theory This field theory isfurthermore likely to be only an approximation to a theory we do not yetknow According to this opinion GR should not be taken too seriously asa guidance for theoretical developments

I think that this opinion derives from a confusion the confusion betweenthe specific form of the EinsteinndashHilbert action and the modification of thenotions of space and time engendered by GR The EinsteinndashHilbert actionmight very well be a low-energy approximation of a high-energy theoryBut the modification of the notions of space and time does not depend onthe specific form of the EinsteinndashHilbert action It depends on its diffeo-morphism invariance and its background independence These properties

1In [22] Gell-Mann says that the main lesson to be learnt from Einstein is ldquoto lsquotakevery seriouslyrsquo ideas that work and see if they can be usefully carried much furtherthan the original proponent suggestedrdquo


(which are briefly illustrated in Section 113 below and discussed in de-tail in Chapter 2) are most likely to hold in the high-energy theory aswell One should not confuse the details of the dynamics of GR with themodifications of the notions of space and time that GR has determinedIf we make this confusion we underestimate the radical novelty of thephysical content of GR The challenge of quantum gravity is precisely tofully incorporate this radical novelty into QFT In other words the taskis to understand what is a general-relativistic QFT or a background-independent QFT

Today many physicists prefer disregarding or postponing these founda-tional issues and instead choose to develop and adjust current theoriesThe most popular strategy towards quantum gravity in particular isto pursue the line of research grown in the wake of the success of thestandard model of particle physics The failure of perturbative quantumGR is interpreted as a replay of the failure of Fermi theory2 Namely asan indication that we must modify GR at high energy With the inputof the grand-unified-theories (GUTs) supersymmetry and the KaluzandashKlein theory the search for a high-energy correction of GR free from badultraviolet divergences has led to higher derivative theories supergravityand finally to string theory

Sometimes the claim is made that the quantum theory of gravity hasalready been found and it is string theory Since this is a book about quan-tum gravity without strings I should say a few words about this claimString theory is based on a physical hypothesis elementary objects areextended rather than particle-like This hypothesis leads to a very richunified theory which contains much phenomenology including (with suit-able inputs) fermions YangndashMills fields and gravitons and is expected bymany to be free of ultraviolet divergences The price to pay for these theo-retical results is a gigantic baggage of additional physics supersymmetryextra dimensions an infinite number of fields with arbitrary masses andspins and so on

So far nothing of this new physics shows up in experiments Super-symmetry in particular has been claimed to be on the verge of beingdiscovered for years but hasnrsquot shown up Unfortunately so far the the-ory can accommodate any disappointing experimental result because it ishard to derive precise new quantitative physical predictions with whichthe theory could be falsified from the monumental mathematical appa-ratus of the theory Furthermore even recovering the real world is noteasy within the theory the search for a compactification leading to the

2Fermi theory was an empirically successful but nonrenormalizable theory of the weakinteractions just as GR is an empirically successful but nonrenormalizable theory ofthe gravitational interaction The solution has been the GlashowndashWeinbergndashSalamelectroweak theory which corrects Fermi theory at high energy


standard model with its families and masses and no instabilities has notyet succeeded as far as I know It is clear that string theory is a very inter-esting hypothesis but certainly not an established theory It is thereforeimportant to pursue alternative directions as well

String theory is a direct development of the standard model and isdeeply rooted in the techniques and the conceptual framework of flatspace QFT As I shall discuss in detail throughout this book manyof the tools used in this framework ndash energy unitary time evolutionvacuum state Poincare invariance S-matrix objects moving in a space-time Fourier transform ndash no longer make sense in the quantum grav-itational regime in which the gravitational field cannot be approxi-mated by a background spacetime ndash perhaps not even asymptotically3

Therefore string theory does not address directly the main challengeof quantum gravity understanding what is a background-independentQFT Facing this challenge directly before worrying about unificationleads instead to the direction of research investigated by loop quantumgravity4

The alternative to the line of research followed by string theory is givenby the possibility that the failure of perturbative quantum GR is not areplay of Fermi theory That is it is not due to a flaw of the GR actionbut instead it is due to the fact that the conventional weak-field quantumperturbation expansion cannot be applied to the gravitational field

This possibility is strongly supported a posteriori by the results of loopquantum gravity As we shall see loop quantum gravity leads to a pictureof the short-scale structure of spacetime extremely different from that ofa smooth background geometry (There are hints in this direction fromstring theory calculations as well [25]) Spacetime turns out to have anonperturbative quantized discrete structure at the Planck scale whichis explicitly described by the theory The ultraviolet divergences are curedby this structure The ultraviolet divergences that appear in the pertur-bation expansion of conventional QFT are a consequence of the fact that

3To be sure the development of string theory has incorporated many aspects of GRsuch as curved spacetimes horizons black holes and relations between different back-grounds But this is far from a background-independent framework such as the onerealized by GR in the classical context GR is not about physics on a curved space-time or about relations between different backgrounds it is about the dynamics ofspacetime A background-independent fundamental definition of string theory is beingactively searched for along several directions but so far the definition of the theoryand all calculations rely on background metric spaces

4It has been repeatedly suggested that loop gravity and string theory might mergebecause loop gravity has developed precisely the background-independent QFT meth-ods that string theory needs [23] Also excitations over a weave (see Section 671)have a natural string structure in loop gravity [24]


we erroneously replace this discrete Planck-scale structure with a smoothbackground geometry

If this is physically correct ultraviolet divergences do not require theheavy machinery of string theory to be cured On the other hand the con-ventional weak-field perturbative methods cannot be applied because wecannot work with a fixed smooth background geometry We must there-fore adapt QFT to the full conceptual novelty of GR and in particularto the change in the notion of space and time induced by GR What arethese changes I sketch an answer below leaving a complete discussion toChapter 2


GR is the discovery that spacetime and the gravitational field are thesame entity What we call ldquospacetimerdquo is itself a physical object in manyrespects similar to the electromagnetic field We can say that GR is thediscovery that there is no spacetime at all What Newton called ldquospacerdquoand Minkowski called ldquospacetimerdquo is unmasked it is nothing but a dy-namical object ndash the gravitational field ndash in a regime in which we neglectits dynamics

In newtonian and special-relativistic physics if we take away the dy-namical entities ndash particles and fields ndash what remains is space and time Ingeneral-relativistic physics if we take away the dynamical entities nothingremains The space and time of Newton and Minkowski are re-interpretedas a configuration of one of the fields the gravitational field This impliesthat physical entities ndash particles and fields ndash are not immersed in spaceand moving in time They do not live on spacetime They live so to sayon one another

It is as if we had observed in the ocean many animals living on anisland animals on the island Then we discover that the island itself is infact a great whale So the animals are no longer on the island just animalson animals Similarly the Universe is not made up of fields on spacetimeit is made up of fields on fields This book studies the far-reaching effectthat this conceptual shift has on QFT

One consequence is that the quanta of the field cannot live in spacetimethey must build ldquospacetimerdquo themselves This is precisely what the quantaof space do in loop quantum gravity

We may continue to use the expressions ldquospacerdquo and ldquotimerdquo to indicateaspects of the gravitational field and I do so in this book We are usedto this in classical GR But in the quantum theory where the field hasquantized ldquogranularrdquo properties and its dynamics is quantized and there-fore only probabilistic most of the ldquospatialrdquo and ldquotemporalrdquo features ofthe gravitational field are lost


Therefore to understand the quantum gravitational field we must aban-don some of the emphasis on geometry Geometry represents the classicalgravitational field but not quantum spacetime This is not a betrayal ofEinsteinrsquos legacy on the contrary it is a step in the direction of ldquorelativ-ityrdquo in the precise sense meant by Einstein Alain Connes has describedbeautifully the existence of two points of view on space the geometricone centered on space points and the algebraic or ldquospectralrdquo one cen-tered on the algebra of dual spectral quantities As emphasized by Alainquantum theory forces us to a complete shift to this second point of viewbecause of noncommutativity In the light of quantum theory continuousspacetime cannot be anything else than an approximation in which wedisregard quantum noncommutativity In loop gravity the physical fea-tures of space appear as spectral properties of quantum operators thatdescribe our (the observersrsquo) interactions with the gravitational field

The key conceptual difficulty of quantum gravity is therefore to accept theidea that we can do physics in the absence of the familiar stage of spaceand time We need to free ourselves from the prejudices associated withthe habit of thinking of the world as ldquoinhabiting spacerdquo and ldquoevolving intimerdquo Chapter 3 describes a language for describing mechanical systemsin this generalized conceptual framework

This absence of the familiar spacetime ldquostagerdquo is called the backgroundindependence of the classical theory Technically it is realized by the gaugeinvariance of the action under (active) diffeomorphisms A diffeomorphismis a transformation that smoothly drags all dynamical fields and particlesfrom one region of the four-dimensional manifold to another (the pre-cise definition of these transformations is given in Chapter 2) In turngauge invariance under diffeomorphism (or diffeomorphism invariance) isthe consequence of the combination of two properties of the action itsinvariance under arbitrary changes of coordinates and the fact that thereis no nondynamical ldquobackgroundrdquo field


Is quantum mechanics5 compatible with the general-relativistic notionsof space and time It is provided that we choose a sufficiently generalformulation For instance the Schrodinger picture is only viable for the-ories where there is a global observable time variable t this conflictswith GR where no such variable exists Therefore the Schrodinger pic-ture makes little sense in a background-independent context But there

5I use the expression ldquoquantum mechanicsrdquo to indicate the theory of all quantumsystems with a finite or infinite number of degrees of freedom In this sense QFT ispart of quantum mechanics


are formulations of quantum theory that are more general than theSchrodinger picture In Chapter 5 I describe a formulation of QM suf-ficiently general to deal with general-relativistic systems (For anotherrelativistic formulation of QM see [26]) Formulations of this kind aresometimes denoted ldquogeneralized quantum mechanicsrdquo I prefer to useldquoquantum mechanicsrdquo to denote any formulation of quantum theory ir-respective of its generality just as ldquoclassical mechanicsrdquo is used to des-ignate formalisms with different degrees of generality such as NewtonrsquosLagrangersquos Hamiltonrsquos or symplectic mechanics

On the other hand most of the conventional machinery of perturbativeQFT is profoundly incompatible with the general-relativistic frameworkThere are many reasons for this

bull The conventional formalism of QFT relies on Poincare invarianceIn particular it relies on the notion of energy and on the existence ofthe nonvanishing hamiltonian operator that generates unitary timeevolution The vacuum for instance is the state that minimizes theenergy Generally there is no global Poincare invariance no generalnotion of energy and no nonvanishing hamiltonian operator in ageneral-relativistic theory

bull At the root of conventional QFT is the physical notion of particleThe theoretical experience with QFT on curved spacetime [27] andon the relation between acceleration and temperature in QFT [28]indicates that in a generic gravitational situation the notion of par-ticle can be quite delicate (This point is discussed in Section 534)

bull Consider a conventional renormalized QFT The physical contentof the theory can be expressed in terms of its n-point functionsW (x1 xn) The n-point functions reflect the invariances of theclassical theory In a general-relativistic theory invariance under acoordinate transformation x rarr xprime = xprime(x) implies immediately thatthe n-point functions must satisfy

W (x1 xn) = W (xprime(x1) xprime(xn)) (11)

and therefore (if the points in the argument are distinct) it must bea constant That is

W (x1 xn) = constant (12)

Clearly we are immediately in a very different framework from con-ventional QFT

bull Similarly the behavior for small |xminus y| of the two-point function ofa conventional QFT

W (x y) =constant

|xminus y|d (13)


expresses the short-distance structure of the QFT More generallythe short-distance structure of the QFT is reflected in the operatorproduct expansion

O(x)Oprime(y) =sum

n

On(x)|xminus y|n (14)

Here |x minus y| is the distance measured in the spacetime metric Onflat space for instance |xminusy|2 = ημν(xμminusyμ)(xνminusyν) In a general-relativistic context these expressions make no sense since there isno background Minkowski (or other) metric ημν In its place there isthe gravitational field namely the quantum field operator itself Butthen if standard operator product expansion becomes meaninglessthe short-distance structure of a quantum gravitational theory mustbe profoundly different from that of conventional QFT As we shallsee in Chapter 7 this is precisely the case

There is a tentative escape strategy to circumvent these difficultieswrite the gravitational field e(x) as the sum of two terms

e(x) = ebackground(x) + h(x) (15)

where ebackground(x) is a background field configuration This may beMinkowski or any other Assume that ebackground(x) defines spacetimenamely it defines location and causal relations Then consider h(x) asthe gravitational field governed by a QFT on the spacetime backgrounddefined by ebackground For instance the field operator h(x) is assumed tocommute at spacelike separations where spacelike is defined in the geom-etry determined by ebackground(x) As a second step one may then considerconditions on ebackground(x) or relations between the formulations of thetheory defined by different choices of ebackground(x) This escape strategyleads to three orders of difficulties (i) Conventional perturbative QFTof GR based on (15) leads to a nonrenormalizable theory To get rid ofthe uncontrollable ultraviolet divergences one has to resort to the compli-cations of string theory (ii) As mentioned loop quantum gravity showsthat the structure of spacetime at the Planck scale is discrete Thereforephysical spacetime has no short-distance structure at all The unphysicalassumption of a smooth background ebackground(x) implicit in (15) maybe precisely the cause of the ultraviolet divergences (iii) The separationof the gravitational field from spacetime is in strident contradiction withthe very physical lesson of GR If GR is of any guide in searching for aquantum theory of gravity the relevant spacetime geometry is the onedetermined by the full gravitational field e(x) and the separation (15) ismisleading


A formulation of quantum gravity that does not take the escape strategy(15) is a background-independent or general covariant QFT The mainaim of this book is develop the formalism for background-independentQFT


I sketch here the physical picture of quantum spacetime that emergesfrom loop quantum gravity (LQG) The basic ideas and assumptions onwhich LQG is based are the following

(i) Quantum mechanics and general relativity QM suitably formulatedto be compatible with general covariance is assumed to be cor-rect The Einstein equations may be modified at high energy butthe general-relativistic notions of space and time are assumed to becorrect The motivation for these two assumptions is the extraordi-nary empirical success they have had so far and the absence of anycontrary empirical evidence

(ii) Background independence LQG is based on the idea that the quan-tization strategy based on the separation (15) is not appropriatefor describing the quantum properties of spacetime

To this we can add

(iii) No unification Nowadays a fashionable idea is that the problemof quantizing gravity has to be solved together with the problemof finding a unified description of all interactions LQG is a solu-tion of the first problem not the second6

(iv) Four spacetime dimensions and no supersymmetry LQG is com-patible with these possibilities but there is nothing in the theorythat requires higher dimensions or supersymmetry Higher space-time dimensions and supersymmetry are interesting theoreticalideas which as many other interesting theoretical ideas can bephysically wrong In spite of 15 years of search numerous pre-liminary announcements of discovery then turned out to be false

6A motivation for the idea that these two issues are connected is the expectation thatwe are ldquonear the end of physicsrdquo Unfortunately the expectation of being ldquonear theend of physicsrdquo has been present all along the three centuries of the history of modernphysics In the present situation of deep conceptual confusion on the fundamentalaspects of the world I see no sign indicating that we are close to the end of ourdiscoveries about the physical world When I was a student it was fashionable toclaim that the problem of finding a theory of the strong interactions had to be solvedtogether with the problem of getting rid of renormalization theory Nice idea Butwrong


and despite repeated proclamations that supersymmetry was go-ing to be discovered ldquonext yearrdquo so far empirical evidence has beensolidly and consistently against supersymmetry This might changebut as scientists we must take the indications of the experimentsseriously

On the basis of these assumptions LQG is a straightforward quantiza-tion of GR with its conventional matter couplings The program of LQGis therefore conservative and of small ambition The physical inputs ofthe theory are just QM and GR well-tested physical theories No majoradditional physical hypothesis or assumption is made (such as elementaryobjects are strings space is made by individual discrete points quantummechanics is wrong GR is wrong supersymmetry extra dimensions )No claim of being the final ldquoTheory Of Everythingrdquo is made

On the other hand LQG has a radical and ambitious side to merge theconceptual insight of GR into QM In order to achieve this we have togive up the familiar notions of space and time The space continuum ldquoonwhichrdquo things are located and the time ldquoalong whichrdquo evolution happensare semiclassical approximate notions in the theory In LQG this radicalstep is assumed in its entirety

LQG does not make use of most of the familiar tools of conventionalQFT because these become inadequate in a background-independent con-text It only makes use of the general tools of quantum theory a Hilbertspace of states operators related to the measurement of physical quanti-ties and transition amplitudes that determine the probability outcome ofmeasurements of these quantities Hilbert space of states and operatorsassociated to physical observables are obtained from classical GR follow-ing a rather standard quantization strategy A quantization strategy isa technique for searching for a solution to a well-posed inverse problemfinding a quantum theory with a given classical limit The inverse prob-lem could have many solutions As noticed presently the difficulty is notto discriminate among many complete and consistent quantum theoriesof gravity We would be content with one

121 Why loops

Among the technical choices to make in order to implement a quantiza-tion procedure is which algebra of field functions to promote to quan-tum operators In conventional QFT this is generally the canonical al-gebra formed by the positive and negative frequency components of thefield modes The quantization of this algebra leads to the creation and


annihilation operators a and adagger The characterization of the positive andnegative frequencies requires a background spacetime

In contrast to this what characterizes LQG is the choice of adifferent algebra of basic field functions a noncanonical algebra basedon the holonomies of the gravitational connection The holonomy (orldquoWilson looprdquo) is the matrix of the parallel transport along a closedcurve

The idea that holonomies are the natural variables in a gauge the-ory has a long history In a sense it can be traced back to the veryorigin of gauge theory in the physical intuition of Faraday Faraday un-derstood electromagnetic phenomena in terms of ldquolines of forcerdquo Twokey ideas underlie this intuition First that the relevant physical vari-ables fill up space this intuition by Faraday is the origin of field the-ory Second that the relevant variables do not refer to what happens ata point but rather refer to the relation between different points con-nected by a line The mathematical quantity that expresses this ideais the holonomy of the gauge potential along the line In the Maxwellcase for instance the holonomy U(Aα) along a loop α is simply theexponential of the line integral along α of the three-dimensional Maxwellpotential A

U(Aα) = e∮α A = exp

int 2π

0ds Aa(α(s))

dαa(s)ds

(16)

In LQG the holonomy becomes a quantum operator that creates ldquoloopstatesrdquo In the loop representation formulation of Maxwell theory forinstance a loop state |α〉 is a state in which the electric field vanisheseverywhere except along a single Faraday line α More precisely it is aneigenstate of the electric field with eigenvalue

Eα(x) =∮

dsdα(s)

dsδ3(x α(s)) (17)

where s rarr α(s) is the Faraday line in space This electric field vanisheseverywhere except on the loop α itself and at every point of α it is tangentto the loop see Figure 11 Notice that the vector distribution field E(x)defined in (17) is divergenceless that is it satisfies Coulomb law

div Eα(x) = 0 (18)


Fig 11 A loop α and the distributional electric field configuration Eα (repre-sented by the arrows)

in the sense of distributions In fact for any smooth function f we have

[div Eα](f) =int

d3x f(x) div Eα(x)

=int

d3x f(x)part

partxa

∮ds

dαa(s)ds

δ3(x α(s))

= minus∮

dsdαa(s)

dspart

partαaf(α(s))

= minus∮

αdf = minus

∮ds

dds

f(α(s)) = 0 (19)

Indeed intuitively Coulomb law requires precisely that an electric fieldat a point ldquocontinuesrdquo in the direction of the field itself namely that itdefines Faraday lines The state |α〉 is therefore a sort of minimal quantumexcitation satisfying (18) it is an elementary quantum excitation of asingle Faraday line

The idea that a YangndashMills theory is truly a theory of these loops hasbeen around for as long as such theories have been studied MandelstamPolyakov and Wilson among many others have long argued that loopexcitations should play a major role in quantum YangndashMills theories andthat we must get to understand quantum YangndashMills theories in terms ofthese excitations In fact much of the development of string theory hasbeen influenced by this idea

In lattice YangndashMills theory namely in the approximation to YangndashMills theory where spacetime is replaced by a fixed lattice loop stateshave finite norm In fact certain finite linear combinations of loop statescalled ldquospin networkrdquo states form a well-defined and well-understood or-thonormal basis in the Hilbert space of a lattice gauge theory

However in a QFT theory over a continuous background the idea offormulating the theory in terms of loop-like excitations has never provedfruitful The difficulty is essentially that loop states over a background areldquotoo singularrdquo and ldquotoo manyrdquo The quantum Maxwell state |α〉 describedabove for instance has infinite norm and an infinitesimal displacement of


a loop state over the background spacetime produces a distinct indepen-dent loop state yielding a continuum of loop states Over a continuousbackground the space spanned by the loop states is far ldquotoo bigrdquo forproviding a basis of the (separable) Hilbert space of a QFT

However loop states are not too singular nor too many in abackground-independent theory This is the key technical point on whichLQG relies The intuitive reason is as follows Spacetime itself is formedby loop-like states Therefore the position of a loop state is relevant onlywith respect to other loops and not with respect to the background Aninfinitesimal (coordinate) displacement of a loop state does not produce adistinct quantum state but only a gauge equivalent representation of thesame physical state Only a finite displacement carrying the loop stateacross another loop produces a physically different state Therefore thesize of the space of the loop states is dramatically reduced by diffeomor-phism invariance most of it is just gauge Equivalently we can think thata single loop has an intrinsic Planck-size ldquothicknessrdquo

Therefore in a general-relativistic context the loop basis becomesviable The state space of the theory called Kdiff is a separable Hilbertspace spanned by loop states More precisely as we shall see in Chapter 6Kdiff admits an orthonormal basis of spin network states which are formedby finite linear combinations of loop states and are defined precisely asthe spin network states of a lattice YangndashMills theory This Hilbert spaceand the field operators that act on it are described in Chapter 6 Theyform the basis of the mathematical structure of LQG

Therefore LQG is the result of the convergence of two lines of think-ing each characteristic of twentieth-century theoretical physics On theone hand the intuition of Faraday Yang and Mills Wilson MandelstamPolyakov and others that forces are described by lines On the otherhand the EinsteinndashWheelerndashDeWitt intuition of background indepen-dence and background-independent quantum states Truly remarkablyeach of these two lines of thinking is the solution of the blocking difficultyof the other On the one hand the traditional nonviability of the loop ba-sis in the continuum disappears because of background independence Onthe other hand the traditional difficulty of controlling diffeomorphism-invariant quantities comes under control thanks to the loop basis

Even more remarkably the spin network states generated by this happymarriage turn out to have a surprisingly compelling geometric interpre-tation which I sketch below


Physical systems reveal themselves by interacting with other systemsThese interactions may happen in ldquoquantardquo energy is exchanged with an


oscillator of frequency ν in discrete packets or quanta of size E = hνIf the oscillator is in the nth energy eigenstate we say that there are nquanta in it If the oscillator is a mode of a free field we say that thereare n ldquoparticlesrdquo in the field Therefore we can view the electromagneticfield as made up of its quanta the photons What are the quanta of thegravitational field Or since the gravitational field is the same entity asspacetime what are the quanta of space

The properties of the quanta of a system are determined by the spectralproperties of the operators representing the quantities involved in ourinteraction with the system The operator associated with the energy ofthe oscillator for instance has a discrete spectrum and the number ofquanta n labels its eigenvalues The set of its eigenstates form a basisin the state space of the quantum system this fact allows us to vieweach state of the system as a quantum superposition of states |n〉 formedby n quanta To understand the quantum properties of space we havetherefore to consider the spectral problem of the operators associatedwith the quantities involved in our interaction with space itself The mostdirect interaction we have with the gravitational field is via the geometricstructure of the physical space A measurement of length area or volumeis in fact according to GR a measurement of a local property of thegravitational field

For instance the volume V of a physical region R is

V =int

Rd3x | det e(x)| (110)

where e(x) is the (triad matrix representing the) gravitational field Inquantum gravity e(x) is a field operator and V is therefore an operatoras well

The volume V is a nonlinear function of the field e and the definitionof the volume operator implies products of local operator-valued distri-butions This can be achieved as a limit using an appropriate regulariza-tion procedure The development of regularization procedures that remainmeaningful in the absence of a background metric is a major technical toolon which LQG is based Using these techniques a well-defined self-adjointoperator V can be defined The computation of its spectral properties isthen one of the main results of LQG and will be derived in Section 665

The spectrum of V turns out to be discrete Therefore the spacetimevolume manifests itself in quanta of definite volume size given by theeigenvalues of the volume operator These quanta of space can be in-tuitively thought of as quantized ldquograinsrdquo of space or ldquoatoms of spacerdquoThe first intuitive picture of quantum space is therefore that of ldquograinsof spacerdquo These have quantized amounts of volume determined by thespectrum of the operator V


j

j

2= 12

s =i1 i2

1 = 1

j3

= 12

Fig 12 A simple spin network

The next element of the picture is the information on which grain isadjacent to which Adjacency (being contiguous being in touch beingnearby) is the basis of spatial relations If two spacetime regions are ad-jacent that is if they touch each other they are separated by a surfaceS Let A be the area of the surface S Area also is a function of the grav-itational field and is therefore represented by an operator like volumeThe spectral problem for this operator has been solved in LQG as wellIt is discussed in detail in Section 662 This spectrum turns out also tobe discrete Intuitively the grains of space are separated by ldquoquanta ofareardquo The principal series of the eigenvalues of the area for instance islabeled by multiplets of half-integers ji i = 1 n and turns out to begiven by

A = 8πγ Gsum

i


where γ the Immirzi parameter is a free dimensionless constant of thetheory

Consider a quantum state of space |s〉 formed by N ldquograinsrdquo of spacesome of which are adjacent to one another Represent this state as anabstract graph Γ with N nodes (By abstract graph I mean here anequivalence class under smooth deformations of graphs embedded in a3-manifold) The nodes of the graph represent the grains of space thelinks of the graph link adjacent grains and represent the surfaces separat-ing two adjacent grains The quantum state is then characterized by thegraph Γ and by labels on nodes and on links the label in on a node nis the quantum number of the volume and the label jl on a link l is thequantum number of the area

A graph with these labels is called an (abstract) ldquospin networkrdquos = (Γ in jl) see Figure 12 In Section 631 we will see that the quantumnumbers in and jl are determined by the representation theory of the localgauge group (SU(2)) More precisely jl labels unitary irreducible repre-sentations and in labels a basis in the space of the intertwiners betweenthe representations adjacent to the node n The area of a surface cutting



n links of the spin network with labels ji i = 1 n is then given by(111)

As shown in Section 631 the (kinematical) Hilbert space Kdiff admitsa basis labeled precisely by these spin networks This is a basis of statesin which certain area and volume operators are diagonal Its physicalinterpretation is the one sketched in Figure 13 a spin network state |s〉describes a quantized three-geometry

A loop state |α〉 is a spin network state in which the graph Γ hasno nodes namely is a single loop α and is labeled by the fundamentalrepresentation of the group In such a state the gravitational field hassupport just on the loop α itself as the electric field in (17)

In LQG physical space is a quantum superposition of spin networks inthe same sense that the electromagnetic field is a quantum superpositionof n-photon states The first and basic prediction of the (free) QFT ofthe electromagnetic field is the existence of the photons and the specificquantitative prediction of the energy and the momentum of the photonsof a given frequency Similarly the first prediction of LQG is the existenceof the quanta of area and volume and the quantitative prediction of theirspectrum

The theory predicts that any sufficiently accurate measurement of areaor volume would measure one of these spectral values So far verifyingthis prediction appears to be outside our technological capacities

Where is a spin network A spin network state does not have a positionIt is an abstract graph not a graph immersed in a spacetime manifold


Only abstract combinatorial relations defining the graph are significantnot its shape or its position in space

In fact a spin network state is not in space it is space It is not localizedwith respect to something else something else (matter particles otherfields) might be localized with respect to it To ask ldquowhere is a spinnetworkrdquo is like asking ldquowhere is a solution of the Einstein equationsrdquo Asolution of the Einstein equations is not ldquosomewhererdquo it is the ldquowhererdquowith respect to which anything else can be localized In the same waythe other dynamical objects such as YangndashMills and fermion fields liveon the spin network state

This is a consequence of diffeomorphism invariance Technically spinnetwork states are first defined as graphs embedded in a three-dimensionalmanifold then the implementation of the diffeomorphism gauge identifiestwo graphs that can be deformed into each other They are gauge equiv-alent This is like identifying two solutions of the Einstein equations thatare related by a change of coordinates Spin networks embedded in amanifold are denoted S and called ldquoembedded spin networksrdquo equiva-lence classes of these under diffeomorphisms are denoted s and are calledldquoabstract spin networksrdquo or s-knots A quantum state of space is deter-mined by an s-knot7

The fact that spin networks do not live in space but rather are spacehas far-reaching consequences Space itself turns out to have a discrete andcombinatorial character Notice that this is not imposed on the theory orassumed It is the result of a completely conventional quantum mechanicalcalculation of the spectrum of the physical quantities that describe thegeometry of space Since there is no spatial continuity at short scale thereis (literally) no room in the theory for ultraviolet divergencies The theoryeffectively cuts itself off at the Planck scale Space is effectively granularat the Planck scale and there is no infinite ultraviolet limit

Chapter 7 describes how YangndashMills and fermion fields can be coupledto the theory This can be obtained by enriching the structure of the spinnetworks s In the case of a YangndashMills theory with gauge group G forinstance links carry an additional quantum number labeling irreduciblerepresentations G The spin network itself behaves like the lattice of lat-tice YangndashMills theory In quantum gravity therefore the lattice itselfbecomes a dynamical variable But notice a crucial difference with re-spect to conventional lattice YangndashMills theory the lattice size is not tobe scaled down to zero it has physical Planck size

In summary spin networks provide a mathematically well-defined andphysically compelling description of the kinematics of the quantum grav-itational field They also provide a well-defined picture of the small-scale

7The expression ldquospin networkrdquo is used in the literature to designate both the embeddedand the abstract ones as well as to designate the quantum states they label


structure of space It is remarkable that this novel picture of space emergessimply from the combination of old YangndashMills theory ideas with general-relativistic background independence


The dynamics of the quantum gravitational field can be described givingamplitudes W (s) for spin network states Let me illustrate here in aheuristic manner the physical interpretation of these amplitudes and theway they are defined in the theory A major feature of this book is thatit is based on a general-relativistic way of thinking about observables +evolution This section sketches this view and may be somewhat harderto follow than the previous ones

Interpretation of the amplitude W (s) The quantum dynamics of a par-ticle is entirely described by the transition probability amplitudes

W (x t xprime tprime) = 〈x|eminus iH0(tminustprime)|xprime〉 = 〈x t|xprime tprime〉 (112)

where |x t〉 is the eigenstate of the Heisenberg position operator x(t)with eigenvalue x H0 is the hamiltonian operator and |x〉 = |x 0〉 Thepropagator W (x txprime tprime) depends on two events (x t) and (xprime tprime) thatbound a finite portion of a classical trajectory The space of the pairs ofevents (x t xprime tprime) is called G in this book

A physical experiment consists of a preparation at time tprime and a mea-surement at time t Say that in a particular experiment we have localizedthe particle in xprime at tprime and then found it in x at time t The set (x t xprime tprime)represents the complete set of data of a specific complete observationalset up including preparation and measurement The space G is the spaceof these data sets In the quantum theory we associate the complex am-plitude W (x t xprime tprime) which is a function on G with any such data set Asemphasized by Feynman this amplitude codes the full quantum dynamicsFollowing Feynman we can compute W (x t xprime tprime) with a sum-over-pathsthat take the values x and xprime at t and tprime respectively

If we measure a different observable than position we obtain statesdifferent from the states |x〉 Let |ψin〉 be the state prepared at time tprimeand let |ψout〉 be the state measured at time t The amplitude associatedto these measurements is

A = 〈ψout|eminusiH0(tminustprime)|ψin〉 (113)

The pair of states (ψin ψout) determines a state ψ = |ψin〉 otimes 〈ψout| inthe space Kttprime which is the tensor product of the Hilbert space of theinitial states and (the dual of) the Hilbert space of the final states The


propagator defines a (possibly generalized) state |0〉 in Kttprime by 〈0|(|xprime〉 otimes〈x|) = W (x t xprime tprime) The amplitude (113) can be written simply as

A = 〈0|ψ〉 (114)

Therefore we can express the dynamics from tprime to t in terms of a singlestate |0〉 in a Hilbert space Kttprime that represents outcomes of measurementson tprime and t The state |0〉 is called the covariant vacuum and should notbe confused with the state of minimal energy

Let us extend this idea to field theory In field theory the analog ofthe data set (x t xprime tprime) is the couple [Σ ϕ] where Σ is a 3d surfacebounding a finite spacetime region and ϕ is a field configuration on ΣThese data define a set of events (x isin Σ ϕ(x)) that bound a finite portionof a classical configuration of the field just as (x t xprime tprime) bound a finiteportion of the classical trajectory of the particle The data from a localexperiment (measurements preparation or just assumptions) must in factrefer to the state of the system on the entire boundary of a finite spacetimeregion The field theoretical space G is therefore the space of surfaces Σand field configurations ϕ on Σ Quantum dynamics can be expressed interms of an amplitude W [Σ ϕ] Following Feynmanrsquos intuition we canformally define W [Σ ϕ] in terms of a sum over bulk field configurationsthat take the value ϕ on the boundary Σ In fact in Section 53 I arguethat the functional W [Σ ϕ] captures the dynamics of a QFT

Notice that the dependence of W [Σ ϕ] on the geometry of Σ codes thespacetime position of the measuring apparatus In fact the relative posi-tion of the components of the apparatus is determined by their physicaldistance and the physical time lapsed between measurements and thesedata are contained in the metric of Σ

Consider now a background-independent theory Diffeomorphism in-variance implies immediately that W [Σ ϕ] is independent of Σ This isthe analog of the independence of W (x y) from x and y mentioned inSection 114 Therefore in gravity W depends only on the boundaryvalue of the fields However the fields include the gravitational field andthe gravitational field determines the spacetime geometry Therefore thedependence of W on the fields is still sufficient to code the relativedistance and time separation of the components of the measuring ap-paratus

What is happening is that in background-dependent QFT we have twokinds of measurements those that determine the distances of the partsof the apparatus and the time lapsed between measurements and the ac-tual measurements of the fieldsrsquo dynamical variables In quantum gravityinstead distances and time separations are on an equal footing with thedynamical fields This is the core of the general-relativistic revolutionand the key for background-independent QFT


We need one final step Notice from (112) that the argument of W isnot the classical quantity but rather the eigenstate of the correspond-ing operator The eigenstates of the gravitational field are spin networksTherefore in quantum gravity the argument of W must be a spin networkrepresenting the possible outcome of a measurement of the gravitationalfield (or the geometry) on a closed 3d surface Thus in quantum gravityphysical amplitudes must be expressed by amplitudes of the form W (s)These give the correlation probability amplitude associated with the out-come s in a measurement of a geometry just as W (x t xprime tprime) does for aparticle

A particularly interesting case is when we can separate the boundarysurface in two components then s = sout cup sin In this case W (sout sin)can be interpreted as the probability amplitude of measuring the quantumthree-geometry sout if sin was observed

Notice that a spin network sin is the analog of (x t) not just x aloneThe time variable is mixed up with the physical variables (Chapter 3) Thenotion of unitary quantum evolution in time is ill defined in this contextbut probability amplitudes remain well defined and physically meaningful(Chapter 5) The quantum dynamical information of the theory is entirelycontained in the spin network amplitudes W (s) Given a configuration ofspace and matter these amplitudes determine a correlation probabilityof observing it

Calculation of the amplitude W (s) In the relativistic formulation of clas-sical hamiltonian theory dynamics is governed by the relativistic hamilto-nian H8 This is discussed in detail in Chapter 3 The quantum dynamicsis governed by the corresponding quantum operator H In quantum grav-ity H is defined on the space of the spin networks There is no externaltime variable t in the theory and the quantum dynamical equation whichreplaces the Schrodinger equation is the equation HΨ = 0 called theWheelerndashDeWitt equation The space of the solutions of the WheelerndashDeWitt equation is denoted H There is an operator P Kdiff rarr H thatprojects Kdiff on the solutions of the WheelerndashDeWitt equation (for amathematically more precise statement see Section 52)

The transition amplitudes W (s sprime) are the matrix elements of the oper-ator P They define the physical scalar product namely the scalar productof the space H

W (s sprime) = 〈s|P |sprime〉Kdiff= 〈s|sprime〉H (115)

Thus the transition amplitude between two states is simply their physicalscalar product (Chapter 5) More generally there is a preferred state |empty〉

8H is sometimes called the ldquohamiltonian constraintrdquo or the ldquosuperhamiltonianrdquo



in Kdiff which is formed by no spin networks It represents a space withzero volume or more precisely no space at all The covariant vacuumstate which defines the dynamics of the theory is defined by |0〉=P |empty〉The amplitude of a spin network is defined by

W (s) = 〈0|s〉 = 〈empty|P |s〉 (116)

The construction of the operator H is a major task in LQG It is delicateand it requires a nontrivial regularization procedure in order to deal withoperator products Chapter 7 is devoted to this construction Remarkablythe limit in which the regularization is removed exists precisely thanksto diffeomorphism invariance (Section 71) This is a second major payoffof background independence At present more than one version of theoperator H has been constructed and it is not yet clear which variant(if any) is correct The remarks that follow refer to all of them

The most remarkable aspect of the hamiltonian operator H is that itacts only on the nodes A state labeled by a spin network without nodes ndashthat is in which the graph Γ is simply a collection of nonintersectingloops ndash is a solution of the WheelerndashDeWitt equation In fact the unex-pected fact that exact solutions of the WheelerndashDeWitt equation couldbe found at all was the first major surprise that raised interest in LQGin the first place in the late 1980s

Acting on a generic state |s〉 the action of the operator H turns out tobe discrete and combinatorial the topology of the graph is changed andthe labels are modified in the vicinity of a node A typical example of theaction of H on a node is illustrated in Figure 14 the action on a nodesplits the node into three nodes and multiplies the state by a number a(that depends on the labels of the spin network around the node) Labelsof links and nodes are not indicated in the figure

Notice the various manners in which the spin network basis is effectivein quantum gravity The states in the spin network basis

(i) diagonalize area and volume(ii) control diff-invariance diffeomorphism equivalence classes of states

are labeled by the s-knots(iii) simplify the action of H reducing it to a combinatorial action on

the nodes


The construction of the hamiltonian operator H completes the defini-tion of the general formalism of LQG in the case of pure gravity This isextended to matter couplings in Chapter 7 In Chapter 8 I describe someof the most interesting applications of the theory In particular I illustratethe application of LQG to cosmology (control of the classical initial sin-gularity inflation) and to black-hole physics (entropy emitted spectrum)I also mention some of its tentative applications in astrophysics


To be able to compute all the predictions of a theory it is not sufficient tohave the general definition of the theory A road towards the calculationof transition amplitudes in quantum gravity is provided by the spinfoamformalism

Following Feynmanrsquos ideas we can give W (s sprime) a representation as asum-over-paths This representation can be obtained in various mannersIn particular it can be intuitively derived from a perturbative expansionsumming over different histories of sequences of actions of H that send sprime

into sA path is then the ldquoworld-historyrdquo of a graph with interactions hap-

pening at the nodes This world-history is a two-complex as in Figure15 namely a collection of faces (the world-histories of the links) facesjoin at edges (the world-histories of the nodes) in turn edges join atvertices A vertex represents an individual action of H An example of avertex corresponding to the action of H of Figure 14 is illustrated inFigure 16 Notice that on moving from the bottom to the top a sectionof the two-complex goes precisely from the graph on the left-hand side ofFigure 14 to the one on the right-hand side Thus a two-complex is likea Feynman graph but with one additional structure A Feynman graph iscomposed by vertices and edges a spinfoam by vertices edges and faces

Faces are labeled by the area quantum numbers jl and edges by thevolume quantum numbers in A two-complex with faces and edges la-beled in this manner is called a ldquospinfoamrdquo and denoted σ Thus a spin-foam is a Feynman graph of spin networks or a world-history of spinnetworks A history going from sprime to s is a spinfoam σ bounded by sprime

and sIn the perturbative expansion of W (s sprime) there is a term associated

with each spinfoam σ bounded by s and sprime This term is the amplitude ofσ The amplitude of a spinfoam turns out to be given by (a measure termμ(σ) times) the product over the vertices v of a vertex amplitude Av(σ)The vertex amplitude is determined by the matrix element of H betweenthe incoming and the outgoing spin networks and is a function of the labels


v2

v1

5

56

7

8

8

1

3

7

63

3

si

sf

s1

Σi

Σf

Fig 15 A spinfoam representing the evolution of an initial spin network si toa final spin network sf via an intermediate spin network s1 Here v1 and v2 arethe interaction vertices

Fig 16 The vertex of a spinfoam

of the faces and the edges adjacent to the vertex This is analogous to theamplitude of a conventional Feynman vertex which is determined by thematrix element of the hamiltonian between the incoming and outgoingstates


The physical transition amplitudes W (s sprime) are then obtained by sum-ming over spinfoams bounded by the spin networks s and sprime

W (s sprime) simsum

σpartσ=scupsprime

μ(σ)prod

v

Av(σ) (117)

More generally for a spin network s representing a closed surface

W (s) simsum

σpartσ=s

μ(σ)prod

v

Av(σ) (118)

In general the Feynman path integral can be derived from Schrodingertheory by exponentiating the hamiltonian operator but it can also be di-rectly interpreted as a sum over classical trajectories of the particle Simi-larly the spinfoam sum (117) can be interpreted as a sum over spacetimesThat is the sum (117) can be seen as a concrete and mathematicallywell-defined realization of the (ill-defined) WheelerndashMisnerndashHawking rep-resentation of quantum gravity as a sum over four-geometries

W (3g 3gprime) simint

partg= 3gcup3gprime[Dg] e

iS[g] (119)

Because of their foamy structure at the Planck scale spinfoams canbe viewed as a mathematically precise realization of Wheelerrsquos intuitionof a spacetime ldquofoamrdquo In Chapter 9 I describe various concrete realiza-tions of (117) as well as the possibility of directly relating (117) with adiscretization of (119)


The search for a quantum theory of gravity raises questions such as Whatis space What is time What is the meaning of ldquobeing somewhererdquoWhat is the meaning of ldquomovingrdquo Is motion to be defined with respectto objects or with respect to space Can we formulate physics withoutreferring to time or to spacetime And also What is matter What iscausality What is the role of the observer in physics

Questions of this kind have played a central role in periods of majoradvances in physics For instance they played a central role for EinsteinHeisenberg Bohr and their colleagues but also for Descartes GalileoNewton and their contemporaries and for Faraday Maxwell and theircolleagues Today this manner of posing problems is often regarded asldquotoo philosophicalrdquo by many physicists

Indeed most physicists of the second half of the twentieth century haveviewed questions of this nature as irrelevant This view was appropriate

13 Conceptual issues 29

for the problems they were facing one does not need to worry aboutfirst principles in order to apply the Schrodinger equation to the heliumatom to understand how a neutron star holds together or to find out thesymmetry group governing the strong interactions During this periodphysicists lost interest in general issues As someone has said during thisperiod ldquodo not ask what the theory can do for you ask what you cando for the theoryrdquo That is do not ask foundational questions just keepdeveloping and adjusting the theory you happen to find in front of youWhen the basics are clear and the issue is problem-solving within a givenconceptual scheme there is no reason to worry about foundations theproblems are technical and the pragmatical approach is the most effectiveone

Today the kind of difficulties that we face have changed To understandquantum spacetime we have to return once more to those foundationalissues We have to find new answers to the old foundational questions Thenew answers have to take into account what we have learned with QM andGR This conceptual approach is not the one of Weinberg and Gell-Mannbut it is the one of Newton Maxwell Einstein Bohr Heisenberg FaradayBoltzmann and many others It is clear from the writings of the latterthat they discovered what they did discover by thinking about generalfoundational questions The problem of quantum gravity will not be solvedunless we reconsider these questions

Several of these questions are discussed in the text Here I only commenton one of these conceptual issues the role of the notion of time


The transition amplitudes W (s sprime) do not depend explicitly on time Thisis to be expected because the physical predictions of classical GR do notdepend explicitly on the time coordinate t either The theory predictscorrelations between physical variables not the way physical variablesevolve with respect to a preferred time variable But what is the meaningof a physical theory in which the time variable t does not appear

Let me tell a story It was Galileo Galilei who first realized that thephysical motion of objects on Earth could be described by mathematicallaws expressing the evolution of observable quantities ABC in timeThat is laws for the functions A(t) B(t) C(t) A crucial contributionby Galileo was to find an effective way to measure the time variable tand therefore provide an operational meaning to these functions In factGalileo gave a decisive contribution to the discovery of the modern clockby realizing as a young man that the small oscillations of a pendulumldquotake equal timerdquo The story goes that Galileo was staring at the slowoscillations of the big chandelier that can still be seen in the marvelous


Cathedral of Pisa9 He checked the period of the oscillations against hispulse and realized that the same number of pulses lapsed during anyoscillation of the chandelier This was the key insight the basis of themodern clock today virtually every clock contains an oscillator Laterin life Galileo used a clock to discover the first quantitative terrestrialphysical law in his historic experiments on descent down inclines

Now the puzzling part of the story is that while Galileo checked thependulum against his pulse not long afterwards doctors were checkingtheir patientrsquos pulse against a pendulum What is the actual meaning ofthe pendulum periods taking ldquoequal timerdquo An equal amount of t lapsesin any oscillation how do we know this if we can access t only via anotherpendulum

It was Newton who cleared up the issue conceptually Newton as-sumes that an unobservable quantity t exists which flows (ldquoabsolute andequal to itselfrdquo) We write equations of motion in terms of this t butwe cannot truly access t we can build clocks that give readings T1(t)T2(t) that according to our equations approximate t with the preci-sion we want What we actually measure is the evolution of other variablesagainst clocks namely A(T1) B(T1) Furthermore we can check clocksagainst one another by measuring the functions T1(T2) T2(T3) Thefact that all these observations agree with what we compute using evo-lution equations in t gives us confidence in the method In particularit gives us confidence that to assume the existence of the unobservablephysical quantity t is a useful and reasonable thing to do

Simply the usefulness of this assumption is lost in quantum gravity Thetheory allows us to calculate the relations between observable quantitiessuch as A(B) B(C) A(T1) T1(A) which is what we see But it doesnot give us the evolution of these observable quantities in terms of anunobservable t as Newtonrsquos theory and special relativity do In a sensethis simply means that there are no good clocks at the Planck scale

Of course in a specific problem we can choose one variable decide totreat it as the independent variable and call it ldquotherdquo time For instance acertain clock time a certain proper time along a certain particle historyetc The choice is largely arbitrary and generally it is only locally meaning-ful A generally covariant theory does not choose a preferred time variable

Here are two examples to illustrate this arbitrariness- Imagine we throw a precise clock upward and compare its lapsed reading tf when it

lands back with the lapsed reading te of a clock remaining on the Earth GR predictsthat the two clocks read differently and provides a quantitative relation between tf

9Nice story Too bad the chandelier was hung there a few decades after Galileorsquos dis-covery


and te Is this about the observable tf evolving in the physical time te or about theobservable te evolving in the physical time tf

10

- The cosmological context is often indicated as one in which a natural choice oftime is available the cosmological time tc is the proper time from the Big Bang alongthe galaxiesrsquo worldlines But an event A happening on Andromeda at the same tc asours happens much later than an event B on Andromeda simultaneous to us in thesense of Einsteinrsquos definition of simultaneity11 So what is happening ldquoright nowrdquo onAndromeda A or B Furthermore the real world is not truly homogeneous when twogalaxies having two different ages relative to the Big Bang or two different massesme merge which of the two has the right time

So long as we remain within classical general relativity a given gravi-tational field has the structure of a pseudo-riemannian manifold There-fore the dynamics of the theory has no preferred time variable but wenevertheless have a notion of spacetime for each given solution But inquantum theory there are no classical field configurations just as thereare no trajectories of a particle Thus in quantum gravity the notion ofspacetime disappears in the same manner in which the notion of trajec-tory disappears in the quantum theory of a particle A single spinfoam canbe thought of as representing a spacetime but the history of the world isnot a single spinfoam it is a sum over spinfoams

The theory is conceptually well defined without making use of the no-tion of time It provides probabilistic predictions for correlations betweenthe physical quantities that we can observe In principle we can checkthese predictions against experiments12 Furthermore the theory providesa clear and intelligible picture of the quantum gravitational field namelyof a ldquoquantum geometryrdquo

Thus there is no background ldquospacetimerdquo forming the stage on whichthings move There is no ldquotimerdquo along which everything flows The worldin which we happen to live can be understood without using the notionof time

mdashmdash


The fact that perturbative quantum general relativity is nonrenormaliz-able has been long believed but was proven only in 1986 by Goroff andSagnotti [29]

10If you are tempted to say that the lapsed reading te of the clock remaining on Earthgives the ldquotrue timerdquo recall that the pseudo-riemannian distance between the twoevents at which the clocks meet is tf not te it is the clock going up and down thatfollows a geodesic

11Thanks to Marc Lachieze-Rey for this observation12The special properties of a time variable may emerge only macroscopically This is

discussed in Sections 34 and 551


For an orientation on current research on quantum gravity see for in-stance the review papers [30ndash33] An interesting panoramic of points ofview on the problem is in the various contributions to the book [34] Ihave given a critical discussion on the present state of spacetime physicsin [35ndash37] A historical account of development of quantum gravity isgiven in Appendix B

As a general introduction to quantum gravity ndash a subject where nothingyet is certain ndash the student eager to learn is strongly advised to study alsothe classic reviews which are rich in ideas and present different points ofview such as John Wheeler 1967 [38] Steven Weinberg 1979 [39] StephenHawking 1979 and 1980 [4041] Karel Kuchar 1980 [42] and Chris Ishamrsquosmagistral syntheses [43ndash45] On string theory classic textbooks are GreenSchwarz and Witten and Polchinksi [46] For a discussion of the difficul-ties of string theory and a comparison of the results of strings and loopssee [47] written in the form of a dialog and [48] For a fascinating pre-sentation of Alain Connesrsquo vision see [49] Lee Smolinrsquos popular-sciencebook [50] provides a readable and enjoyable introduction to LQG

LQG has inspired novels and short stories Blue Mars by Kim StanleyRobinson [51] contains a description of the future evolution and merg-ing of loop gravity and strings I recommend the science fiction novelSchild Ladder by Greg Egan [52] which opens with one of the clearestpresentations of the picture of space given by loop gravity (Greg is a tal-ented writer and also a scientist who is contributing to the development ofLQG) and for those who can read Italian Anna prende il volo by EnricoPalandri [53] a charming novel with a gentle meditation on the meaning ofthe disappearance of time Literature has the capacity of delicately merg-ing the novel hard views that science develops into the common discourseof our civilization

2General Relativity

Lev Landau has called GR ldquothe most beautifulrdquo of the scientific theories The theoryis first of all a description of the gravitational force Nowadays it is very extensivelysupported by terrestrial and astronomical observations and so far it has never beenquestioned by an empirical observation

But GR is far more than that It is a complete modification of our understanding ofthe basic grammar of nature This modification does not apply solely to gravitationalinteraction it applies to all aspects of physics In fact the extent to which Einsteinrsquosdiscovery of this theory has modified our understanding of the physical world and thefull reach of its consequences have not yet been completely unraveled

This chapter is not an introduction to GR nor an exhaustive description of thetheory For this I refer the reader to the classic textbooks on the subject Here I givea short presentation of the formalism in a compact and modern form emphasizingthe reading of the theory which is most useful for quantum gravity I also discuss indetail the physical and conceptual basis of the theory and the way it has modified ourunderstanding of the physical world

21 Formalism


Let M be the ldquospacetimerdquo four-dimensional manifold Coordinates onM are written as x x where x = (xμ) = (x0 x1 x2 x3) Indicesμ ν = 0 1 2 3 are spacetime tangent indices

bull The gravitational field e is a one-form

eI(x) = eIμ(x) dxμ (21)

with values in Minkowski space Indices I J = 0 1 2 3 label the com-ponents of a Minkowski vector They are raised and lowered with theMinkowski metric ηIJ

33


I call ldquogravitational fieldrdquo the tetrad field rather than Einsteinrsquos metric field gμν(x)There are three reasons for this (i) the standard model cannot be written in terms of gbecause fermions require the tetrad formalism (ii) the tetrad field e is nowadays moreutilized than g in quantum gravity and (iii) I think that e represents the gravitationalfield in a more conceptually clean way than g (see Section 223) The relation with themetric formalism is given in Section 215

bull The spin connection ω is a one-form with values in the Lie algebra ofthe Lorentz group so(3 1)

ωIJ(x) = ωI

μJ(x) dxμ (22)

where ωIJ = minusωJI It defines a covariant partial derivative Dμ on allfields that have Lorentz (I) indices

DμvI = partμv

I + ωIμJ vJ (23)

and a gauge-covariant exterior derivative D on forms For instance for aone-form uI with a Lorentz index

DuI = duI + ωIJ and uJ (24)

The torsion two-form is defined as

T I = DeI = deI + ωIJ and eJ (25)

A tetrad field e determines uniquely a torsion-free spin connection ω =ω[e] called compatible with e by

T I = deI + ω[e]IJ and eJ = 0 (26)

The explicit solution of this equation is given below in (291) or (292)

bull The curvature R of ω is the Lorentz algebra valued two-form1

RIJ = RI

J μν dxμ and dxν (27)

defined by2

RIJ = dωI

J + ωIK and ωK

J (28)

1Generally I write spacetime indices μν before internal Lorentz indices IJ But for thecurvature I prefer to stay closer to Riemannrsquos notation

2Sometimes the curvature of a connection ωIJ is written as RI

J = DωIJ If we naively

use the definition (24) for D we get an extra 2 in the quadratic term The point isthat the indices on the connection are not vector indices That is (24) defines theaction of D on sections of a vector bundle and a connection is not a section of a vectorbundle

21 Formalism 35

We have then immediately from (24)

D2uI = RIJ and uJ (29)

and from this equation and (26)

RIJ and eJ = 0 (210)

A region where the curvature is zero is called ldquoflatrdquo Equations (25) and(28) are called the Cartan structure equations

bull The Einstein equations ldquoin vacuumrdquo are

εIJKL (eI andRJK minus 23λ eI and eJ and eK) = 0 (211)

The equation (26) relating e and ω and the Einstein equations (211)are the field equations of GR in the absence of other fields They are theEulerndashLagrange equations of the action

S[e ω] =1

16πG

intεIJKL(

14eIandeJandR[ω]KLminus 1

12λ eIandeJandeKandeL) (212)

where G is the Newton constant3 and λ is the cosmological constantwhich I often set to zero below

bull Inverse tetrad Using the matrix eμI (x) defined to be the inverse of the matrixeIμ(x) we define the Ricci tensor

RIμ = RIJ

μν eνJ (213)

and the Ricci scalar

R = RIμ eμI (214)

and write the vacuum Einstein equations (211) as

RIμ minus 1

2ReIμ + λeIμ = 0 (215)

3The constant 16πG has no effect on the classical equations of motion (211) Howeverit governs the strength of the interaction with the matter fields described below andit also determines the quantum properties of the system In this it is similar to themass constant m in front of a free-particle action the classical equations of motion(x = 0) do not depend on m but the quantum dynamics of the particle does Forinstance the rate at which a wave packet spreads depends on m Similarly we willsee that the quanta of pure gravity are governed by this constant


bull Second-order formalism Replacing ω with ω[e] in (212) we get the equivalentaction

S[e] =1

16πG

intεIJKL (

1

4eI and eJ andR[ω[e]]KL minus 1

12λ eI and eJ and eK and eL) (216)

The formalisms in (212) where e and ω are independent is called the first-order for-malism The two formalism are not equivalent in the presence of fermions we do notknow which one is physically correct because the effect of gravity on single fermions ishard to measure

bull Selfdual formalism Consider the selfdual ldquoprojectorrdquo P iIJ given by

P ijk =

1

2εijk P i

0j = minusP ij0 =

i

2δij (217)

where i = 1 2 34 This verifies the two properties

1

2εIJ

KLP iKL = iP i

IJ P IJi P i

KL = P IJKL equiv 1

2δ I[K δLJ] +

i

4εIJKL (218)

where P IJKL is the projector on selfdual tensors Define the complex SO(3) connection

Aiμ = P i

IJ ωIJμ (219)

Equivalently

Ai = ωi + iω0i (220)

(We write ωi = 12εijkωjk See pg xxii) We can use the complex selfdual connection

Ai (three complex one-forms) instead of the real connection ωIJ (six real one-forms)

as the dynamical variable for GR (This is equivalent to describing a system with tworeal degrees of freedom x and y in terms of a single complex variable z = x + iy) Interms of Ai the vacuum Einstein equations read

PiIJ eI and (F i minus 2

3λ P i

KLeK and eL) = 0 (221)

where F i = dAi + εijkAjAk is the curvature of A5 These are the EulerndashLagrange

equations of the action

S[eA] =1

16πG

int(minusiPiIJ eI and eJ and F i minus 1

12λ εIJKL eI and eJ and eK and eL) (222)

which differs from the action (212) by an imaginary term that does not change theequations of motion The selfdual formalism is often used in canonical quantizationbecause it simplifies the form of the hamiltonian theory If we replace the imaginaryunit i in (217) with a real parameter γ (222) is called the Holst action [54] and givesrise to the Ashtekar-Barbero-Immirzi formalism γ is called the Immirzi parameter

Plebanski formalism The Plebanski selfdual two-form is defined as

Σi = P iIJ eI and eJ (223)

That isΣ1 = e2 and e3 + i e0 and e1 (224)

4The complex Lorentz algebra splits into two complex so(3) algebras called the selfdualand anti-selfdual components so(3 1C) = so(3C) oplus so(3C) The projector (217)reads out the selfdual component

5Because of the split mentioned in the previous footnote the curvature of the selfdualcomponent of the connection is the selfdual component of the curvature

21 Formalism 37

and so on cyclically A straightforward calculation shows that Σ satisfies

DΣi equiv dΣi + Aij and Σj = 0 (225)

where we write Aij = εijkA

k See pg xxii The algebraic equations for a triplet ofcomplex two-forms Σi

3 Σi and Σj = δij Σk and Σk = minusδij Σk and Σk Σi and Σ

j= 0 (226)

are solved by (223) where eI is an arbitrary real tetrad The GR action can thus bewritten as

S[Σ A] =minusi

16πG

int (Σi and F i +

1

3λ Σk and Σk)

(227)

where Σi satisfies the Plebanski constraints (226) The Plebanski formalism is often

used as a starting point for spinfoam models


In the general-relativistic parlance ldquomatterrdquo is anything which is not thegravitational field As far as we know the world is made up of the grav-itational field YangndashMills fields fermion fields and presumably scalarfields

bull Maxwell The electromagnetic field is described by the one-form fieldA the Maxwell potential

A(x) = Aμ(x) dxμ (228)

Its curvature is the two-form F = dA with components Fμν = partμAν minuspartνAμ Its dynamics is governed by the action

SM[eA] =14

intF lowast and F (229)

bull YangndashMills The above generalizes to a nonabelian connection A ina YangndashMills group G A defines a gauge covariant exterior derivative Dand curvature F The action is

SYM[eA] =14

inttr[F lowast and F ] (230)

where tr is a trace on the algebra

bull Scalar Let ϕ(x) be a scalar field possibly with values in a representa-tion of G The YangndashMills field A defines the covariant partial derivative

Dμϕ = partμϕ + AAμLAϕ (231)

where LA are the generators of the gauge algebra in the representationsto which ϕ belongs The action that governs the dynamics of the field is

Ssc[eA ϕ] =int

d4x e(ηIJ eμI Dμϕ eνJ Dνϕ + V (ϕ)

) (232)

where e is the determinant of eIμ and V (ϕ) is a self-interaction potential


bull Fermion A fermion field ψ is a field in a spinor representation ofthe Lorentz group possibly with values in a representation of G Thespin connection ω and the YangndashMills field A define the covariant partialderivative

Dμψ = partμψ + ωIμJL

JIψ + AA

μLAψ (233)

where LJI and LA are the generators of the Lorentz and gauge algebras

in the representations to which ψ belongs Define

Dψ = γIeμI Dμψ (234)

where γI are the standard Dirac matrices The action that governs thedynamics of the fermion field is

Sf [e ωA ϕ ψ] =int

d4x e(ψ Dψ + Y (ϕ ψ ψ)

)+ complex conjugate

(235)where the second term is a polynomial interaction potential with a scalarfield

bull The ldquolagrangian of the worldrdquo the standard model As far as we knowthe world can be described in terms of a set of fields e ωA ψ ϕ whereG = SU(3) times SU(2) times U(1) and ψ and ϕ are in suitable multiplets andis governed by the action

S[e ωA ψ ϕ] = SGR[e ω] + SYM[eA] + Sf [e ωA ψ] + Ssc[eA ϕ]= SGR[e ω] + Smatter[e ωA ϕ ψ] (236)

with suitable polynomials V and Y The equations of motion that followfrom this action by varying e are the Einstein equations (211) with asource term namely

εIJKL (eI andRJK minus 23λ eI and eJ and eK) = 2πG TL (237)

where the energy-momentum three-form

TI =det e3

TμI εμνρσdxν and dxρ and dxσ (238)

is defined by

TI(x) =δSmatter

δeI(x) (239)

Equivalently the Einstein equations (237) can be written as

RIμ minus 1

2ReIμ + λeIμ = 8πG T I

μ (240)

21 Formalism 39

T Iμ(x) is called the energy-momentum tensor It is the sum of the individ-

ual energy-momentum tensors of the various matter terms6

bull Particles The trajectory xμ(s) of a point particle is an approximate notion Macro-scopic objects have finite size and elementary particles are quantum entities and there-fore have no trajectories At macroscopic scales the notion of a point-particle trajectoryis nevertheless very useful

In the absence of nongravitational forces the equations of motion for the worldlineγ s rarr xμ(s) of a particle are determined by the action

S[e γ] = m

intds

radicminusηIJvI(s)vJ(s) (241)

wherevI(s) = eIμ(x(s))vμ(s) (242)

and vμ is the particle velocity

vμ(s) = xμ(s) equiv dxμ(s)

ds (243)

This action is independent of the way the trajectory is parametrized and thereforedetermines the path not its parametrization With the parametrization choice vIv

I =minus1 the equations of motion are


whereΓσμν = eρJe

Jσ(eρIpart(μeIν) + eνIpart[μe

Iρ] + eμIpart[νe

Iρ]) (245)

is called the LevindashCivita connection In an arbitrary parametrization the equations ofmotion are

xμ + Γμνρ xν xρ = I(s) xμ (246)

where I(s) is an arbitrary function of s

Minkowski solution Consider a regime in which we can assume that theNewton constant G is small that is a regime in which we can neglectthe effect of matter on the gravitational field Assume also that withinour approximation the cosmological constant λ is negligible The Ein-stein equations (211) then admit (among many others) the particularlyinteresting solution

eIμ(x) = δIμ ωIμJ(x) = 0 (247)

which is called the Minkowski solution This solution is everywhere flatAssume that the gravitational field is in this configuration What are

the equations of motion of the matter interacting with this particular

6The energy-momentum tensor defined as the variation of the action with respect tothe gravitational field may differ by a total derivative from the one conventional inMinkowski space defined as the Noether current of translations


gravitational field These are easily obtained by inserting the Minkowskisolution (247) into the matter action (236)

S[Aϕ ψ] = Smatter[e = δ ω = 0 A ϕ ψ] (248)

The action S[Aϕ ψ] is the action of the standard model used in high-energy physics This action is usually written in terms of the spacetimeMinkowski metric ημν This metric is obtained from the Minkowski value(247) of the tetrad field For instance in the action of a scalar field (232)the combination ηIJeμI (x)eνJ(x) becomes

ηIJeμI (x)eνJ(x) = ηIJδμI δνJ = ημν (249)

on this solutionThe Minkowski metric ημν of special-relativistic physics is nothing but

a particular value of the gravitational field It is one of the solutions ofthe Einstein equation within a certain approximation


The general definition of a system with a gauge invariance and the onewhich is most useful for understanding the physics of gauge systems is thefollowing which is due to Dirac Consider a system of evolution equationsin an evolution parameter t The system is said to be ldquogaugerdquo invariantif evolution is under-determined that is if there are two distinct solu-tions that are equal for t less than a certain t see Figure 21 These twosolutions are said to be ldquogauge equivalentrdquo Any two solutions are saidto be gauge equivalent if they are gauge equivalent (as above) to a thirdsolution The gauge group G is a group that acts on the physical fields andmaps gauge-equivalent solutions into one another Since classical physics isdeterministic under-determined evolution equations are physically consis-tent only under the stipulation that only quantities invariant under gaugetransformations are physical predictions of the theory These quantitiesare called the gauge-invariant observables

The equations of motion derived by the action (236) are invariant underthree groups of gauge transformations (i) local YangndashMills gauge trans-formations (ii) local Lorentz transformations and (iii) diffeomorphismtransformations They are described below Gauge-invariant observablesmust be invariant under these three groups of transformations

(i) Local G transformations G is the YangndashMills group A local G transformationis labeled by a map λ M rarr G It acts on ϕψ and the connection A in the

21 Formalism 41

t

j

t

~j (t)j(t)

Fig 21 Dirac definition of gauge two different solutions of the equations ofmotion must be considered gauge equivalent if they are equal for t lt t

well-known form while e and ω are invariant

λ ϕ(x) rarr Rϕ(λ(x)) ϕ(x) (250)

ψ(x) rarr Rψ(λ(x)) ψ(x) (251)

Aμ(x) rarr R(λ(x)) Aμ(x) + λ(x)partμλminus1(x) (252)

eIμ(x) rarr eIμ(x) (253)

ωIμJ(x) rarr ωI

μJ(x) (254)

Here Rϕ and Rψ are the representations of G to which ϕ and ψ belong and Ris the adjoint representation

(ii) Local Lorentz transformations A local Lorentz transformation is labeled bya map λ M rarr SO(3 1) It acts on ϕψ and the connection ω precisely as aYangndashMills local transformation with YangndashMills group G=SO(3 1) Scalars ϕbelong to the trivial representation fermions ψ belong to the spinor representa-tions S The gravitational field e transforms in the fundamental representationExplicitly writing an element of SO(3 1) as λI

J we have

λ ϕ(x) rarr ϕ(x) (255)

ψ(x) rarr S(λ(x)) ψ(x) (256)

Aμ(x) rarr Aμ(x) (257)

eIμ(x) rarr λIJ(x) eJμ(x) (258)

ωIμJ(x) rarr λI

K(x)ωKμL(x)λL

J(x) + λ IK (x)partμλ

KJ(x) (259)

(iii) Diffeomorphisms Third and most important is the invariance under diffeo-morphisms A diffeomorphism gauge transformation is labeled by a smoothinvertible map φ M rarr M (that is by a ldquodiffeomorphismrdquo of M)7 It actsnonlocally on all the fields by pulling them back according to their form char-

7There is an unfortunate terminological imprecision A map φ M rarr M is called adiffeomorphism The associated transformations (260)ndash(264) on the fields are alsooften loosely called a diffeomorphism (also in this book) instead of diffeomorphismgauge transformations This tends to generate confusion


acter ϕ and ψ are zero forms e ω and A are one-forms8

φ ϕ(x) rarr ϕ(φ(x)) (260)

ψ(x) rarr ψ(φ(x)) (261)

Aμ(x) rarr partφν(x)

partxμAν(φ(x)) (262)

eIμ(x) rarr partφν(x)

partxμeIν(φ(x)) (263)

ωIμJ(x) rarr partφν(x)

partxμωIνJ(φ(x)) (264)

These three groups of transformations send solutions of the equationsof motion into other solutions of the equations of motion They are gaugetransformations because we can take these transformations to be the iden-tity before a given coordinate time t and different from the identity af-terwards Therefore they are responsible for the under-determination ofthe evolution equations Following Diracrsquos argument given above physicalpredictions of the theory must be given by quantities invariant under allthree of these transformations

In particular let a local quantity in spacetime be a quantity dependenton a fixed given point x Notice that such a quantity cannot be invariantunder a diffeomorphism Therefore no local quantity in spacetime (in thissense) is a gauge-invariant observable in GR The meaning of this fact andthe far-reaching consequences of diffeomorphism invariance are discussedbelow in Section 232


At each point x of the spacetime manifold M the gravitational field eIμ(x)defines a map from the tangent space TxM to Minkowski space The mapsends a vector vμ in TxM into the Minkowski vector uI = eIμ(x)vμ TheMinkowski length |u| =

radicminusu middot u =radic

minusηIJuIuJ defines a norm |v| of thetangent vector vμ

|v| equiv |u| =radicminusηIJ(eIμ(x)vμ) (eJν (x)vν) (265)

8Under this definition internal Lorentz spinor and gauge indices do not transformunder a diffeomorphism Alternatively one should consider fiber-preserving diffeo-morphisms of the Lorentz and gauge bundle This alternative can be viewed as math-ematically more clean and physically more attractive because it makes more explicitthe fact that local inertial frames or local gauge choices at different spacetime pointscannot be identified (see later) However the mathematical description of a diffeo-morphism becomes more complicated while the two choices are ultimately physicallyequivalent due to the gauge invariance under local Lorentz and gauge transforma-tions The proper mathematical transformation of a spinor under diffeomorphisms isdiscussed in [55] and [56]

21 Formalism 43

|v| is called the ldquophysical lengthrdquo of the tangent vector v The tangentvector v is called timelike (spacelike or lightlike) if u is timelike (spacelikeor lightlike)

This fact allows us to assign a size to any d-dimensional surface in M At any point x on the surface the gravitational field maps the tangentspace of the surface into a surface in Minkowski space This surface carriesa volume form which can be pulled back to the tangent space of x andthen to the surface itself and integrated In particular

The length L of a curve γ s rarr xμ(s) is the line integral of the norm ofits tangent

L[e γ] =int

|dγ| =int

ds |u(s)| =int

dsradic

minusηIJ uI(s)uJ(s) (266)

whereuI(s) = eIμ(γ(s))

dxμ(s)ds

(267)

This can be written as the line integral of the norm of the one-formeI(x) = eIμ(x)dxμ along γ

L[e γ] =int

γ|e| (268)

The length is independent of the parametrization and the orien-tation of γ A curve is called timelike if its tangent is everywheretimelike Notice that the action of a particle (241) is nothing butthe length of its path in spacetime

S[e γ] = m L[e γ] (269)

The area A of a two-dimensional surface S σ= (σi) rarr xμ(σi) i= 1 2immersed in M is

A[eS] =int ∣∣d2S

∣∣ =int

Sd2σ

radicdet (ui middot uj) (270)

whereuIi (σ) = eIμ(γ(σ))

partxμ(σ)partσi

(271)

and the determinant is over the i j indices That is

A[eS] =int

d2σradic

(u1 middot u1)(u2 middot u2) minus (u1 middot u2)2 (272)

A surface is called spacelike if its tangents are all spacelike


The volume V of a three-dimensional region R σ = (σi) rarr xμ(σi) i =1 2 3 immersed in M is

V[eR] =int ∣

∣d3R∣∣ =

int

Rd3σ

radicn middot n (273)

wherenI = εIJKL uJ1u

K2 uL3 (274)

is normal to the surface A region is called spacelike if n is every-where timelike

The quantities L A and V are particular functions of the grav-itational field e The reason they have these geometric names isdiscussed below in Section 223


In GR quantities close to observations such as lengths and areas arenonlocal in the sense that they depend on finite but extended regions inspacetime such as lines and surfaces Another natural nonlocal quantitywhich plays a central role in the quantum theory is the holonomy U ofthe gravitational connection (ω or its selfdual part A) along a curve γ

Definition of the holonomy Given a connection A in a group G overa manifold M the holonomy is defined as follows Let a curve γ be acontinuous piecewise smooth map from the interval [0 1] into M

γ [0 1] minusrarr M (275)s minusrarr xμ(s) (276)

The holonomy or parallel propagator U [A γ] of the connection A alongthe curve γ is the element of G defined by

U [A γ](0) = 11 (277)dds

U [A γ](s) minus γμ(s)Aμ

(γ(s)

)U [A γ](s) = 0 (278)

U [A γ] = U [A γ](1) (279)

where γμ(s) equiv dxμ(s)ds is the tangent to the curve (In the mathematicalliterature the term ldquoholonomyrdquo is generally used for closed curves only Inthe quantum gravity literature it is commonly employed for open curvesas well) The formal solution of this equation is

U [A γ] = P expint 1

0ds γμ(s) Ai

μ

(γ(s)

)τi equiv P exp

int

γA (280)

21 Formalism 45

where τi is a basis in the Lie algebra of the group G and the path orderedP is defined by the power series expansion

P expint 1

0dsA

(γ(s)

)

=infinsum

n=0

int 1

0ds1

int s1

0ds2 middot middot middot

int snminus1

0dsnA

(γ(sn)

)middot middot middotA

(γ(s1)

) (281)

The connection A is a rule that defines the meaning of parallel-transporting a vector in a representation R of G from a point of M toa nearby point the vector v at x is defined to be parallel to the vectorv +R(Aadxμ)v at x+ dx A vector is parallel-transported along γ to thevector R(U(A γ))v

An important property of the holonomy is that it transforms homoge-neously under the gauge transformation (252) of A That is U [Aλ γ] =λ(xγf )U [A γ]λminus1(xγi ) where xγif are the initial and final points of γ

A technical remark that we shall need later on the holonomy of anycurve γ is well defined even if there are (a finite number of) points whereγ is nondifferentiable and A is ill defined The reason is that we canbreak γ into components where everything is differentiable and define theholonomy of γ as the product of the holonomies of the components whichare well defined by continuity

Physical interpretation of the holonomy Consider two left-handed neutri-nos that meet at the spacetime point A separate and then meet again atthe spacetime point B Assume their spins are parallel at A and evolve un-der the sole influence of the gravitational field What is their relative spinat B A left-handed neutrino lives in the selfdual representation of theLorentz group and therefore its spin is parallel-transported by the selfdualconnection A Let γ1 and γ2 be the worldlines of the two neutrinos fromA to B and let γ = γminus1

2 γ1 be the loop formed by the two worldlines Ifthe first neutrino has spin ψ at B the second has spin ψprime = U(A γ)ψ Byhaving the two neutrinos interact we can in principle measure a quantitysuch as α = 2Re〈ψ|ψprime〉 which (assuming |ψ| = 1) gives the trace of theholonomy α = tr U [A γ]

Metric notation Einstein wrote GR in terms of the metric field HereI give the translation to metric variables Notice however that this isnecessarily incomplete since the fermion equations of motion cannot bewritten in terms of the metric field


The metric field g is a symmetric tensor field defined by

gμν(x) = eIμ(x) eJν (x) ηIJ (282)

At each point x of M g defines a scalar product in the tangent space TxM

(u v) = gμν(x)uμvν u v isin TxM (283)

and therefore maps TxM into T lowastxM In other words gμν and its inverse gμν can be used

to raise and lower tangent indices The fact that eμI (x) equiv ηIJgμνeJν (x) is the inverse

matrix of eIμ(x) is then a result not a definition

The metric-preserving linear connection Γ is the field Γρμν(x) defined by

Γρμν = eρI(partμe

Iν + ωI

μJ eJν ) (284)

It defines a covariant partial derivative Dμ on all fields that have tangent (μ) indices

Dμvν = partμv

ν + Γνμρv

ρ (285)

Together with ω it defines a covariant partial derivative Dμ on all objects thathave Lorentz as well as tangent indices In particular notice that (284) yieldsimmediately

DμeIν = partμe

Iν + ωI

μJ eJν minus Γρμν eIρ = 0 (286)

The antisymmetric part T ρμν = Γρ

μν minus Γρνμ of the linear connection gives the torsion

T I = eIρTρμνdxμdxν defined in (25)

The LevindashCivita connection is the (metric-preserving) linear connection determinedby e and ω[e] That is it is defined by

partμeIν + ω[e]IμJ eJν minus Γρ

μν eIρ = 0 (287)

whose solution is (245) It is torsion-free Notice that the antisymmetric part of thisequation is the first Cartan structure equation with vanishing torsion namely (26)which is sufficient to determine ω[e] as a function of e

The LevindashCivita connection is uniquely determined by g it is the unique torsion-freelinear connection that is metric preserving namely that satisfies

Dμgνρ = 0 (288)

or equivalently

partμgνρ minus Γσμνgσρ minus Γσ

μρgνσ = 0 (289)

This equation is solved by (245) or

Γρμν =

1

2gρσ(partμgσν + partνgμσ minus partσgμν) (290)

Notice that equations (287) and (290) allow us to write the explicit solution of theGR equation of motion (26)

ω[e]IμJ = eνJ(partμeIν minus Γρ

μνeIρ) (291)

21 Formalism 47

where Γ is given by (290) and g by (282) Explicitly this gives with a bit of algebra

ω[e]IJμ = 2 eν[Ipart[μeν]J] + eμKeνIeσJpart[σeν]

K (292)

The Riemann tensor can be defined via

Rμνρσ eIμ = RI

J ρσ eJν (293)

The Ricci tensor is

Rμν = RIμ eIν (294)

where RIμ is defined in (213) The energy-momentum tensor (see footnote 6 after (240))

Tμν = T Iμ eIν (295)

In terms of these quantities the Einstein equations (240) read

Rμν minus 1

2Rgμν + λgμν = 8πG Tμν (296)

The Minkowski solution isgμν(x) = ημν (297)

where we see clearly that the spacetime Minkowski metric is nothing but a particularvalue of the gravitational field With a straightforward calculation the action (212)reads

S[g] =1

16πG

int(R + λ)

radicminus det g d4x (298)

The matter action cannot be written in metric variables

Riemann geometry The tensor g equips the spacetime manifold M with a metric struc-ture it defines a distance between any two points and this distance is a smooth functionon M (More precisely it defines a pseudo-metric structure as distance can be imag-inary) Riemann studied the structure defined by (M g) called today a riemannianmanifold and defined the Riemann curvature tensor as a generalization of Gauss the-ory of curved surfaces to an arbitrary number of dimensions Riemann presented thismathematical theory as a general theory of ldquogeometryrdquo that generalizes Euclidean ge-ometry Einstein utilized this mathematical theory for describing the physical dynamicsof the gravitational field In retrospect the reason this was possible is because as un-derstood by Einstein the euclidean structure of the physical space in which we liveis determined by the local gravitational field Therefore elementary physical geometryis simply a description of the local properties of the gravitational field as revealed bymatter (rigid bodies) interacting with it This point is discussed in more detail belowin Section 223

mdashmdashmdashndash

The basic equations of GR presented in this section do not look too dif-ferent from the equations of a prerelativistic9 field theory such as QED orthe standard model But the similarity can be very misleading The phys-ical interpretation of a general-relativistic theory is very different fromthe interpretation of a prerelativistic one In particular the meaning of

9Recall that in this book ldquorelativisticrdquo means general relativistic


the coordinates xμ is different than in prerelativistic physics and thegauge-invariant observables are not related to the fields as they are inprerelativistic physics

The process of understanding the physical meaning of the GR formal-ism has taken many decades and perhaps it is not entirely concluded yetFor several decades after Einsteinrsquos discovery of the theory for instanceit was not clear whether or not the theory predicted gravitational wavesThe prevailing opinion was that wave solutions were only a coordinate ar-tifact and did not represent physical waves capable of carrying energy andmomentum or as Bondi put it capable of ldquoboiling a glass of waterrdquo Thisopinion was wrong of course Einstein himself badly misinterpreted themeaning of the Schwarzschild singularity Wrong high-precision measure-ments of the EarthndashMoon distance have been in the literature for a whilebecause of a mistake due to a conceptual confusion between physical andcoordinate distance

I do not want to give the impression that GR is ldquofoggyrdquo Quite thereverse the fact that in all these and similar instances consensus haseventually emerged indicates that the conceptual structure of GR is se-cure But to understand this conceptual structure to understand how touse the equations of GR correctly and how to relate the quantities ap-pearing in these equations to the numbers measured in the laboratoryor observed by the astronomers is definitely a nontrivial problem Moregenerally the problem is to understand what precisely GR says about theworld Clarity in this respect is essential if we want to understand thequantum physics of the theory

In order to shed light on this problem it is illuminating to retrace theconceptual path and the problems that led to the discovery of the theoryThis is done in the following Section 22 The impatient reader may skipSection 22 and jump to Section 23 where the interpretation of GR iscompactly presented (but impatience slows understanding)


The roots of GR are in two distinct problems Einsteinrsquos genius was tounderstand that the two problems solve each other

221 Einsteinrsquos first problem a field theory for the newtonianinteraction

It was Newton who discovered dynamics But to a large extent it wasDescartes who a generation earlier fixed the general rules of the modernscience of nature or the Scientia Nova as it was called at the time One of


Descartesrsquo prescriptions was the elimination of all the ldquoinfluences from farawayrdquo that plagued mediaeval science According to Descartes physicalinteractions happen only between contiguous entities ndash as in collisionspushes and pulls Newton violated this prescription describing gravity asthe instantaneous ldquoaction-at-a-distancerdquo of the force

F = Gm1m2

d2 (299)

Newton did not introduce action-at-a-distance with a light heart he callsit ldquorepugnantrdquo His violation of the cartesian prescriptions was one of thereasons for the strong initial opposition to newtonianism For many hislaw of gravitation sounded too much like the discredited ldquoinfluences fromthe starsrdquo of the ineffective science of the Middle Ages But the empiricalsuccess of Newtonrsquos dynamics and gravitational theory was so immensethat most worries about action-at-a-distance dissipated

Two centuries later it is another Briton who finds the way to ad-dress the problem afresh in an effort to understand electric and magneticforces Faraday introduces a new notion10 which is going to revolution-ize modern physics the notion of field For Faraday the field is a setof lines filling space The Faraday lines begin and end on charges inthe absence of charges each line closes forming a loop In his wonderfulbook which is one of the pillars of modern physics and has virtually noequations Faraday discusses whether the field is a real physical entity11

Maxwell formalizes Faradayrsquos powerful physical intuition into a beautiful

10Many ideas of modern science have been resuscitated from hellenistic science [57]Is the FaradayndashMaxwell notion of field a direct descendant of the notion of πνευμα(pneuma) that appears for instance in Hipparchus as the carrier of the attractionof the Moon on the oceans causing the tides and which also appears in contextsrelated to magnetism [58] Did Faraday know this notion

11ldquoWith regards to the great point under consideration it is simply whether the linesof force have a physical existence or not I think that the physical nature of thelines must be grantedrdquo [59] Strictly speaking we can translate the problem in modernterms as to whether the field has degrees of freedom independent from the chargesor not But this doesnrsquot diminish the ontological significance of Faradayrsquos questionwhich seems to me transparent in these lines Faradayrsquos continuation is lovely ldquoAndthough I should not have raised the argument unless I had thought it both importantand likely to be answered ultimately in the affirmative I still hold the opinion withsome hesitation with as much indeed as accompanies any conclusion I endeavor todraw respecting points in the very depths of sciencerdquo I think that Faradayrsquos greatnessshines in this ldquohesitationrdquo which betrays his full awareness of the importance of thestep he is taking (virtually all of modern fundamental physics comes out of theselines) as well as the full awareness of the risk of taking any major novel step


mathematical theory ndash a field theory At each spacetime point Maxwellelectric and magnetic fields represent the tangent to the Faraday lineThere is no action-at-a-distance in the theory the Coulomb descriptionof the electric force between two charges namely the instantaneous action-at-a-distance law

F = kq1 q2d2

(2100)

is understood to be correct only in the static limit A charge q1 at distanced from another charge q2 does not produce an instantaneous force on q2because if we move q1 rapidly away it takes a time t = dc before q2begins to feel any change This is the time the interaction takes to moveacross space at a finite speed in a manner remarkably consistent withDescartesrsquo prescription

When Einstein studies physics Maxwell theory is only three decadesold In his writings Einstein rhapsodizes on the beauty of Maxwell the-ory and the profound impression it made upon him Given the formalsimilarity of the Newton and Coulomb forces (299) and (2100) it iscompletely natural to suspect that (299) also is only correct in the staticlimit Namely that the gravitational force is not instantaneous either ifa neutron star rushing at great speed from the deep sky smashed awaythe Sun it would take a finite time before any effect be felt on EarthThat is it is natural to suspect that there is a field theory behind New-ton theory as well Einstein set out to find this field theory GR is what hefound

Special relativity In fact the need for a field theory behind Newton law(299) is not just suggested by the CoulombndashMaxwell analogy it is indi-rectly required by Maxwell theory The reason is that Maxwell theory notonly eliminated the apparent action-at-a-distance of Coulomb law (2100)but it also led to a reorganization of the notions of space and time whichin turn renders any action-at-a-distance inconsistent This reorganizationof the notions of space and time is special relativity a key step towardsGR

In spite of its huge empirical success Maxwell theory had an appar-ent flaw if taken as a fundamental theory12 it is not galilean invariantGalilean invariance is a consequence of the equivalence of inertial frames ndashat least it had always been understood as such Inertial frame equivalenceor the fact that velocity is a relative notion is one of the pillars of dy-namics The story goes that in the silent halls of Warsawrsquos University an

12Rather than as a phenomenological theory of the disturbances of a mechanical etherwhose dynamics is still to be found


old and grave professor stormed out of his office like a madman shoutingldquoEureka Eureka The new Archimedes is bornrdquo when he saw Einsteinrsquos1905 paper offering the solution of this apparent contradiction The wayEinstein solves the problem is an example of theoretical thinking at itsbest I think it should be kept in mind as an exemplar when we considerthe apparent contradictions between GR and QM

Einstein maintains his confidence in the galilean discovery that physicsis the same in all moving inertial frames and also maintains his confidencethat Maxwell equations are correct in spite of the apparent contradictionHe realizes that there is contradiction only because we implicitly hold athird assumption By dropping this third assumption the contradictiondisappears The third assumption regards the notion of time It is the ideathat it is always meaningful to say which of two distant events A andB happens first Namely that simultaneity is well defined in a mannerindependent of the observer Einstein observes that this is a prejudice wehave on the structure of reality We can drop this prejudice and accept thefact that the temporal ordering of distant events may have no meaningIf we do so the picture returns to consistency

The success of special relativity was rapid and the theory is todaywidely empirically supported and universally accepted Still I do notthink that special relativity has really been fully absorbed even nowthe large majority of cultivated people as well as a surprisingly highnumber of theoretical physicists still believe deep in their heart thatthere is something happening ldquoright nowrdquo on Andromeda that there isa single universal time ticking away the life of the Universe Do you myreader

An immediate consequence of special relativity is that action-at-a-distance is not just ldquorepugnantrdquo as Newton felt it is a nonsense Thereis no (reasonable) sense in which we can say that the force due to themass m1 acts on the mass m2 ldquoinstantaneouslyrdquo If special relativity iscorrect (299) is not just likely to be the static limit of a field theoryit has to be the static limit of a field theory When the neutron starhits the Sun there is no ldquonowrdquo at which the Earth could feel the effectThe information that the Sun is no longer there must travel from Sun toEarth across space carried by an entity This entity is the gravitationalfield

Maxwell rarr Einstein Therefore shortly after having worked out the keyconsequences of special relativity Einstein attacks what is obviously thenext problem searching the field theory that gives (299) in the staticlimit His aim is to do for (299) what Faraday and Maxwell had done for(2100) The result in brief is the following expressed in modern language


Maxwellrsquos solution to the problem is tointroduce the one-form field Aμ(x)

The force on the particles is

xμ = eFμν xν (2101)

where F is constructed with the firstderivatives of AA satisfies the (Maxwell) field equations

partμFνμ = Jν (2102)

a system of second-order partial differ-ential equations for A with the chargecurrent Jν as sourceMore generally the field equations canbe obtained as EulerndashLagrange equa-tions of the action

S[Amatt] =1

4

intF lowast and F

+Smatt[Amatt] (2103)

where F is the curvature of A

Smatt is obtained from the matter actionby replacing derivatives with covariantderivatives

It follows that the source of the fieldequations is

Jμ =δ

δAμSmatt[Amatt] (2104)

Einsteinrsquos solution is to introduce thefield eIμ(x) a one-form with value inMinkowski spaceThe force on the particles is (eq (244))


where Γ is constructed with the firstderivatives of e (equation (245))e satisfies the (Einstein) field equations(eq (237) here with λ = 0)

RIμ minus 1

2eIμR = 8πG T I

μ (2106)

a system of second order partial differ-ential equations for e with the energymomentum tensor T I

μ as sourceMore generally the field equations canbe obtained as EulerndashLagrange equa-tions of the action ((236) in second or-der form)

S[ematt] =1

16πG

inteIandeJandRKLεIJKL

+ Smatt[ematt] (2107)

where R is the curvature of the connec-tion ω compatible with eSmatt is obtained from the matter ac-tion by replacing derivatives with co-variant derivatives and the Minkowskimetric with the gravitational metricIt follows that the source of the fieldequations is (237)

T Iμ =

δ

δeIμSmatt[ematt] (2108)

The structural similarity between the theories of Maxwell and Einsteintheories is evident However this is only half of the story


To understand Einsteinrsquos second problem we have to return again to theorigin of modern physics In the western culture there are two traditionalways of understanding what is ldquospacerdquo as an entity or as a relation

ldquoSpace is an entityrdquo means that space still exists when there is nothingelse besides space It exists by itself and objects move in it Thisis the way Newton describes space and is called absolute spaceIt is also the way spacetime (rather than space) is understood in


special relativity Although considered since ancient times (in thedemocritean tradition) this way of understanding space was not thetraditional dominant view in western culture The dominant viewfrom Aristotle to Descartes was to understand space as a relation

ldquoSpace is a relationrdquo means that the world is made up of physical objectsor physical entities These objects have the property that they canbe in touch with one another or not Space is this ldquotouchrdquo orldquocontiguityrdquo or ldquoadjacencyrdquo relation between objects Aristotle forinstance defines the spatial location of an object as the (internal)boundary of the set of the objects that surround it This is relationalspace

Strictly connected to these two ways of understanding space there aretwo ways of understanding motion

ldquoAbsolute motionrdquo If space is an entity motion can be defined as goingfrom one part of space to another part of space This is how Newtondefines motion

ldquoRelative motionrdquo If space is a relation motion can only be defined asgoing from the contiguity of one object to the contiguity of anotherobject This is how Descartes13 and Aristotle14 define motion

For a physicist the issue is which of these two ways of thinking aboutspace and motion allows a more effective description of the world

For Newton space is absolute and motion is absolute15 This is a sec-ond violation of cartesianism Once more Newton does not take this step

13ldquoWe can say that movement is the transference of one part of matter or of one bodyfrom the vicinity of those bodies immediately contiguous to it and considered at restinto the vicinity of some othersrdquo (Descartes Principia Philosophiae Section II-25p 51) [60]

14Aristotle insists that motion is relative He illustrates the point with the example of aman walking on a boat The man moves with respect to the boat which moves withrespect to the water of the river which moves with respect to the ground Aristotlersquosrelationalism is tempered by the fact that there are preferred objects that can be usedas a preferred reference the Earth at the center of the Universe and the celestialspheres in particular one of the fixed stars Thus we can say that something ismoving ldquoin absolute termsrdquo if it moves with respect to the Earth However thereare two preferred frames in ancient cosmology the Earth and the fixed stars andthe two rotate with respect to each other The thinkers of the Middle Ages did notmiss this point and discussed at length whether the stars rotate around the Earthor the Earth rotates under the stars Remarkably in the fourteenth century Buridanconcluded that neither view is more true than the other on grounds of reason andOresme studied the rotation of the Earth more than a century before Copernicus

15ldquoSo it is necessary that the definition of places and hence local motion be referredto some motionless thing such as extension alone or space in so far as space is seentruly distinct from moving bodiesrdquo [61] This is in open contrast with Descartesdefinition given in footnote 13


with a light heart he devotes a long initial section of the Principia toexplain the reasons of his choice The strongest argument in Newtonrsquosfavor is entirely a posteriori his theoretical construction works extraor-dinarily well Cartesian physics was never as effective But this is notNewtonrsquos argument Newton resorts to empirical evidence discussing afamous experiment with a bucket

Newtonrsquos bucket Consider a ldquobucket full of water hung by a long cordso often turned about that the cord is strongly twistedrdquo Whirl thebucket so that it starts rotating and the cord untwisting At first

(i) the bucket rotates (with respect to us) and the water remainsstill The surface of the water is flatThen the motion of the bucket is transmitted to the water byfriction and thus the water starts rotating together with thebucket At some time

(ii) the water and the bucket rotate together The surface of thewater is no longer flat it is concave

We know from experience that the concavity of the water is caused byrotation Rotation with respect to what Newtonrsquos bucket experimentshows something subtle about this question If motion is change of placewith respect to the surrounding objects as Descartes demands then wemust say that in (i) water rotates (with respect to the bucket whichsurrounds it) while in (ii) water is still (with respect to the bucket) Butobserves Newton the concavity of the surface appears in (ii) not in (i)It appears when the water is still with respect to the bucket not whenthe water moves with respect to the bucket Therefore the rotation thatproduces the physical effect is not the rotation with respect to the bucketIt is the rotation with respect to what

It is rotation with respect to space itself answers Newton The concav-ity of the water surface is an effect of the absolute motion of the waterthe motion with respect to absolute space not to the surrounding bodiesThis claims Newton proves the existence of absolute space

Newtonrsquos argument is subtle and for three centuries nobody had been able to defeatit To understand it correctly we should lay to rest a common misunderstanding Rela-tionalism namely the idea that motion can be defined only in relation to other objectsshould not be confused with galilean relativity Galilean relativity is the statementthat ldquorectilinear uniform motionrdquo is a priori indistinguishable from stasis Namely thatvelocity (just velocity) is relative to other bodies Relationalism on the other handholds that any motion (however zig-zagging) is a priori indistinguishable from stasisThe very formulation of galilean relativity assumes a nonrelational definition of motionldquorectilinear and uniformrdquo with respect to what

Now when Newton claimed that motion with respect to absolute space is real andphysical he in a sense overdid it insisting that even rectilinear uniform motion is


absolute This caused a painful debate because there are no physical effects of inertialmotion and therefore the bucket argument fails for this particular class of motions16

Therefore inertial motion and velocity are to be considered relative in newtonian me-chanics

What Newton needed for the foundation of dynamics ndash and what we are discussinghere ndash is not the relativity of inertial motion it is whether accelerated motion exem-plified by the rotation of the water in the bucket is relative or absolute The questionhere is not whether or not there is an absolute space with respect to which velocity canbe defined The question is whether or not there is an absolute space with respect towhich acceleration can be defined Newtonrsquos answer supported by the bucket argumentwas positive Without this answer Newtonrsquos main law

F = ma (2109)

wouldnrsquot even make sense

Opposition to Newtonrsquos absolute space was even stronger than oppo-sition to his action-at-a-distance Leibniz and his school argued fierilyagainst Newton absolute motion and Newtonrsquos use of absolute accelera-tion17 Doubts never really disappeared down through subsequent cen-turies and a lingering feeling remained that something was wrong inNewtonrsquos argument At the end of the nineteenth century Ernst Machreturned to the issue suggesting that Newtonrsquos bucket argument could bewrong because the water does not rotate with respect to absolute space itrotates with respect to the full matter content of the Universe I will com-ment on this idea and its influence on Einstein in Section 241 But asfor action-at-a-distance the immense empirical triumph of newtonianismcould not be overcome

Or could it After all in the early twentieth century 43 seconds of arcin Mercuryrsquos orbit were observed which Newtonrsquos theory didnrsquot seem tobe able to account for

Generalize relativity Einstein was impressed by galilean relativity Thevelocity of a single object has no meaning only the velocity of objectswith respect to one another is meaningful Notice that in a sense thisis a failure of Newtonrsquos program of revealing the ldquotrue motionsrdquo It is aminor but significant failure For Einstein this was a hint that there issomething wrong in the newtonian (and special-relativistic) conceptualscheme

16Newton is well aware of this point which is clearly stated in the Corollary V ofthe Principia but he chooses to ignore it in the introduction to Principia I thinkhe did this just to simplify his argument which was already hard enough for hiscontemporaries

17Leibniz had other reasons of complaint with Newton The two were fighting overthe priority for the invention of calculus ndash scientistsrsquo frailties remain the same in allcenturies


In spite of its immense empirical success Newtonrsquos idea of an abso-lute space has something deeply disturbing in it As Leibniz Mach andmany others emphasized space is a sort of extrasensorial entity that actson objects but cannot be acted upon Einstein was convinced that theidea of such an absolute space was wrong There can be no absolutespace no ldquotrue motionrdquo Only relative motion and therefore relative ac-celeration must be physically meaningful Absolute acceleration shouldnot enter physical equations With special relativity Einstein had suc-ceeded in vindicating galilean relativity of velocities from the challenge ofMaxwell theory He was then convinced that he could vindicate the entirearistotelianndashcartesian relativity of motion In Einsteinrsquos terms ldquothe lawsof motion should be the same in all reference frames not just in the iner-tial framesrdquo Things move with respect to one another not with respect toan absolute space there cannot be any physical effect of absolute motion

According to many contemporary physicists this is excessive weightgiven to ldquophilosophicalrdquo thinking which should not play a role in physicsBut Einsteinrsquos achievements in physics are far more effective than the onesobtained by these physicists

223 The key idea

The question addressed in Newtonrsquos bucket experiment is the followingThe rotation of the water has a physical effect ndash the concavity of thewater surface with respect to what does the water ldquorotaterdquo Newtonargues that the relevant rotation is not the rotation with respect to thesurrounding objects (the bucket) therefore it is rotation with respect toabsolute space Einsteinrsquos new answer is simple and fulgurating

The water rotates with respect to a local physical entity the gravitational field

It is the gravitational field not Newtonrsquos inert absolute space that tellsobjects if they are accelerating or not if they are rotating or not Thereis no inert background entity such as newtonian space there are onlydynamical physical entities Among these are the fields Among the fieldsis the gravitational field

The flatness or concavity of the water surface in Newtonrsquos bucket is notdetermined by the motion of the water with respect to absolute spaceIt is determined by the physical interaction between the water and thegravitational field

The two lines of Einsteinrsquos thinking about gravity (finding a field the-ory for the newtonian interaction and getting rid of absolute acceleration)meet here Einsteinrsquos key idea is that Newton has mistaken the gravita-tional field for an absolute space


What leads Einstein to this idea Why should newtonian accelerationbe defined with respect to the gravitational field The answer is givenby the special properties of the gravitational interaction18 These canbe revealed by a thought experiment called Einsteinrsquos elevator I presentbelow a modern and more realistic version of Einsteinrsquos elevator argument

An ldquoelevatorrdquo argument newtonian cosmology Here is a simple physical situation thatillustrates that inertia and gravity are the same thing The model is simple but com-pletely realistic It leads directly to the physical intuition underlying GR

In the context of newtonian physics consider a universe formed by a very largespherical cloud of galaxies Assume that the galaxies are ndash and remain ndash uniformlydistributed in space with a time-dependent density ρ(t) and that they attract eachother gravitationally Let C be the center of the cloud Consider a galaxy A (say ours)at a distance r(t) from the center C As is well known the gravitational force on A dueto the galaxies outside a sphere of radius r around C cancels out and the gravitationalforce due to the galaxies inside this sphere is the same as the force due to the samemass concentrated in C Therefore the gravitational force on A is

F = minusGmA

43πr3(t) ρ(t)

r2(t) (2110)

ord2r

dt2= minusG

4

3π r(t)ρ(t) (2111)

If the density remains spatially constant it scales uniformly as rminus3 That is ρ(t) =ρ0r

minus3(t) where ρ0 is a constant equal to the density at r(t) = 1 Therefore

d2r

dt2= minus4

3πGρ0

1

r2(t)= minus c

r2(t) (2112)

where

c =4πGρO

3(2113)

is a constant Equation (2112) is the Friedmann cosmological equation which governsthe expansion of the universe (It is the same equation that one obtains from full GRin the spatially flat case)

In the newtonian model we are considering the galaxy C is in the center of theuniverse and defines an inertial frame while the galaxy A is not in the center and isnot inertial Assume that the cloud is so large that its boundary cannot be observedfrom C or A If you are in one of these two galaxies how can you tell in which youare That is how can you tell whether you are in the inertial reference frame C or inthe accelerated frame A

The answer is very remarkably that you cannot Since the entire cloud expandsor contracts uniformly the picture of the local sky looks uniformly expanding or con-tracting precisely in the same manner from all galaxies But you cannot detect if youare in the inertial galaxy C or in the accelerated galaxy A by local experiments eitherIndeed to detect if you are in an accelerated frame you have to observe inertial forces

18Gravity is ldquospecialrdquo in the sense that newtonian absolute space is a configuration ofthe gravitational field Once we get rid of the notion of absolute space the gravita-tional interaction is no longer particularly special It is one of the fields forming theworld But it is a very different world from that of Newton and Maxwell


such as the ones that make the water surface of Newtonrsquos bucket concave The A frameacceleration is

a =c

r2(t)u (2114)

where u is a unit vector pointing towards C Therefore there is an inertial force

Finertial = minus c

r2(t)u (2115)

on all moving masses This is the force that should allow us to detect that the frame isnot inertial However all masses feel besides the local forces Flocal also the cosmologicalgravitational pull towards C

Fcosmological =c

r2(t)u (2116)

so that their motion in the accelerated A frame is governed by

ma = Flocal + Finertial + Fcosmological (2117)

= Flocal (2118)

because (2115) and (2116) cancel out exactly Therefore the local dynamics in Alooks precisely as if it were inertial The parabola of a falling stone in A seen from theaccelerated A frame looks as a straight line There is no way of telling if you are thecenter and no way of telling if you are inertial or not

How do we interpret this impossibility of detecting the inertial frameAccording to newtonian physics the dynamics in C or A should be com-pletely different But this difference is not physically observable In thenewtonian conceptual scheme A is noninertial there are gravitationalforces and inertial forces but there is a sort of conspiracy that hides bothof them In fact the situation is completely general in a sufficiently smallregion inertial and gravitational forces cancel to any accuracy in a free-falling reference system19 It is clear that there should be a better wayof understanding this physical situation without resorting to all theseunobservable forces

The better way is to drop the newtonian preferred global frame andto realize each galaxy has its own local inertial reference frame We candefine local inertial frame by the absence of observable inertial effects asin newtonian physics Each galaxy then has its local inertial frame These

19This is the equivalence principle By the way Newton the genius knew it ldquoIf bodiesmoved among themselves are urged in the direction of parallel lines by equal acceler-ative forces they will all continue to move among themselves after the same manneras if they had not been urged by those forcesrdquo (Newton Principia Corollary VI tothe ldquoLaws of Motionrdquo) [62] Newton uses this corollary for computing the complicatedmotion of the Moon in the Solar System In the frame of the Earth inertial forcesand the solar gravity cancel out with good approximation and the Moon follows akeplerian orbit


frames are determined by the gravitational force That is it is gravitythat determines at each point what is inertial Inertial motion is suchwith respect to the local gravitational field not with respect to absolutespace

Gravity determines then the way the frames of different galaxies fallwith respect to one another The gravitational field expresses the rela-tion between the various inertial frames It is the gravitational field thatdetermines inertial motion Newtonrsquos true motion is not motion with re-spect to absolute space it is motion with respect to a frame determinedby the gravitational field It is motion relative to the gravitational fieldEquation (2109) governs the motion of objects with respect to the grav-itational field

The form of the gravitational field Recall that Einsteinrsquos problem wasto describe the gravitational field The discussion above indicates thatthe gravitational field can be viewed as the field that determines at eachpoint of spacetime the preferred frames in which motion is inertial Letus write the mathematics that expresses this intuition

Return to the cloud of galaxies Since we have dropped the idea ofa global inertial reference system let us coordinatize events in the cloudwith arbitrary coordinates x= (xμ) The precise physical meaning of thesecoordinates is discussed in detail in the next section Let xμA be coordinatesof a particular event A say in our galaxy Since these coordinates arearbitrarily chosen motion described in the coordinates xμ is in generalnot inertial in our galaxy For instance particles free from local forcesdo not follow straight lines But we can find a locally inertial referenceframe around A Let us denote the coordinates it defines as XI and takethe event A as the origin so that XI(A) = 0 The coordinates XI can beexpressed as functions

XI = XI(x) (2119)

of the arbitrary coordinates x In the x coordinates the noninertiality ofthe motion in A is gravity Gravity in A is the information of the changeof coordinates that takes us to inertial coordinates This information iscontained in the functions (2119) But only the value of these functions ina small neighborhood around A is relevant because if we move away thelocal inertial frame will change Therefore we can Taylor-expand (2119)and keep only the first nonvanishing term As XI(A) = 0 to first non-vanishing order we have

XI(x) = eIμ(xA) xμ (2120)


where we have defined

eIμ(xA) =partXI(x)partxμ

∣∣∣∣x=x(A)

(2121)

The quantity eIμ(xA) contains all the information we need to know thelocal inertial frame in A The construction can be repeated at each pointx The quantity

eIμ(x) =partXI(x)partxμ

∣∣∣∣x

(2122)

where XI are now inertial coordinates at x is the gravitational field atx This is the form of the field introduced in Section 211

The gravitational field eIμ(x) is therefore the jacobian matrix of thechange of coordinates from the x coordinates to the coordinates XI thatare locally inertial at x The field eIμ(x) is also called the ldquotetradrdquo fieldfrom the Greek word for ldquofourrdquo or the ldquosoldering formrdquo because it ldquosol-dersrdquo a Minkowski vector bundle to the tangent bundle or followingCartan the ldquomoving framerdquo although there is nothing moving about it

Transformation properties If the coordinate system XI defines a localinertial system at a given point so does any other local coordinate sys-tem Y J = ΛJ

IXI where Λ is a Lorentz transformation Therefore the

index I of eIμ(x) transforms as a Lorentz index under a local Lorentztransformation and the two fields eIμ(x) and

eprimeJμ(x) = ΛJI(x)eIμ(x) (2123)

represent the same physical gravitational field Thus this description ofgravity has a local Lorentz gauge invariance

What happens if instead of using the physical coordinates x we hadchosen coordinates y = y(x) The chain rule determines the field eprimeIν(y)that we would have found had we used coordinates y

eprimeIν(y) =partxμ(y)partyν

eIμ(x(y)) (2124)

The transformation properties (2123) and (2124) are precisely the trans-formation properties (258) and (263) under which the GR action is in-variant

These transformation laws are also the ones of a one-form field valued in a vectorbundle P over the spacetime manifold M whose fiber is Minkowski space M associatedwith a principal SO(3 1) Lorentz bundle This is a natural geometric setting for thegravitational field The connection ω defined in Section 211 is a connection of this


bundle This setting realizes the physical picture of a patchwork of Minkowski spacessuggested by the cloud of galaxies carrying Lorentz frames at each galaxy More pre-cisely the gravitational field can be viewed as map e TM rarr P that sends tangentvectors to Lorentz vectors

Matter Finally consider a particle moving in spacetime along a worldlinexμ(τ) If a particle has velocity vμ = dxμdτ at a point x its velocity inlocal Minkowski coordinates XI at x is

uI =partXI(x)partxμ

∣∣∣∣x

vμ = eIμ(x)vμ (2125)

In this local Minkowski frame the infinitesimal action along the trajectoryis

dS = mradicminusηIJuIuJ dτ (2126)

Therefore the action along the trajectory is the one given in (241) Thesame argument applies to all matter fields the action is a sum over space-time of local terms which can be inferred from their Minkowski spaceequivalent

Metric geometry In Section 214 we saw that the gravitational field e defines a metricstructure over spacetime One is often tempted to give excessive significance to thisstructure as if distance was an essential property of reality But there is no a priorikantian notion of distance needed to understand the world We could have developedphysics without ever thinking about distances and still have retained the completepredictive and descriptive power of our theories

What is the physical meaning of the spacetime metric structure What do we meanwhen we say that two points are 3 centimeters apart or two events are 3 seconds apart

The answer is in the dynamics of matter interacting with the gravitational field Letus first consider Minkowski space Consider two objects A and B that are 3 centimetersapart This means that if we put a ruler between the two points the part of the rulerthat fits between the two is marked 3 cm The shape of the ruler is determined bythe Maxwell and Schrodinger equations at the atomic level These equations containthe Minkowski tensor ηIJ They have stable solutions in which the molecules maintainpositions (better vibrate around equilibrium positions) at a fixed ldquodistancerdquo L fromone another L is determined by the constants in these equations This means that themolecules maintain positions at points with coordinate distances ΔxI such that

ηIJΔxIΔxJ = L2 (2127)

We exploit this peculiar behavior of condensed matter for coordinatizing spacetimelocations That is ldquodistancerdquo is nothing but a convenient manner for labeling locationsdetermined by material objects (the ruler) whose dynamics is governed by certainequations We could avoid mentioning distance by saying a number N =3 [cm]L of


molecules obeying the Maxwell and Schrodinger equations with given initial values fitbetween A and B

Consider now the same situation in a gravitational field e Again the fact that twopoints A and B are 3 centimeters apart means that we can fit the N molecules of theruler between A and B But now the dynamics of the molecules is determined by theirinteraction with the gravitational field The Maxwell and Schrodinger equations havestable solutions in which the molecules keep themselves at coordinate distances Δxμ

such thatηIJe

Iμ(x)eJν (x)ΔxμΔxν = L2 (2128)

Thus a measure of distance is a measurement of the local gravitational field performedexploiting the peculiar way matter interacts with gravity

The same is true for temporal intervals Consider two events A and B that happenin time The meaning that 3 seconds have elapsed between A and B is that a second-ticking clock has ticked three times in this time interval The physical system that weuse as a clock interacts with the gravitational field The pace of the clock is determinedby the local value of e Thus a clock is nothing but a device measuring an extensivefunction of the gravitational field along a worldline going from A to B

Imagine that a particle falls along a timelike geodesic from A to B We know fromspecial relativity that the increase of the action of the particle in the particle frame is

dS = mdt (2129)

where m is the particle mass Therefore a clock comoving with the particle will measurethe quantity

T =1

mS =

int B

A

dτradic

minusηIJeIμeJν xμxν (2130)

Thus a clock is a device for measuring a function T of the gravitational field Ingeneral any metric measurement is nothing but a measurement of a nonlocal functionof the gravitational field

This is true in an arbitrary gravitational field e as well as in flat space In flatspace we can use these measurements for determining positions with respect to thegravitational field Since the flat-space gravitational field is Newton absolute spacethese measurements locate points in spacetime


Before getting to the last and main step in Einsteinrsquos discovery of GRwe need the notion of active diffeomorphism I introduce this notion withan example

Consider the surface of the Earth and call it M At each point P isin M on Earthsay the city of Paris there is a certain temperature T (P ) The temperature is a scalarfunction T M rarr R on the Earthrsquos surface Imagine a simplified model of weatherevolution in which the only factor determining temperature change was the displace-ment of air due to wind By this I mean the following Fix a time interval say we callT the temperature on May 1st and T the temperature on May 2nd During this timeinterval the winds move the air which is over a point Q = φ(P ) to the point P Ifsay Q is the French village of Quintin this means that the winds have blown the airof Quintin to Paris Assume the temperature T (P ) of Paris on May 2nd is equal to thetemperature T (Q) of Quintin the day before The ldquowindrdquo map φ is a map from the


Earthrsquos surface to itself which associates with each point P the point Q from whichthe air has been blown by the wind From May 1st to May 2nd the temperature fieldchanges then as follows

T (P ) rarr T (P ) = T (φ(P )) (2131)

Assuming it is smooth and invertible the map φ M rarr M is an active diffeomorphismThe scalar field T on M is transformed by this active diffeomorphism as in (2131)it is ldquodraggedrdquo along the surface of the Earth by the diffeomorphism φ Notice thatcoordinates play no role in all this

Now imagine that we choose certain geographical coordinates x to coordinatize thesurface of the Earth For instance latitude and longitude namely the polar coordinatesx = (θ ϕ) with ϕ = 0 being Greenwich Using these coordinates the temperature isrepresented by a function of the coordinates T (x) The May 1st temperature T (x) andthe May 2nd temperature T (x) are related by

T (x) = T (φ(x)) (2132)

For instance if the wind has blown uniformly westward by 220prime degrees (Quintin is220prime west of Paris) then

T (θ ϕ) = T (θ ϕ + 220prime) (2133)

Of course there is nothing sacred about this choice of coordinates For instance theFrench might resent that the origin of the coordinates is Greenwich and have it passthrough Paris instead Thus the French would describe the same temperature fieldthat the British describe as T (θ ϕ) by means of different polar coordinates defined byϕ = 0 being Paris Since Paris is 220prime degrees East of Greenwich for the French thetemperature field on May 1st is

T prime(θ ϕ) = T (θ ϕ + 220prime) (2134)

This is a change of coordinates or a passive diffeomorphismNow the two equations (2133) and (2134) look precisely the same But it would

be silly to confuse them In (2133) T (θ ϕ) is the temperature on May 2nd while in(2134) T prime(θ ϕ) is the temperature on May 1st but written in French coordinatesIn summary the first equation represents a change in the temperature field due tothe wind the second equation represents a change in convention The first equationdescribes an ldquoactive diffeomorphismrdquo the second a change of coordinates also calleda ldquopassive diffeomorphismrdquo

Given a manifold M an active diffeomorphism φ is a smooth invertiblemap from M to M A scalar field T on M is a map T M rarr R Givenan active diffeomorphism φ we define the new scalar field T transformedby φ as

T (P ) = T (φ(P )) (2135)

Coordinates play no role in thisA coordinate system x on a d-dimensional manifold M is an invertible

differentiable map from (an open set of) M to Rd Given a field T on M this map determines the function t Rd rarr R defined by t(x) = T (P (x))called ldquothe field T in coordinates xrdquo20 A passive diffeomorphism is an

20In the physics literature the two maps T M rarr R and t = T xminus1 Rd rarr R


invertible differentiable map φ Rd rarr Rd that defines a new coordi-nate system xprime on M by x(P ) = φ(xprime(P )) The value of the field T incoordinates xprime is given by

tprime(xprime) = t(φ(xprime)) (2136)

Beware the formal similarity between (2135) and (2136)The above extends immediately to all structures on M For instance

an active diffeomorphism φ carries a one-form field e on M to the newone-form field e = φlowaste the pull-back of e under φ and so on

In particular a metric d M timesM rarr R+ is an assignment of a distanced(AB) between any two points A and B of M An active diffeomorphismdefines the new metric d given by d(AB) equiv d(φminus1(A) φminus1(B)) Thetwo metrics d and d are isometric but distinct21 An equivalence class ofmetrics under active diffeomorphisms is sometimes called a ldquogeometryrdquoGiven a coordinate system we can represent a (Riemannian) metric dby means of a tensor field on Rd Riemannrsquos metric tensor gμν(x) orequivalently the tetrad field eIμ(x) Under a change of coordinate systemthe same metric is represented by a different gμν(x) or eIμ(x)

The example of the Earthrsquos temperature given above illustrates a pecu-liar relation between active and passive diffeomorphisms given two tem-perature fields T and T related by an active diffeomorphism we can al-ways find a coordinate transformation such that in the new coordinatesT is represented by the same function as T in the old coordinates Thissimple mathematical observation is at the root of Einsteinrsquos argumentsthat I will describe below (The argument will be essentially that a the-ory that does not distinguish coordinate systems cannot distinguish fieldsrelated by active diffeomorphisms either)

More precisely the relation between active and passive diffeomorphismsis as follows The group of the active diffeomorphisms acts on the space

are always indicated with the same symbol generating confusion between active andpassive diffeomorphisms In this paragraph I use distinct notations In the rest of thetext however I shall adhere to the standard notation and indicate the field and itscoordinate representation with the same symbol

21Here is an example of isometric but distinct metrics The 2001 Shell road-map saysthat the distances between New York (NY) Chicago (C) and Kansas City (KC) ared(NY C) = 100 miles d(C KC) = 50 miles d(KC NY) = 100 miles while the 2002Lonely Planet tourist guide claims that these distances are d(NY C) = 100 milesd(C KC) = 100 miles d(KC NY) = 50 miles Obviously these are not the samedistances But they are isometric the two are transformed into each other by theactive diffeomorphism φ(NY) = C φ(C) = KC φ(KC) = NY


Space of metrics dSpace of functions g (x)mν

Orbit of the passive

diffeomorphism groupOrbit of the active

diffeomorphism group

Coordinate system S

Coordinate system S prime

Fig 22 Active and passive diffeomorphisms

of metrics d The group of passive diffeomorphisms acts on the spaceof functions gμν(x) The orbits of the first group are in natural one-to-one correspondence with the orbits of the second However the relationbetween the individual metrics d and the individual functions gμν(x) de-pends on the coordinate system chosen The situation is illustrated inFigure 22


Around 1912 using the idea that any motion is relative Einstein hadfound the form of the gravitational field as well as the equations of motionsof matter in a given gravitational field This was already a remarkableachievement but the field equations for the gravitational field were stillmissing In fact the best part of the story had yet to come

Two problems remained open the field equations and understandingthe physical meaning of the coordinates xμ introduced above Einsteinstruggled with these two problems during the years 1912ndash1915 trying sev-eral solutions and changing his mind repeatedly Einstein has called this


search his ldquostruggle with the meaning of the coordinatesrdquo The strugglewas epic The result turned out to be amazing In Einsteinrsquos words it wasldquobeyond my wildest expectationsrdquo

To increase Einsteinrsquos stress Hilbert probably the greatest mathemati-cian at the time was working on the same problem trying to be first tofind the gravitational field equations The fact that Hilbert with his farsuperior mathematical skills could not find these equations first testifiesto the profound differences between fundamental physical problems andmathematical problems

In his search for the field equations Einstein was guided by severalpieces of information First the static limit of the field equations mustyield the Newton law as the static limit of Maxwell theory yields theCoulomb law Second the source of Coulomb law is charge and the chargedensity is the temporal component of four-current Jμ(x) which is thesource of Maxwell equations The source of the Newtonian interaction ismass Einstein had understood with special relativity that mass is in facta form of energy and that the energy density is the temporal compo-nent of the energy-momentum tensor Tμν(x) Therefore Tμν(x) had tobe the likely source of the field equations Third the introduction of thegravitational field was based on the use of arbitrary coordinates there-fore there should be some form of covariance under arbitrary changes ofcoordinates in the field equations Einstein searched for covariant second-order equations as relations between tensorial quantities since these areunaffected by coordinate change He learned from Riemannian geometrythat the only combination of second derivatives of the gravitational fieldthat transforms tensorially is the Riemann tensor Rμ

νρσ(x) This was infact Riemannrsquos major result Einstein knew all this in 1912 To deriveEinsteinrsquos field equations (297) from these ideas is a simple calculationpresented in all GR textbooks and which a good graduate student cantoday repeat easily Still Hilbert couldnrsquot do it and Einstein got stuckfor several years What was the problem

The problem was ldquothe meaning of the coordinatesrdquo Here is the story

1 Einstein for general covariance At first Einstein demands the fieldequations for the gravitational field eIμ(x) to be generally covariant onM This means that if eIμ(x) is a solution then eprimeIν(y) defined in (2124)should also be a solution For Einstein this requirement (unheard of atthe time) was the formalization of the idea that the laws of nature must bethe same in all reference frames and therefore in all coordinate systems

2 Einstein against general covariance In 1914 however Einstein con-vinces himself that the field equations should not be generally covariant


t = 0

M

AB

M

AB

f

e e~

(a) (b)

Fig 23 The active diffeomorphism φ drags the nonflat (wavy) gravitationalfield from the point B to the point A

[63] Why Because Einstein rapidly understands the physical conse-quences of general covariance and he initially panics in front of themThe story is very instructive because it reveals the true magic hiddeninside GR Einsteinrsquos argument against general covariance is the follow-ing22

Consider a region of spacetime containing two spacetime points A andB Let e be a gravitational field in this region Say that around the pointA the field is flat while at the point B it is not (see Figure 23(a)) Nextconsider a map φ from M to M that maps the point A to the point BConsider the new field e = φlowaste which is pulled back by this map Thevalue of the field e at A is determined by the value of e at B and thereforethe field e will not be flat around A (see Figure 23(b))

Now if e is a solution of the equations of motion and if the equationsof motion are generally covariant then e is also a solution of the equationsof motion This is because of the relation between active diffeomorphismsand changes of coordinates we can always find two different coordinatesystems on M say x and y such that the function eIμ(x) that represents ein the coordinate system x is the same function as the function eIμ(y) thatrepresents e in the coordinate systems y Since the equations of motion

22At first Einstein got discouraged about generally covariant field equations becauseof a mistake he was making while deriving the static limit the calculation yieldedthe wrong limit But this is of little importance here given the powerful use thatEinstein has been routinely capable of making of general conceptual arguments


are the same in the two coordinate systems the fact that this functionsatisfies the Einstein equations implies that e as well as e are physicalsolutions

Let me repeat the argument in a different form We have found in theprevious section that if eIμ(x) is a solution of the Einstein equations thenso is eprimeIν(y) defined in (2124) But the function eprimeIν can be interpreted intwo distinct manners First as the same field as e expressed in a differentcoordinate system Second as a different field e expressed in the samecoordinate system That is we can define the new field as

eIμ(x) = eprimeIμ(x) (2137)

This new field e is genuinely different from e In general it will not beflat around A In particular the scalar curvature R of e at A is

R|A = R(xA) = R(φ(xA)) = R|B (2138)

In other words if the equations of motion are generally covariant they arealso invariant under active diffeomorphisms

Given this Einstein makes the following famous observation

The ldquoholerdquo argument Assume the gravitational field-equations aregenerally covariant Consider a solution of these equations in whichthe gravitational field is e and there is a region H of the uni-verse without matter (the ldquoholerdquo represented as the white regionin Figure 23) Assume that inside H there is a point A where eis flat and a point B where it is not flat Consider a smooth mapφ M rarr M which reduces to the identity outside H and such thatφ(A) = B and let e = φlowaste be the pull-back of e under φ The twofields e and e have the same past are both solutions of the fieldequations but have different properties at the point A Thereforethe field equations do not determine the physics at the spacetimepoint A Therefore they are not deterministic But we know that(classical) gravitational physics is deterministic Therefore either

(i) the field equations must not be generally covariant or(ii) there is no meaning in talking about the physical spacetime

point A

On the basis of this argument Einstein searched for nongenerally covari-ant field equations for three years in a frantic race against Hilbert

3 Einsteinrsquos return to general covariance Then rather suddenly in 1915Einstein published generally covariant field equations What had hap-pened Why had Einstein changed his mind Is there a mistake in the


t = 0

M

AB

M

AB

f

e x x~

e xa xb~~

a b

Fig 24 The diffeomorphism moves the nonflat region as well as the intersectionpoint of the two particles a and b from the point B to the point A

hole argument No the hole argument is correct The correct physicalconclusion however is (ii) not (i) This point hit Einstein like a flashof lightning the precise conceptual discovery to which all his previousthoughts had led

Einsteinrsquos way out from the difficulty raised by the hole argument isto realize that there is no meaning in referring to ldquothe point Ardquo or ldquotheevent Ardquo without further specifications

Let us follow Einsteinrsquos explanation in detail

Spacetime coincidences Consider again the solution e of the field equa-tions but assume that in the universe there are also the two particles aand b Say that the worldlines (xa(τa) xb(τb)) of the two particles intersectat the spacetime point B see Figure 24

Now for given initial conditions the worldlines of the particles are de-termined by the gravitational field They are geodesics of e or if otherforces are involved they satisfy the geodesic equation with an addi-tional force term Consider the field e = φlowaste The particlesrsquo worldlines(xa(τa) xb(τb)) are no longer solutions of the particlesrsquo equations of mo-tion in this gravitational field If the gravitational field is e instead of ethe particlesrsquo motions over M will be different But it is easy to find themotion of the particles determined by e precisely because the completeset of equations of motion is generally covariant Therefore an active dif-feomorphism on the gravitational field and the particles sends solutionsinto solutions Thus the motion of the particles in the field e is given by


the worldlines

xa(τa) = φminus1(xa(τa)) xb(τb) = φminus1(xb(τb)) (2139)

Then the particles a and b no longer intersect in B They intersect inA = φminus1(B)

Now instead of asking whether or not the field is flat at A let us askwhether or not the field is flat at the point where the particles meetClearly the result is the same for the two cases (e xa xb) and (e xa xb)Formally assuming the intersection point is at τa = τb = 0

R|inters = R(xa(0)) = R(φ(xa(0)))= R(φ(φminus1(xa(0)))) = R(xa(0)) = R|inters (2140)

This prediction is deterministic There are not two contradictory predic-tions therefore there is determinism so long as we restrict ourselves tothis kind of prediction Einstein calls ldquospacetime coincidencesrdquo this wayof determining points

Einstein observes that this conclusion is general the theory does notpredict what happens at spacetime points (like newtonian and special-relativistic theories do) Rather it predicts what happens at locations de-termined by the dynamical elements of the theory themselves In Einsteinrsquoswords

All our space-time verifications invariably amount to a determinationof space-time coincidences If for example events consisted merely inthe motion of material points then ultimately nothing would be ob-servable but the meeting of two or more of these points Moreover theresults of our measuring are nothing but verifications of such meetingsof the material points of our measuring instruments with other mate-rial points coincidences between the hands of a clock and points onthe clock dial and observed point-events happening at the same placeat the same time The introduction of a system of reference serves noother purpose than to facilitate the description of the totality of suchcoincidences [64]

The two solutions (e xa xb) and (e xa xb) are only distinguished by theirlocalization on the manifold They are different in the sense that they as-cribe different properties to manifold points However if we demand thatlocalization is defined only with respect to the fields and particles them-selves then there is nothing that distinguishes the two solutions physi-cally In fact concludes Einstein the two solutions represent the samephysical situation The theory is gauge invariant in the sense of Diracunder active diffeomorphisms there is a redundancy in the mathematicalformalism the same physical world can be described by different solutionsof the equations of motion


It follows that localization on the manifold has no physical meaningThe physical picture is completely different from the example of the tem-perature field on the Earthrsquos surface illustrated in the previous section Inthat example the cities of Paris and Quintin were real distinguishable en-tities independent from the temperature field In GR general covarianceis compatible with determinism only assuming that individual spacetimepoints have no physical meaning by themselves It is like having only thetemperature field without the underlying Earth

What disappears in this step is precisely the background spacetime thatNewton believed to have been able to detect with great effort beyond theapparent relative motions

Einsteinrsquos step toward a profoundly novel understanding of nature isachieved Background space and spacetime are effaced from this new un-derstanding of the world Motion is entirely relative Active diffeomor-phism invariance is the key to implement this complete relativizationReality is not made up of particles and fields on a spacetime it is madeup of particles and fields (including the gravitational field) that can onlybe localized with respect to one another No more fields on spacetimejust fields on fields Relativity has become general

23 Interpretation

General covariance makes the relation between formalism and experimentfar more indirect than in conventional field theories

Take Maxwell theory as an example We assume that there is a back-ground spacetime We have special objects at our disposal (the walls ofthe lab the Earth) that define an inertial frame to a desired approxima-tion These objects allow us to designate locations relative to backgroundspacetime We have two kinds of measuring devices (a) meters and clocksthat measure distance and time intervals from these reference objects and(b) devices that measure the electric and magnetic fields The reading ofthe devices (a) gives us xμ The reading of the devices (b) gives us Fμν We measure the two and say that the field has the value Fμν at the pointxμ The theory can predict the value Fμν at the point xμ

We cannot do the same in GR The theory does not predict the value ofthe field at the point xμ So how do we compare theory and observations


As discussed at the end of the previous section a physical state does notcorrespond to a solution e(x) of Einsteinrsquos equations but to an equivalenceclass of solutions under active diffeomorphisms Therefore the quantitiesthat the theory predicts are all and only the quantities that are well


defined on these equivalence classes That is only the quantities that areinvariant under diffeomorphisms These quantities are independent fromthe coordinates xμ

In concrete applications of the theory these quantities are generallyobtained by solving away the coordinates x from solutions to the equationsof motion Here are a few examples

Solar System Consider the dynamics of the Solar System The vari-ables are the gravitational field e(x) and the worldlines of the plan-ets xn(τn) Fix a solution (e(x) xn(τn)) to the equations of motionWe want to derive physical predictions from this solution and com-pare them with observations Choose for simplicity τn = x0 so thatthe solution is expressed by (e(x) xn(x0)) Consider the worldlineof the Earth Compute the distance dn(x0) between the Earth andthe planet n defined as the proper time elapsed along the Earthrsquosworldline while a null geodesic (a light pulse) leaving the Earth atx0 travels from Earth to the planet and back

The functions (dn(x0)) can be computed from the given solutionsto the equations of motion Consider a space C with coordinates(dn) The functions (dn(x0)) define a curve γ on this space

We can associate a measuring device with each dn a laser ap-paratus that measures the distance to planet n These quantitiescan be measured together We obtain the event (dn) which can berepresented by a point in C The theory predicts that this point willfall on the curve γ A sequence of these events can be comparedwith the curve γ and in this way we can test the given solutionsto the equations of motion against experience (In the terminologyof Chapter 3 the quantities dn are partial observables) Notice thatthis can be done with arbitrary precision and that distant starsinertial systems preferred coordinates or choice of time variableplay no role

Clocks Consider the gravitational field around the Earth Consider twoworldlines Let the first be the worldline of an object fixed on theEarthrsquos surface Let the second be the worldline of an object in freefall on a keplerian orbit around the Earth that is a satellite Fixan arbitrary initial point P on the worldline of the orbiting objectand let T1 be the proper time from P along this worldline Send alight signal from P to the object on Earth let Q be the point onthe Earthrsquos worldline when the signal is received and let T2 be theproper time from Q along this worldline Then let T2(T1) be thereception proper time on Earth of a signal sent at T1 proper timein orbit GR allows us to compute the function T2(T1) for any T1


It is easy to associate measuring devices to T1 and T2 these area clock on Earth and a clock in orbit If the orbiting object sendsa signal at fixed proper times T1 the reception times T2 can becompared with the predictions of the theory Here T1 and T2 arethe partial observables I let you decide which one of the two is theldquotrue time variablerdquo

Solar System with a clock We can add a clock to the Solar Systemmeasurements described above Fixing arbitrarily an initial eventon Earth (a particular eclipse the birth of Jesus or the death ofJohn Lennon) we can compute the proper time T (x0) lapsed fromthis event along the Earthrsquos worldline The partial observable Tcan be added to the partial observables dn giving the set (dn T ) ofpartial observables If we do so it may be convenient to express thecorrelations (dn T ) as functions dn(T ) A complete gauge-invariantobservable fully predicted by the theory is the value dn(T ) of aplanet distance at a certain given Earth proper time T from theinitial event Notice that T is not a coordinate It is a complicatednonlocal function of the gravitational field to which a measuringdevice (measuring a partial observable) has been attached The useof a clock on Earth to determine a local temporal localization isjust a matter of convenience

Binary pulsar Consider a binary-star system in which one of the twostars is a pulsar Because of a Doppler effect the frequency of thepulsing signal oscillates with the orbital period of the system Thisfact allows us to count the number of pulses in each orbit Let Nn

be the number of pulses we receive in the nth orbit A theoreticalmodel of the pulsar allows us to compute the expected decrease inorbital period due to gravitational wave emission and therefore theexpected sequence Nn which can be compared with the observedone Doing this with sufficient care won JH Taylor and RA Hulsethe 1993 Nobel Prize

Notice that in all these examples the coordinates xμ have disappearedfrom the observable quantities This is true in general A theoretical modelof a physical system is made using coordinates xμ but then observablequantities are independent of the coordinates xμ23


In the mathematical formalism of GR we utilize the ldquospacetimerdquo man-ifold M coordinatized by x However a state of the universe does not

23Unless we gauge-fix them to given partial observables see Section 246


correspond to a configuration of fields on M It corresponds to an equiva-lence class of field configurations under active diffeomorphisms An activediffeomorphism changes the localization of the field on M by dragging itaround Therefore localization on M is just gauge it is physically irrele-vant

In fact M itself has no physical interpretation it is just a mathematicaldevice a gauge artifact Pre-general-relativistic coordinates xμ designatepoints of the physical spacetime manifold ldquowhererdquo things happen (see adetailed discussion below in Section 245) in GR there is nothing of thesort The manifold M cannot be interpreted as a set of physical ldquoeventsrdquoor physical spacetime points ldquowhererdquo the fields take value It is meaning-less to ask whether or not the gravitational field is flat around the point Aof M because there is no physical entity ldquospacetime point Ardquo Contraryto Newton and to Minkowski there are no spacetime points where parti-cles and fields live There are no spacetime points at all The Newtoniannotions of space and time have disappeared

In Einsteinrsquos words

the requirement of general covariance takes away from space andtime the last remnant of physical objectivity [64]

Einstein justifies this conclusion in the immediate continuation of thistext which is the paragraph I quoted at the end of the previous sectionwith the observation that all observations are spacetime coincidences

In newtonian physics if we take away the dynamical entities whatremains is space and time In general-relativistic physics if we take awaythe dynamical entities nothing remains The space and time of Newtonand Minkowski are reinterpreted as a configuration of one of the fieldsthe gravitational field

Concretely this radically novel understanding of spatial and temporalrelations is implemented in the theory by the invariance of the field equa-tions under diffeomorphisms Because of background independence ndash thatis since there are no nondynamical objects that break this invariance inthe theory ndash diffeomorphism invariance is formally equivalent to generalcovariance namely the invariance of the field equations under arbitrarychanges of the spacetime coordinates x and t

Diffeomorphism invariance implies that the spacetime coordinates xand t used in GR have a different physical meaning to the coordinatesx and t used in prerelativistic physics In prerelativistic physics x andt denote localization with respect to appropriately chosen reference ob-jects These reference objects are chosen in such a way that they make thephysical influence of background spacetime manifest In particular theirmotion can be chosen to be inertial In GR on the other hand the space-time coordinates x and t have no physical meaning physical predictionsof GR are independent of the coordinates x and t

24 Complements 75

A physical theory should not describe the location in space and theevolution in time of dynamical objects It describes relative location andrelative evolution of dynamical objects Newton introduced the notion ofbackground spacetime because he needed the acceleration of a particle tobe well defined (so that F = ma could make sense) In the newtoniantheory and in special relativity a particle accelerates when it does sowith respect to a fixed spacetime in which the particle moves In generalrelativity a particle (a dynamical object) accelerates when it does so withrespect to the local values of the gravitational field (another dynamicalobject) There is no meaning for the location of the gravitational field orthe location of the particle only the relative location of the particle withrespect to the gravitational field has physical meaning

What remains of the prerelativistic notion of spacetime is a relationbetween dynamical entities we can say that two particlesrsquo worldlines ldquoin-tersectrdquo that a field has a certain value ldquowhererdquo another field has a certainvalue or that we measure two partial observables ldquotogetherrdquo This is pre-cisely the modern realization of Descartesrsquo notion of contiguity and it isthe basis of spatial and temporal notions in GR

As Whitehead put it we cannot have spacetime without dynamicalentities anymore than saying that we can have the catrsquos grin withoutthe cat The world is made up of fields Physically these do not liveon spacetime They live so to say on one another No more fields onspacetime just fields on fields It is as outlined in the metaphor in Section113 where we no longer had animals on the island just animals on thewhale animals on animal Our feet are no longer in space we have to ridethe whale

24 Complements

I close this chapter by discussing a certain number of issues related to the interpretationof GR

241 Mach principles

The ideas of Ernst Mach had a strong influence on Einsteinrsquos discovery of GR Machpresented a number of acute criticisms to Newtonrsquos motivations for introducing absolutespace and absolute time In particular he pointed out that in Newtonrsquos bucket argumentthere is a missing element he observed that the inertial reference frame (the referenceframe with respect to which rotation has detectable physical effects) is also the referenceframe in which the fixed stars do not rotate Mach then suggested that the inertialreference frame is not determined by absolute space but rather it is determined by theentire matter content of the Universe including distant stars He suggested that if wecould repeat the experiment with a very massive bucket the mass of the bucket wouldaffect the inertial frame and the inertial frame would rotate with the bucket

In the light of GR the observation is certainly pertinent and it is clear that the ar-gument may have played a role in Einsteinrsquos dismissal of Newtonrsquos argument However


for some reason the precise relation between Machrsquos suggestion and GR has generateda vast debate Machrsquos suggestion that inertia is determined by surrounding matter hasbeen called ldquothe Mach principlerdquo and much ink has been employed to discuss whetheror not GR implements this principle whether or not ldquoGR is machianrdquo Remarkably inthe literature one finds arguments and proofs in favor as well as against the conclusionthat GR is machian Why this confusion

Because there is no well-defined ldquoMach principlerdquo Mach provided a very importantbut vague suggestion that Einstein developed into a theory not a precise statement thatcan be true or false Every author that has discussed ldquothe Mach principlerdquo has actuallyconsidered a different principle Some of these ldquoMach principlesrdquo are implemented inGR others are not

In spite of the confusion or perhaps thanks to it the discussion on how machianGR is sheds some light on the physical content of GR Here I list several versions ofthe Mach principle that have been considered in the literature and for each of theseI comment on whether this particular Mach principle is True or False in GR In thefollowing ldquomatterrdquo means any dynamical entity except the gravitational field

bull Mach principle 1 Distant stars can affect the local inertial frameTrue Because matter affects the gravitational field

bull Mach principle 2 The local inertial frame is completely determined by thematter content of the UniverseFalse The gravitational field has independent degrees of freedom

bull Mach principle 3 The rotation of the inertial reference frame inside the bucketis in fact dragged by the bucket and this effect increases with the mass of thebucketTrue In fact this is the LensendashThirring effect a rotating mass drags the inertialframes in its vicinity

bull Mach principle 4 In the limit in which the mass of the bucket is large theinternal inertial reference frame rotates with the bucketDepends It depends on the details of the way the limit is taken

bull Mach principle 5 There can be no global rotation of the UniverseFalse Einstein believed this to be true in GR but Godelrsquos solution is a counter-example

bull Mach principle 6 In the absence of matter there would be no inertiaFalse There are vacuum solutions of the Einstein field equations

bull Mach principle 7 There is no absolute motion only motion relative to some-thing else therefore the water in the bucket does not rotate in absolute terms itrotates with respect to some dynamical physical entityTrue This is the basic physical idea of GR

bull Mach principle 8 The local inertial frame is completely determined by thedynamical fields in the UniverseTrue In fact this is precisely Einsteinrsquos key idea


In contemporary philosophy of science there is an interesting debate on the inter-pretation of GR The two traditional theses about space ndash absolute and relational ndashsuitably edited to take into account scientific progress continue under the names

24 Complements 77

of substantivalism and relationalism Here I present a few considerations on theissue

GR changes the notion of spacetime in physics in the sense of relationalism In pre-relativistic physics spacetime is a fixed nondynamical entity in which physics happensIt is a sort of structured container which is the home of the world In relativistic physicsthere is nothing of the sort There are only interacting fields and particles The onlynotion of localization which is present in the theory is relative dynamical objects canbe localized only with respect to one another This is the notion of space defendedby Aristotle and Descartes against which Newton wrote the initial part of PrincipiaNewton had two points the physical reality of inertial effects such as the concavityof the water in the bucket and the immense empirical success of his theory based onabsolute space Einstein provided an alternative interpretation for the cause of the con-cavity ndash interaction with the local gravitational field ndash and a theory based on relationalspace that has better empirical success than Newton theory After three centuriesthe European culture has returned to a fully relational understanding of space andtime

At the basis of cartesian relationalism is the notion of ldquocontiguityrdquo Two objects arecontiguous if they are close to one another Space is the order of things with respectto the contiguity relation At the basis of the spacetime structure of GR is essentiallythe same notion Einsteinrsquos ldquospacetime coincidencesrdquo are analogous to Descartes ldquocon-tiguityrdquo

A substantivalist position can nevertheless still be defended to some extent Ein-steinrsquos discovery is that newtonian spacetime and the gravitational field are the sameentity This can be expressed in two equivalent ways One states that there is no space-time there is only the gravitational field This is the choice I have made in this bookThe second states that there is no gravitational field it is spacetime that has dynamicalproperties This choice is common in the literature I prefer the first because I find thatthe differences between the gravitational field and other fields are more accidental thanessential But the choice between the two points of view is only a matter of choice ofwords and thus ultimately personal taste If one prefers to keep the name ldquospacetimerdquofor the gravitational field then one can still hold a substantivalist position and claimthat according to GR spacetime is an entity not a relation Furthermore localizationcan be defined with respect to the gravitational field and therefore the substantivalistcan say that spacetime is an entity that defines localization For an articulation of thisthesis see for instance [65]

However this is a very weakened substantivalist position One is free to call ldquospace-timerdquo anything with respect to which we define position But to what extent is space-time different from any arbitrary continuum of objects used to define position New-tonrsquos acute formulation of his substantivalism already mentioned in footnote 15 abovecontains a precise characterization of ldquospacerdquo

so it is necessary that the definition of places and hence of localmotion be referred to some motionless thing such as extension aloneor ldquospacerdquo in so far as space is seen to be truly distinct from movingbodies24

The characterizing feature of space is that of being truly distinct from moving bodiesthat is in modern terms and after the FaradayndashMaxwell conceptual revolution that of

24I Newton De Gravitatione et aequipondio fluidorum [61]


being truly distinct from dynamical entities such as particles or fields This is clearlynot the case for the spacetime of GR If the modern substantivalist is happy to give upNewtonrsquos strong substantivalism and identify the thesis that ldquospacetime is an entityrdquowith the thesis that ldquospacetime is the gravitational field which is a dynamical entityrdquothen the distinction between substantivalism and relationalism is completely reducedto one of semantics

When two opposite positions in a long-standing debate have come so close that theirdistinction is reduced to semantics one can probably say that the issue is solved I thinkone can say that in this sense GR has solved the long-standing issue of the relationalversus substantivalist interpretations of space

243 Has general covariance any physical content Kretschmannrsquosobjection

Virtually any field theory can be reformulated in a generally covariant form An exampleof a generally covariant reformulation of a scalar field theory on Minkowski spacetime ispresented below This fact has led some people to wonder whether general covariancehas any physical significance at all The argument is as follows if any theory canbe formulated in a general covariant language then general covariance is not a principlethat selects a particular class of theories therefore it has no physical content Thisargument was presented by Kretschmann shortly after Einsteinrsquos publication of GRIt is heard among some philosophers of science and sometimes used also by somephysicists that dismiss the conceptual novelty of GR

I think that the argument is wrong The non sequitur is the idea that a formal prop-erty that does not restrict the class of admissible theories has no physical significanceWhy should that be Formalism is flexible and we can artificially give a theory a cer-tain formal property especially if we accept byzantine formulations But it does notfollow from this that the use of one formalism or another is irrelevant Physics is thesearch for the more effective formalism to read Nature The relevant question is notwhether general covariance restricts the class of admissible theories but whether GRcould have been conceived or understood at all without general covariance Let meillustrate this point with the example of rotational invariance

Kretschmannrsquos objection applied to rotational symmetry Ancient physics assumed thatspace has a preferred direction The ldquouprdquo and the ldquodownrdquo were considered absolutelydefined This changes with newtonian physics where space has rotational symmetryall spatial directions are a priori equivalent and only contingent circumstances ndash suchas the presence of a nearby mass like the Earth ndash can make one direction particularPhysicists often say that rotational invariance limits the admissible forces But strictlyspeaking this is not true Kretschmannrsquos objection applies equally well to rotationalinvariance given a theory which is not rotationally invariant we can reformulate it asa rotationally invariant theory just by adding some variable For instance consider aphysical theory T in which all bodies are subject to a force in the z-direction F = minusgwhere g is a constant (such as gravity) This is a nonrotationally-invariant theory Nowconsider another theory T prime in which there is a dynamical vector quantity v of lengthunity and a force F = gv The theory T prime is rotationally invariant but in each solutionthe vector v will take a particular value in a particular direction Calling z this directionwe have precisely the same phenomenology as theory T

24 Complements 79

The example shows that we can express a nonrotationally invariant theory T ina rotationally invariant formalism T prime Therefore rotational invariance does not trulyrestrict the class of admissible theories Shall we conclude with Kretschmann thatrotational invariance has no physical significance

Obviously not Modern physics has made real progress with respect to ancientphysics in understanding that space is rotationally invariant Where is the progress Itis in the fact that the discovery of the rotational invariance of space puts us in a farmore effective position for understanding Nature We can say that we have discoveredthat in general there is no preferred ldquouprdquo and ldquodownrdquo in the Universe Equivalentlywe can say that a rotationally invariant physical formalism is far more effective forunderstanding Nature than a nonrotationally invariant one

There are two key issues here First it would have been difficult to find newtoniantheory within a conceptual framework in which the ldquouprdquo and the ldquodownrdquo are consideredabsolute Second reformulating the theory T in the rotationally invariant form T prime

modifies our understanding of it we have to introduce the dynamical vector v Fromthe point of view of the two theories T and T prime the vector v is a byzantine constructionwithout much sense But notice that from the point of view of understanding Naturethe introduction of v points to the physically correct direction we are led to investigatethe nature and the dynamics of this vector v is indeed the local gravitational fieldand this is precisely the right track towards a more effective understanding of NatureThis is the strength of having understood rotational invariance

In fact if there is rotational invariance in the Universe there should be a rota-tionally invariant manner of understanding ancient physics which in its limited ex-tent was effective Theory T prime above represents precisely this better understanding ofancient physics More than that the reinterpretation itself indicates a new effectiveway of understanding the world In conclusion the fact that the effective but non-rotationally invariant theory T admits the byzantine rotationally invariant formula-tion T prime is not an argument for the physical irrelevance of rotational invariance Farfrom that it is something that is required for us to have confidence in rotationalinvariance

On the one hand rotational invariance is interesting because it enlarges not be-cause it restricts the kind of physics we can naturally describe On the other handrotational invariance does drastically reduce the kind of theories that we are willingto consider Not because it forbids us to write certain theories ndash such as theory T prime

ndash but because if we want to describe a theory such as T we have to pay a priceHere the introduction of the vector v It is up to the theoretician to judge whetherthis price is worth paying that is whether v is in fact a physical entity worthwhileconsidering

The value of a novel idea or a novel language in theoretical physics is not in the factthat old physics cannot be expressed in the new language It is simply in the fact thatit is more effective for describing reality A physical theoretical framework is a map ofreality If the symbols of the map are better chosen the map is more effective A newlanguage by itself rarely truly restricts the kind of theories that can be expressed Butit renders certain theories far simpler and others awkward It orients our investigationon Nature This and nothing else is scientific knowledge

Let me come back to general covariance Like rotational invariance general covari-ance is a novel language which expresses a general physical idea about the worldIt is possible to express Newtonian physics in a generally covariant language It isalso possible to express GR physics in a nongenerally covariant language (by gauge-fixing the coordinates) But newtonian physics expressed in a covariant language or GR


expressed in a noncovariant language are both monsters formulated in a form far moreintricate than what is possible Nobody would have found them

What Einstein discovered is that two classes of entities previously considered dis-tinct are in fact entities of the same kind Newton taught us that (an effective wayto understand the world is to think that) the world is made up of two clearly distinctclasses of entities of very different nature The first class is formed by space and timeThe second class includes all dynamical entities moving in space and in time In new-tonian physics these two classes of entities are different in many respects and enterthe formalism of physical models in very different manners Einstein has understoodthat (a more effective way to understand the world is to think that) the world is notmade up of two distinct kinds of entities There is only one type of entity dynamicalfields General covariance is the language for describing a world without distinctionbetween the spacetime entities and the dynamical entities It is the language that doesnot assume this distinction

We can reinterpret prerelativistic physics in a generally covariant language It sufficesto rewrite the newtonian absolute space and absolute time as a dynamical field andthen write generally covariant equations that fix them to their flat-space values But ifwe do so we are not denying the physical content of Einsteinrsquos idea On the contrary weare simply reinterpreting the world in Einsteinrsquos terms In other words we are showingthe strength not the weakness of general covariance Furthermore in so doing weintroduce a new physical field and we find ourselves in the funny situation of havingto write equations of motion for this field that constrain it to a single value Thus wehave a theory where one of the dynamical fields is strangely constrained to a singlevalue This immediately suggests that perhaps we can relax these equations and allowa full dynamics for this field If we do so we are directly on the track of GR Again farfrom showing the physical irrelevance of general covariance this indicates its enormouscognitive strength

I think that the mistake behind Kretschmannrsquos argument is an excessively legalisticreading of the scientific enterprise It is the mistake of taking certain common physi-cistsrsquo statements too literally Physicists often write that a certain symmetry or a certainprinciple ldquouniquely determinesrdquo a certain theory At a close reading these statementsare almost always much exaggerated The uniqueness only holds under a vast number ofother assumptions that are left implicit and which are facts or ideas the physicist con-siders natural and does not bother detailing The typical physicist carelessly dismissescounter-examples by saying that they would be unphysical implausible or completelyartificial The connection between general physical ideas general principles intuitionssymmetries is a burning melt of powerful ideas not the icy demonstration of a math-ematical theorem What is at stake is finding the most effective language for thinkingthe world not writing axioms It is language in formation not bureaucracy25

25Historically the entire issue might be the result of a misunderstanding Kretschmannattacked Einstein in a virulent form In particular he attacked Einsteinrsquos coincidencessolution of the hole argument Now Einstein probably learned the idea that coinci-dences are the only observables precisely from Kretschmann but didnrsquot give muchcredit to Kretschmann for this I suppose this should have made Kretschmann quitebitter I think that Kretschmannrsquos subtext in saying that general covariance is emptywas not that general covariance was no progress with respect to old physics it wasthat general covariance was no progress with respect to what he himself had alreadyrealized before Einstein

24 Complements 81

Generally covariant flat-space field theory Consider the field theory of a free masslessscalar field φ(x) on Minkowski space The theory is defined by the action

S[φ] =

intd4x ηαβpartαφ partβφ (2141)

The equation of motion is the flat-space KleinndashGordon equation

ηαβpartαpartβφ = 0 (2142)

and the theory is obviously not generally covariantA trivial way to reformulate this theory in generally covariant language is to intro-

duce the tetrad field eαμ(x) and write the equations

partμ(e ηαβeμαeνβpartνφ) = 0 (2143)

Rαβμν = 0 (2144)

The solution of (2144) is that e is flat Since the system is covariant we can choose agauge in which eαμ(x) = δαμ In this gauge (2143) becomes (2141)

A more interesting way is as follows Consider a field theory for five scalar fieldsΦA(x) where A = 1 5 Use the notation

VA = εABCDE partμΦBpartνΦCpartρΦDpartσΦEεμνρσ (2145)

where εμνρσ and εABCDE are the 4-dimensional and 5-dimensional completely antisym-metric pseudo-tensors Consider the theory defined by the action

S[ΦA] =

intd4x V minus1

5 (V4V4 minus V3V3 minus V2V2 minus V1V1) (2146)

where V5 is assumed never to vanish The theory is invariant under diffeomorphismsIndeed VA transforms as a scalar density (because εμνρσ is a scalar density) hence theintegrand is a scalar density and the integral is invariant For α = 1 2 3 4 define thematrix

Eαμ (x) = partμΦα(x) (2147)

its inverse Eμα and its determinant E Varying Φ5 we obtain the equation of motion

partμ(E ηαβEμαE

νβpartνΦ5) = 0 (2148)

This is the massless KleinndashGordon equation (2143) interacting with a gravitationalfield Eα

μ Varying Φα we do not obtain independent equations We obtain the energy-momentum conservation law implied by (2148) The fact that there is only one inde-pendent equation is a consequence of the fact that there is a four-fold gauge invarianceWe can choose a gauge in which

Φa(x) = xa (2149)

We then have immediately Eaμ = δaμ and (2148) becomes (2142) The other four

equations are

parta(partaΦ5partbΦ

5 minus 1

2δab partcΦ

5partcΦ5) = 0 (2150)

Even better we may not fix the gauge and consider the gauge-invariant function offour variables φ(Xa) defined by

φ(Φa(x)) = Φ5(x) (2151)

This function satisfies the Minkowski-space KleinndashGordon equation (2142)How to interpret such a theory The theory (2141) is not generally covariant there-

fore its coordinates x are (partial) observables The theory is defined by five partial


observables four xμ and φ To interpret the theory we must have measuring proceduresassociated with these five quantities The relation between these observables is governedby (2141) On the other hand the theory (2146) is generally covariant therefore thecoordinates x are not observable The theory is defined by five partial observablesthe five φA We must have measuring procedures associated with these five quantitiesThe relation between these observables is governed again by (2141) Therefore in thetwo cases we have the same partial observables identified by ΦA harr (xa φ) relatedby the same equation

There is only one subtle but important difference between theory (2146) and theory(2141) Theory (2141) separates the five partial observables (x φ) into two sets theindependent ones (x) and the dependent one (φ) Theory (2146) treats the five partialobservables ΦA on an equal footing Thus in a strict sense theory (2141) containsone extra item of information a distinction between dependent and independent par-tial observables Because of this difference the two theories reflect two quite differentinterpretations of the world The first describes a worldrsquos ontology split into spacetimeand matter The second describes a world where the spacetime structure is interpretedas relational


The concept of time used in natural language carries many properties Within a giventheoretical framework (say newtonian mechanics) time maintains some of these prop-erties and loses others In different theoretical frameworks time has different proper-ties The best-known example is probably the directionality of time absent in me-chanics present in thermodynamics But many other features of time lack in onetheory and are present in others For instance a property of time in newtonian me-chanics is uniqueness there is a unique time interval between any two events Con-versely in special relativity there are as many time variables as there are Lorentzobservers (x0 xprime0 ) Another attribute of time in newtonian mechanics is globalityevery solution of the equations of motion ldquopassesrdquo through every value of newtoniantime t once and only once In some cosmological models on the other hand thereis no choice of time variable with such a property there is ldquono timerdquo if we demandthat being global is an essential property of time In other words we use the wordldquotimerdquo to denote quite different concepts that may or may not include this or thatproperty

Here I describe a simple classification of possible attributes of time Below I identifyand list nine properties of time Then I describe and tabulate ten separate levels ofincreasing complexity of the notion of time corresponding to an increasing number ofproperties Theories typically fall in one of these levels according to the set of attributesthat the theory ascribes to the notion of time it uses The ten-fold arrangement isconventional the main point I intend to emphasize is that a single clear and purenotion of ldquotimerdquo does not exist

Properties of time Consider an infinite set S without any structure Add to S a topol-ogy and a differential structure dx Thus S becomes a manifold assume that thismanifold is one-dimensional and denote the set S together with its differentiable struc-ture as the line L = (Sdx) Next assume we add a metric structure d to L denotethe resulting metric line as M = (Sdx d) Next fix an ordering lt (a direction) inM Denote the resulting oriented line as the affine line A = (Sdx dlt) Next fix a

24 Complements 83

preferred point of A as the origin 0 the resulting space is isomorphic to the real lineR = (Sdx dlt 0)

The real line R is the traditional metaphor for the idea of time Time is frequentlyrepresented by a variable t in R The structure of R corresponds to an ensemble ofproperties that we naturally associate to the notion of time as follows (a) The existenceof a topology on the set of the time instants namely the existence of a notion of two timeinstants being close to each other and the fact that time is ldquoone-dimensionalrdquo (b) Theexistence of a metric Namely the possibility of stating that two distinct time intervalsare equal in magnitude time is ldquometricrdquo (c) The existence of an ordering relationbetween time instants Namely the possibility of distinguishing the past direction fromthe future direction (d) The existence of a preferred time instant the present theldquonowrdquo To capture these properties in mathematical language we describe time as areal line R An affine line A describes time up to the notion of present a metric lineM describes time up to the notions of present and pastfuture distinction a line Ldescribes time up to the notion of metricity

In newtonian mechanics we begin by representing time as a variable in R but thenthe equations are invariant both under t rarr minust and under t rarr t + a Thus the theoryis actually defined in terms of a variable t in a metric line M Newtonian mechanicsin fact incorporates both the notions of topology of the set of time instants and (ina very essential way) the fact that time is metric but it does not make any use ofthe notion of present nor the direction of time This is well known Note that Newtontheory is not inconsistent with the introduction of the notions of a present and of time-directionality it simply does not make any use of these notions These notions are notpresent in Newton theory

The properties listed above do not exhaust the different ways in which the notionof time enters physical theories the development of theoretical physics has modifiedsubstantially the natural notion of time A first modification was introduced by specialrelativity Einsteinrsquos definition of the time coordinate of distant events yields a notion oftime which is observer dependent An invariant structure can be maintained at the priceof relaxing the 1d character of time and the 3d character of space in favor of a notion of4d spacetime Alternatively we may say that the notion of a single time is replaced bya three-parameter family of times tv one for each Lorentz observer Therefore the timewe use in special relativity is not unique as is the time in newtonian mechanics Ratherthan a single line we have a three-parameter family of lines (the straight lines throughthe origin that fill the light cone of Minkowski space) Denote this three-parameter setof lines as M3

Times in GR There are several distinct possibilities of identifying ldquotimerdquo in GR Eachsingles out a different notion of time Each of these notions reduces to the standardnonrelativistic or special-relativistic time in appropriate limits but each lacks at leastsome of the properties of nonrelativistic time The most common ways of identifyingtime within GR are the following

Coordinate time x0 Coordinate time can be arbitrarily rescaled and does not pro-vide a way of identifying two time intervals as equal in duration Therefore it isnot metric in the sense defined above In addition the possibility of changingthe time coordinate freely from point to point implies that there is an infinite-dimensional choice of equally good coordinate times Finally unlike prerelativis-tic time x0 is not an observable quantity Denote the set of all the possiblecoordinate times as Linfin

Proper time τ This notion of time is metric But it is very different from the notion of


time in special relativity for several reasons First it is determined by the grav-itational field Second we have a different time for each worldline or infinitesi-mally for every speed at every point For an infinitesimal timelike displacementdxμ at a point x the infinitesimal time interval is dτ =

radicminusgμν(x) dxμdxν This

notion of time is a radical departure from the notion of time used in special rela-tivity because it is determined by the dynamical fields in the theory A solutionof Einsteinrsquos equations defines a point in the phase space Γ of GR It assignsa metric structure to every worldline Therefore this notion of time is given bya function from the phase space Γ multiplied by the set of the worldlines wlinto the metric structures d wl times wl rarr R+ Denote this function as minfin Callldquointernalrdquo a notion of time affected by the dynamics

Before GR dynamics could be expressed as evolution in a single time vari-able which has metric properties and could be measured In general-relativisticphysics this concept of time splits into two distinct concepts we can still viewthe dynamics as evolution in a time variable x0 but this time has no metricproperties and is not observable alternatively there is a notion of time thathas metric properties τ but the dynamics of the theory cannot be expressed asevolution in τ Is there a way to go around this split and view GR as a dynam-ical theory in the sense of a theory expressing evolution in an observable metrictime

Clock time The dynamics of GR determines how observable quantities evolve withrespect to one another We can always choose one observable quantity tc de-clare it the independent one and describe how the other observables evolve asfunctions of it A typical example of this clock time is the radius of a spatiallycompact universe in relativistic cosmology R Formally clock time is a functionon the extended configuration space C of the theory (see Chapter 3) Denote thisnotion of time as the clock time τc C rarr R

Under this definition of time GR becomes similar to a standard hamiltoniandynamical theory A clock time however generally behaves as a clock only incertain states or for a limited amount of time The radius of the universe forinstance fails as a good time variable when the universe recollapses In gen-eral a clock time lacks temporal globality In fact several results are knownconcerning obstructions to defining a function tc that behaves as ldquoa good timerdquoglobally [66]

Notice that some of these relativistic notions of time are in a sense opposite to theprerelativistic case while in newtonian theory time evolution is captured by a functionfrom the metric line M (time) to the configuration or phase space now the notion oftime is captured by a function from the configuration or phase space to the metric lineThis inversion is the mathematical expression of the physical idea that the flow of timeis affected or determined by the dynamics of the system itself

Finally none of the ways of thinking of time in classical GR can be uncriticallyextended to the quantum regime Quantum fluctuations of physical clocks and quan-tum superposition of different metric structures make the very notion of time fuzzy atthe Planck scale As will be discussed in the second part of this book a fundamentalconcept of time may be absent in quantum gravity

Notions of time Notice that properties of time progressively disappear in going towardmore fundamental physical theories At the opposite end of the spectrum there are

24 Complements 85

properties associated with the notion of time used in the natural languages which arenot present in physical theories They play a role in other areas of natural investigationsI mention these properties for the sake of completeness These are for instance memoryexpectations and the psychological perception of free will

To summarize I have identified the following properties of the notion of time

1 Existence of memory and expectations2 Existence of a preferred instant of time the present the now3 Directionality the possibility of distinguishing the past from the future direction4 Uniqueness the feature that is lost in special and general relativity where we

cannot identify a preferred time variable5 The property of being external the independence of the notion of time from the

dynamical variables of the theory6 Spatial globality the possibility of defining the same time variable in all space

points7 Temporal globality the fact that every motion goes through every value of the

time variable once and only once8 Metricity the possibility of saying that two time intervals have equal duration9 One-dimensionality namely the possibility of arranging the time instants in a

one-dimensional manifold

This discussion suggests a sequence of notions of time which I list here in order ofdecreasing complexity

Time of natural language This is the notion of time of everyday language which in-cludes all the features just listed This notion of time is not necessarily nonscien-tific for instance any scientific approach to say the human brain should makeuse of this notion of time

Time-with-a-present This is the notion of time that has all the features just listedincluding the existence of a preferred instant the present but not the notionsof memory and expectations which are notions usually more related to complexsystems (brain) than to time itself The notion of present is generally considereda feature of time itself This notion of time is the one to which often people referwhen they refer to the ldquoflow of timerdquo or Eddingtonrsquos ldquovivid perception of theflow of timerdquo [67] This notion of time can be described by the structure of aparametrized line R

Thermodynamical time If we maintain the distinction between a future direction anda past direction but we give up the notion of present we obtain the notionof time typical of thermodynamics Since thermodynamics is the first physicalscience that appears in this list this is maybe a good place to emphasize that thenotion of present of the ldquonowrdquo is completely absent from the description of theworld in physical terms This notion of time can be described by the structureof an affine line A

Newtonian time In newtonian mechanics there is no preferred direction of time Noticethat in the absence of a preferred direction of time the notions of cause and effectare interchangeable This notion of time can be described by the structure of ametric line M

Special-relativistic time If we give up uniqueness we have the time used in specialrelativity different Lorentz observers have a different notion of time Special-relativistic time is still external spatially and temporally global metrical andone-dimensional but it is not unique There is a three-parameter set of quantitiesthat share the status of time This notion of time can be described by the three-parameter set of metric lines M3


Table 21 Notions of time

Time notion Property Example Form

natural language time memory brain time-with-a-present present biology Rthermodynamical time direction thermodynamics Anewtonian time unique newtonian mechanics Mspecial-relativistic time external special relativity M3

cosmological time spatially global cosmological time mproper time temporally global worldline proper time minfin

clock time metric clocks in GR cparameter time one-dimensional coordinate time Linfin

no-time none quantum gravity none

Cosmological time By this I indicate a time which is spatially and temporally globalmetrical and one-dimensional but it is not external namely it is dynamicallydetermined by the theory Proper time in cosmology is the typical example Itis the most structured notion of time that occurs in GR Denote it by m

Proper time By this I indicate a time which is temporally global metrical and one-dimensional but it is not spatially global as the notion of proper time alongworldlines in GR It can be represented by a function minfin defined on the carte-sian product of the phase space and the ensemble of the worldlines

Clock time By this I indicate a time which is metrical and one-dimensional but itis not temporally global A realistic matter clock in GR defines a time in thissense This notion of time can be described by a function c on the phase space

Parameter time By this we mean a notion of time which is not metric and not ob-servable The typical example is the coordinate-time in GR Another exampleof parameter time is the evolution parameter in the parametrized formulationof the dynamics of a relativistic particle Parameter time is described by anunparametrized line L or by an infinite set Linfin of unparametrized lines

No-time Finally this is the bottom level in the analysis it is not a time concept butrather I indicate by no-time the idea that a predictive physical theory can bewell defined also in the absence of any notion of time

The list must not be taken rigidly It is summarized in the Table 21There is a interesting feature that emerges from the above analysis the hierarchical

arrangement While some details of this arrangement may be artificial neverthelessthe analysis points to a general fact moving from theories of ldquospecialrdquo objects likethe brain or living beings toward more general theories that include larger portionsof Nature we make use of a physical notion of time that is less specific and has lessdeterminations If we observe Nature at progressively more fundamental levels andwe seek for laws that hold in more general contexts then we discover that these lawsrequire or admit an increasingly weaker notion of time

This observation suggests that ldquohigh levelrdquo features of time are not present at thefundamental level but ldquoemergerdquo as features of specific physical regimes like the notionof ldquowater surfacerdquo emerges in certain regimes of the dynamics of a combination of waterand air molecules (see for instance [68])

24 Complements 87

Notions of time with more attributes are high-level notions that have no meaning inmore general situations The uniqueness of newtonian time for instance makes senseonly in the special regime in which we consider an ensemble of bodies moving slowlywith respect to each other Thus the notion of a unique time is a high-level notionthat makes sense only for some regimes in Nature For general systems most featuresof time are genuinely meaningless


The precise meaning of the coordinates x = (x t) in newtonian and special-relativisticphysics is far from obvious Let me recall it here in order to clarify the precise differencebetween these and the relativistic coordinates

Newton is well aware that the motions we observe are relative motions and stressesthis point in Principia His point is not that we can directly observe absolute motionHis point is that we can infer the absolute motion or ldquotrue motionsrdquo or motion withrespect to absolute space from its physical effects (such as the concavity of the waterin the bucket) starting from our observation of relative motions

For instance we observe and describe motions with respect to Earth but fromsubtle effects such as Foucaultrsquos pendulum we infer that these are not true motionsThe experiment of the bucket is an example of the possibility of revealing true motion(rotation of the water with respect to space) disentangling it from relative motion(rotation with respect to the bucket) by means of an observable effect (the concavityof the water surface)26

For Newton the coordinates x that enter his main equation

F = md2x(t)

dt2(2152)

are the coordinates of absolute space However since we cannot directly observe spacethe only way we can coordinatize space points is by using physical objects The co-ordinates x of the object A moving along the trajectory x(t) are therefore definedas distances from a chosen system O of objects which we call a ldquoreference framerdquoBut then x are not the coordinates of absolute space So how can equation (2152)work

The solution of the difficulty is to use the capacity of unveiling ldquotrue motionrdquo thatNewton has pointed out in order to select the objects forming the reference frame Owisely There are ldquogoodrdquo and ldquobadrdquo reference frames The good ones are the ones inwhich no effect such as the concavity of the water surface of Newtonrsquos bucket can be

26Newton accords deep significance to the fact that we can unveil true motion Hedescribes relative motion as the way reality is observed by us and true motion as theway reality might be directly ldquoperceivedrdquo or ldquosensedrdquo by God This is why Newtoncalls space ndash the entity with respect to which true motion happens ndash the ldquoSensoriumof Godrdquo true motion is motion ldquowith respect to Godrdquo or ldquoas perceived by GodrdquoThere is a platonic tone in this idea that reason finds the way to the veiled divinetruth beyond appearances I wouldnrsquot read this as so removed from modernity asit is often portrayed There isnrsquot all that much difference between Newtonrsquos inquiryinto Godrsquos way of ldquosensing the worldrdquo and the modern search for the most effectiveway of conceptualizing reality Newtonrsquos God plays a mere linguistical role herethe role of denoting a major enterprise upgrading our own conceptual structure forunderstanding reality


observed within a desired accuracy Equation (2152) is correct to the desired accuracyif we use coordinates defined with respect to these good frames In other words thephysical content of (2152) is actually quite subtle

There exist reference objects O with respect to which the motion ofany other object A is correctly described by (2152)

This is a statement that begins to be meaningful only when a sufficiently large numberof moving objects is involved

Notice also that for this construction to work it is important that the objects Oforming the reference frame are not affected by the motion of the object A Thereshouldnrsquot be any dynamical interaction between A and O

Special relativity does not change much of this picture Since absolute simultaneitymakes no sense if the event A is distant from the clock in the origin its time t is illdefined Einsteinrsquos idea is to define a procedure for assigning a t to distant events usingclocks moving inertially

At clock time te send a light signal that reaches the event Receivethe reflected signal back at tr The time coordinate of the event isdefined to be tA = 1

2 (te + tr)

It is important to emphasize that this is a useful definition not a metaphysical state-ment that the event A happens ldquoright at the time whenrdquo the observer clock displaystA

Special relativity replaces Newtonrsquos absolute space and absolute time with a singleentity Minkowskirsquos absolute spacetime while the notion of inertial system and themeaning of the coordinates are the same as in newtonian mechanics

Summarizing these coordinates have the following properties

(i) Coordinates describe position with respect to physical reference objects (referenceframes)

(ii) Space coordinates are defined by the distance from the reference bodies Timecoordinates are defined with respect to isochronous clocks

(iii) Reference objects are appropriately chosen they are such that the reference systemthey define is inertial

(iv) Inertial frames reveal the structure of absolute spacetime itself

(v) The object A whose dynamics is described by the coordinates does not interactwith the reference objects O There is no dynamical coupling between A and O

Relativistic coordinates do not have any of these properties The fact that the two areindicated with the same notation xμ is only an unfortunate historical accident


Instead of working with arbitrary unphysical coordinates xμ we can choose to coordina-tize spacetime events with coordinates Xμ having an assigned physical interpretationFor instance we can describe the Universe by giving a name X to each galaxy andchoosing X0 as the proper time from the Big Bang along the galaxy worldline If wedo so the defining properties of the coordinates X must be added to the formalismWe must add a certain number of equations for the gravitational field the equations ofmotions of the objects used to fix the coordinates (the galaxies in the example) Theseadditional equations gauge-fix general covariance

24 Complements 89

The gauge-fixing can also be partial For instance a common choice is

e00(X) = 1 ei0(X) = 0 e0

a(X) = 0 (2153)

where i = 1 2 3 and a = 1 2 3 This corresponds to partially fixing the coordinates byrequiring that X0 measures proper time that equal X0 surfaces are locally instantaneitysurfaces in the sense of Einstein for the constant X lines and that the local Lorentzframes are chosen so that these lines are still

If the coordinates are fully specified the set formed by these physical gauge-fixingequations and the equations of motion has no residual gauge invariance that is ini-tial data determine evolution uniquely This procedure can be implemented in manypossible ways since there are arbitrarily many ways of fixing physical coordinates andnone is a priori better than any other In spite of this arbitrariness this procedure isoften convenient when the physical situation suggests a natural coordinate choice asin the cosmological context mentioned

Physical coordinates Xμ defined by matter filling space can only be effectively usedin the cosmological context because it is only at the cosmological scale that matter fillsspace In a system in which there are empty regions such as the Solar System thesephysical coordinates are not available An interesting alternative choice is provided bythe GPS coordinates described below

The physical coordinates Xμ are partial observables and we can associate measuringdevices with them

Undetermined physical coordinates Finally there is a third interpretation of the co-ordinates of GR which is intermediate between arbitrary coordinates xμ and physicalcoordinates Xμ Imagine that a region of the universe is filled with certain light objectswhich may not be in free fall We can use these objects to define physical coordinatesXμ but also choose to ignore the equations of motion of these objects We obtain asystem of equations for the gravitational field and other matter expressed in termsof coordinates Xμ that are interpreted as the spacetime location of reference objectswhose dynamics we have chosen to ignore

This set of equations is under-determined the same initial conditions can evolve intodifferent solutions However the interpretation of such under-determination is simplythat we have chosen to neglect part of the equations of motion Different solutions withthe same initial conditions represent the same physical configuration of the fields butexpressed say in one case with respect to free-falling reference objects in the othercase with respect to reference objects on which a force has acted at a certain momentand so on This procedure has the disadvantage of being useless in quantum theorywhere we cannot assume that something is observable and at the same time neglect itsdynamics

In conclusion one should always be careful in talking about general-relativistic co-ordinates whether one is referring to

(i) arbitrary mathematical coordinates x

(ii) physical coordinates X with an interpretation as positions with respect to objectswhose equations of motion are taken into account

(iii) physical coordinates with an interpretation as positions with respect to objectswhose equations of motion are ignored

The system of equations of motion is nondeterministic in (i) and (iii) deterministic in(ii) The coordinates are partial observables in (ii) and (iii) but not in (i) Confusionabout observability in GR follows from confusing these three different interpretationsof the coordinates The following is an example of physical coordinates


GPS observables In the literature there are many attempts to define useful physicalcoordinates It is easier to define physical coordinates in the presence of matter thanin the context of pure GR Ideally we can consider GR interacting with four scalarmatter fields Assume that the configuration of these fields is sufficiently nondegener-ate Then the components of the gravitational field at points defined by given valuesof the matter fields are gauge-invariant observables This idea has been developed ina number of variants such as dust-carrying clocks and others (see [69ndash71] and refer-ences therein) The extent to which the result is realistic or useful is questionable Itis rather unsatisfactory to understand the theory in terms of fields that do not existor phenomenological objects such as dust and it is questionable whether these pro-cedures could make sense in the quantum theory where the aim is to describe Planckscale dynamics Earlier attempts to write a complete set of gauge-invariant observ-ables are in the context of pure GR [72] The idea is to construct four scalar functionsof the gravitational field (say scalar polynomials of the curvature) and use these tolocalize points The value of a fifth scalar function in a point where the four scalarfunctions have a given value is a gauge-invariant observable This works but the resultis mathematically very intricate and physically very unrealistic It is certainly possiblein principle to construct detectors of such observables but I doubt any experimenterwould get funded for a proposal to build such an apparatus

There is a simple way out based on GR coupled with a minimal and very realisticamount of additional matter Indeed this way out is so realistic that it is in fact realit is essentially already implemented by existing technology the Global PositioningSystem (GPS) which is the first technological application of GR or the first large-scaletechnology that needs to take GR effects into account [73]

Consider a generally covariant system formed by GR coupled with four small bodiesThese are taken to have negligible mass they will be considered as point particlesfor simplicity and called ldquosatellitesrdquo Assume that the four satellites follow timelikegeodesics that these geodesics meet in a common (starting) point O and at O theyhave a given (fixed) speed ndash the same for all four ndash and directions as the four vertices ofa tetrahedron The theory might include any other matter Then (there is a region R ofspacetime for which) we can uniquely associate four numbers sα α = 1 2 3 4 to eachspacetime point p as follows Consider the past lightcone of p This will (generically)intersect the four geodesics in four points pα The numbers sα are defined as thedistance between pα and O (That is the proper time along the satellitesrsquo geodesic)We can use the sα as physically defined coordinates for p The components gαβ(s) of themetric tensor in these coordinates are gauge-invariant quantities They are invariantunder four-dimensional diffeomorphisms (because these deform the metric as well asthe satellitesrsquo worldlines) They define a complete set of gauge-invariant observables forthe region R

The physical picture is simple and its realism is transparent Imagine that the fourldquosatellitesrdquo are in fact satellites each carrying a clock that measures the proper timealong its trajectory starting at the meeting point O Imagine also that each satellitebroadcasts its local time with a radio signal Suppose I am at the point p and have anelectronic device that simply receives the four signals and displays the four readings seeFigure 25 These four numbers are precisely the four physical coordinates sα definedabove Current technology permits us to perform these measurements with an accu-racy well within the relativistic regime [73 74] If we then use a rod and a clock andmeasure the physical 4-distances between sα coordinates we are directly measuring thecomponents of the metric tensor in the physical coordinate system In the terminologyof Chapter 3 the sα are partial observables while the gαβ(s) are complete observables

24 Complements 91

O

p

Σ

t

x

s2s1

Fig 25 s1 and s2 are the GPS coordinates of the point p Σ is a Cauchy surfacewith p in its future domain of dependence

As shown below the physical coordinates sα have nice geometrical properties theyare characterized by

gαα(s) = 0 α = 1 4 (2154)

Surprisingly in spite of the fact that they are defined by what looks like a rather non-local procedure the evolution equations for gαβ(s) are local These evolution equationscan be written explicitly using the ArnowittndashDeserndashMisner (ADM) variables (see [131]of Chapter 3 for details) Lapse and Shift turn out to be fixed local functions of thethree metric

In what follows I first introduce the GPS coordinates sα in Minkowski space ThenI consider a general spacetime I assume the Einstein summation convention only forcouples of repeated indices that are one up and one down Thus in (2154) α is notsummed While dealing with Minkowski spacetime the spacetime indices μ ν are raisedand lowered with the Minkowski metric Here I write an arrow over three- as well asfour-dimensional vectors Also here I use the signature [+minusminusminus] in order to havethe same expressions as in the original article on the subject

Consider a tetrahedron in three-dimensional euclidean space Let its center be atthe origin and its four vertices vα where the vectors vα have unit length |vα|2 = 1 andvα middot vβ = minus13 for α = β Here α = 1 2 3 4 is an index that distinguishes the fourvertices and should not be confused with vector indices With a convenient orientationthese vertices have cartesian coordinates (a = 1 2 3)

v1a = (0 0 1) v2a = (2radic

23 0 minus13) (2155)

v3a = (minusradic

23radic

23 minus13) v4a = (minusradic

23 minusradic

23 minus13) (2156)

Let us now go to a four-dimensional Minkowski space Consider four timelike 4-vectorsWα of length unity | Wα|2 = 1 representing the normalized 4-velocities of four par-ticles moving away from the origin in the directions vα at a common speed v Their


Minkowski coordinates (μ = 0 1 2 3) are

Wαμ =1radic

1 minus v2(1 v vαa) (2157)

Fix the velocity v by requiring the determinant of the matrix Wαμ to be unity (Thischoice fixes v at about one-half the speed of light a different choice changes only a fewnormalization factors in what follows) The four by four matrix Wαμ plays an importantrole in what follows Notice that it is a fixed matrix whose entries are certain givennumbers

Consider one of the four 4-vectors say W = W 1 Consider a free particle inMinkowski space that starts from the origin with 4-velocity W Call it a ldquosatelliterdquo Itsworldline l is x(s) = s W Since W is normalized s is precisely the proper time alongthe worldline Consider now an arbitrary point p in Minkowski spacetime with coordi-nates X Compute the value of s at the intersection between l and the past lightconeof p This is a simple exercise giving

s = X middot W minusradic

( X middot W )2 minus | X|2 (2158)

Now consider four satellites moving out of the origin at 4-velocity Wα If they radiobroadcast their position an observer at the point p with Minkowski coordinates Xreceives the four signals sα

sα = X middot Wα minusradic

( X middot Wα)2 minus | X|2 (2159)

Introduce (nonlorentzian) general coordinates sα on Minkowski space defined by thechange of variables (2159) These are the coordinates read out by a GPS device inMinkowski space The jacobian matrix of the change of coordinates is given by

partsα

partxμ= Wα

μ minus Wαμ ( X middot Wα) minusXμradic( X middot Wα)2 minus | X|2

(2160)

where Wαμ and Xμ are Wαμ and Xμ with the spacetime index lowered with the

Minkowski metric This defines the tetrad field eαμ(s)

eαμ(s(X)) =partsα

partxμ(X) (2161)

The contravariant metric tensor is given by gαβ(s) = eαμ(s)eμβ(s) Using the relation

| Wα|2 = 1 a straightforward calculation shows that

gαα(s) = 0 α = 1 4 (2162)

This equation has the following nice geometrical interpretation Fix α and considerthe one-form field ωα =dsα In sα coordinates this one-form has components ωα

β = δαβ

and therefore ldquolengthrdquo |ωα|2 = gβγωαβω

αγ = gαα But the ldquolengthrdquo of a one-form is

proportional to the volume of the (infinitesimal now) 3-surface defined by the formThe 3-surface defined by dsα is the surface sα = constant But sα = constant is the setof points that read the GPS coordinate sα namely that receive a radio broadcastingfrom a same event pα of the satellite α namely that are on the future lightcone of pαTherefore sα = constant is a portion of this lightcone it is a null surface and thereforeits volume is zero And so |ωα|2 = 0 and gαα = 0

24 Complements 93

Since the sα coordinates define sα = constant surfaces that are null we denote themas ldquonull GPS coordinatesrdquo It is useful to introduce another set of GPS coordinates aswell which have the traditional timelike and spacelike character We denote these assμ call them ldquotimelike GPS coordinatesrdquo and define them by

sα = Wαμ sμ (2163)

This is a simple algebraic relabeling of the names of the four GPS coordinates suchthat sμ=0 is timelike and sμ=a is spacelike In these coordinates the gauge condition(2162) reads

Wαμ Wα

ν gμν(s) = 0 (2164)

This can be interpreted geometrically as follows The (timelike) GPS coordinates arecoordinates sμ such that the four one-form fields

ωα = Wαμ dsμ (2165)

are nullLet us now jump from Minkowski space to full GR Consider GR coupled with four

satellites of negligible mass that move geodesically and whose worldlines emerge froma point O with directions and velocity as above Locally around O the metric can betaken to be minkowskian therefore the details of the initial conditions of the satellitesrsquoworldlines can be taken as above The phase space of this system is the one of pureGR plus ten parameters giving the location of O and the Lorentz orientation of theinitial tetrahedron of velocities The integration of the satellitesrsquo geodesics and of thelightcones can be arbitrarily complicated in an arbitrary metric However if the metricis sufficiently regular there will still be a region R in which the radio signals broadcastby the satellites are received (In the case of multiple reception the strongest one can beselected That is if the past lightcone of p intersects l more than once generically therewill be one intersection which is at shorter luminosity distance) Thus we still havewell-defined physical coordinates sα on R Equation (2162) holds in these coordinatesbecause it depends only on the properties of the light propagation around p We definealso timelike GPS coordinates sμ by (2163) and we get condition (2164) on the metrictensor

To study the evolution of the metric tensor in GPS coordinates it is easier to shiftto ADM variables NNa γab These are functions of the covariant components of themetric tensor defined in general by

ds2 = gμνdxμdxν = N2dt2 minus γab(dxa minusNadt)(dxb minusNbdt) (2166)

Equivalently they are related to the contravariant components of the metric tensor by

gμνvμvν = minusγabvavb + (nμvμ)2 (2167)

where γab is the inverse of γab and nμ = (1NNaN) Using these variables the gaugecondition (2164) reads

Wαa Wα

b γab = (Wαμ nμ)2 (2168)

Notice now that this can be solved for the Lapse and Shift as a function of the 3-metric(recall that Wα

μ are fixed numbers) obtaining

nμ = Wμα q

α (2169)


where Wμα is the inverse of the matrix Wα

μ and

qα =radic

Wαa Wα

b γab (2170)

Or explicitly

N =1

W 0αqα

Na =W a

αqα

W 0αqα

(2171)

The geometrical interpretation is as follows We want the one-form ωα defined in (2165)to be null namely its norm to vanish But in the ADM formalism this norm is the sumof two parts the norm of the pull-back of ωα on the constant time ADM surfacewhich is qα given in (2170) and depends on the three metric plus the square of theprojection of ωα on nμ We can thus obtain the vanishing of the norm by adjusting theLapse and Shift We have four conditions (one per α) and we can thus determine Lapseand Shift from the 3-metric In other words whatever the 3-metric we can alwaysadjust Lapse and Shift so that the gauge condition (2164) is satisfied But in theADM formalism the arbitrariness of the evolution in the Einstein equations is entirelycaptured by the freedom in choosing Lapse and Shift Since here Lapse and Shift areuniquely determined by the 3-metric evolution is determined uniquely if the initialdata on a Cauchy surface are known Therefore the evolution in the GPS coordinates0 of the GPS components of the metric tensor gμν(s) is governed by deterministicequations the ADM evolution equation with Lapse and Shift determined by (2170)ndash(2171) Notice also that evolution is local since the ADM evolution equations as wellas the (2170)ndash(2171) are local27

How can the evolution of the quantities gμν(s) be local The conditions on thenull surfaces described in the previous paragraph are nonlocal Coordinate distancestypically yield nonlocality imagine we define physical coordinates in the Solar Systemusing the cosmological time tc and the spatial distances xS xE xJ (at fixed tc) from saythe Sun the Earth and Jupiter The metric tensor gμν(tc xS xE xJ) in these coordinatesis a gauge-invariant observable but its evolution is highly nonlocal To see this imaginethat in this moment (in cosmological time) Jupiter is swept away by a huge cometThen the value of gμν(tc xS xE xJ) here changes instantaneously without any localcause the value of the coordinate xJ has changed because of an event happening faraway Whatrsquos special about the GPS coordinates that avoids this nonlocality Theanswer is that the value of a GPS coordinate at a point p does in fact depend onwhat happens ldquofar awayrdquo as well Indeed it depends on what happens to the satelliteHowever it only depends on what happened to the satellite when it was broadcastingthe signal received in p and this is in the past of p If p is in the past domain ofdependence of a partial Cauchy surface Σ then the value of gμν(s) in p is completelydetermined by the metric and its derivative on Σ namely evolution is causal becausethe entire information needed to set up the GPS coordinates is in the data in Σ seeFigure 25 Explicitly the sα = constant surfaces around Σ can be uniquely integratedahead all the way to p They certainly can as they represent just the evolution of alight front This is how local evolution is achieved by these coordinates

Summarizing I have introduced a set of physical coordinates determined by certainmaterial bodies Geometrical quantities such as the components of the metric tensorexpressed in physical coordinates are of gauge-invariant observables There is no needto introduce a large unrealistic amount of matter or to construct complicated andunrealistic physical quantities out of the metric tensor Four particles are sufficient to

27This does not imply that the full set of equations satisfied by gμν(s) must be localsince initial conditions on s0 = 0 satisfy four other constraints besides the ADM ones

24 Complements 95

Fig 26 A simple apparatus to measure the gravitational field Two GPS de-vices reading sμL and sμR respectively connected by a 1 meter rod If for instancesμR = sμL for μ = 0 2 3 then the local value of g11(s) is g11(s) = (s1

R minus s1L)minus2m2

coordinatize a (region of a) four-geometry Furthermore the coordinatization procedureis not artificial it is the real one utilized by existing technology

The components of the metric tensor in (timelike) GPS coordinates can be measuredas follows (see Figure 26) Take a rod of physical length L (small with respect to thedistance along which the gravitational field changes significantly) with two GPS de-vices at its ends (reading timelike GPS coordinates) Orient the rod (or search amongrecorded readings) so that the two GPS devices have the same reading s of all coor-dinates except for s1 Let δs1 be the difference in the two s1 readings Then we havealong the rod

ds2 = g11(s)δs1δs1 = L2 (2172)

Therefore

g11(s) =

(L

δs1

)2

(2173)

Nondiagonal components of gab(s) can be measured by simple generalizations of thisprocedure The g0b(s) are then algebraically determined by the gauge conditions Ina thought experiment data from a spaceship traveling in a spacetime region could beused to produce a map of values of the GPS components of the metric tensor Insteadof using a rod which is a rather crude device for measuring distances one could senda light pulse forward and back between the two GPS devices kept at fixed spatial sμ

coordinates If T is the (physical) time for flying back and forward measured by aprecise clock on one device then g11(s) = (cT2δs1)2 This is valid so long as T andL are small compared to the distances over which the gab(s) change by amounts of theorder of the experimental errors

The individual components of the metric tensor expressed in physical coordinatesare measurable The statement that ldquothe curvature is measurable but the metric is notmeasurablerdquo which is often heard is incorrect Both metric and curvature in physicalcoordinates are measurable and predictable Neither metric nor curvature in arbitrarynonphysical coordinates are measurable

The GPS coordinates are partial observables (see Chapter 3) The complete ob-servables are the quantities gμν(s) for any given value of the coordinates sμ Thesequantities are diffeomorphism invariant are uniquely determined by the initial dataand in a canonical formulation are represented by functions on the phase space thatcommute with all constraints

The GPS observables are a straightforward generalization of Einsteinrsquos ldquospacetimecoincidencesrdquo In a sense they are precisely Einsteinrsquos point coincidences Einsteinrsquosldquomaterial pointsrdquo are just replaced by photons (light pulses) the spacetime point sα is


characterized as the meeting point of four photons designated by the fact of carryingthe radio signals sα

mdashmdash


There are many beautiful classic textbooks on GR Two among the bestoffering remarkably different points of view on the theory are Weinberg[75] and Wald [76] The first stresses the similarity between GR and flat-space field theory the second on the contrary emphasizes the geometricreading of GR Here I have followed a third path I place emphasis onthe change of the notions of space and time needed for general-relativisticphysics (which affects quantization dramatically) but I put little emphasison the geometric interpretation of the gravitational field (which is goingto be largely lost in the quantum theory)

Relevant mathematics is nicely presented for instance in the text byChoquet-Bruhat DeWitt-Morette and Dillard-Bleick [77] and in [16] Onthe large empirical evidence in favor of GR piled up in the recent yearssee Ciufolini and Wheeler [78]

The tetrad formalism and its introduction into quantum gravity aremainly due to Cartan to Weyl [80] and to Schwinger [80] the first-orderformalism to Palatini The Plebanski two-form was introduced in [81] Theselfdual connection which is at the root of Ashtekarrsquos canonical theory(see Chapter 4) was introduced by Amitaba Sen [82] The lagrangianformulation for the selfdual connections was given in [83] A formulationof GR based on the sole connection is discussed in [84]

Interesting reconstructions of Einsteinrsquos path towards GR are in [8586]Kretschmannrsquos objection to the significance of general covariance ap-peared in [87] On this see also Andersonrsquos book [88] An account ofthe historical debate on the interpretation of space and motion is JulianBarbourrsquos [89] a wonderful historical book In the philosophy of sciencethe debate was reopened by a 1987 paper on the hole argument by JohnEarman and John Norton [90] On the contemporary version of this de-bate see [65 91ndash93] On the physical side of the discussion of what isldquoobservablerdquo in GR see [71]

The discussion of the different notions of time follows [94] A surprisingand inspiring book on the subject is Fraser [95] a book that will convincethe reader that the notion of time is far from being a monolithic conceptThe literature on the problem of time in quantum gravity is vast I list onlya few pointers here distinguishing various problems origin of the ldquoarrowof timerdquo and the cosmological time asymmetry [96] disappearance of thecoordinate-time variable in canonical quantum gravity [97] possibility of


a consistent interpretation of quantum mechanics for systems withoutglobal time [269899] problems in choosing an ldquointernal timerdquo in generalrelativity and the properties that such an internal time should have [66]see also [100] The presentation of the GPS observables follows [101] seealso [102103]

3Mechanics

In its conventional formulation mechanics describes the evolution of states and ob-servables in time This evolution is governed by a hamiltonian This is also true forspecial-relativistic theories where evolution is governed by a representation of thePoincare group which includes a hamiltonian This conventional formulation is notsufficiently broad because general-relativistic systems ndash in fact the world in which welive ndash do not fit into this conceptual scheme Therefore we need a more general formula-tion of mechanics than the conventional one This formulation must be based on notionsof ldquoobservablerdquo and ldquostaterdquo that maintain a clear meaning in a general-relativistic con-text A formulation of this kind is described in this chapter

The conventional structure of conventional nonrelativistic mechanics already pointsrather directly to the relativistic formulation described here Indeed many aspects ofthis formulation are already utilized by many authors For instance Arnold [104] iden-tifies the (presymplectic) space with coordinates (t qi pi) (time lagrangian variablesand their momenta) as the natural home for mechanics Souriau has developed a beau-tiful and little-known relativistic formalism [105] Probably the first to consider thepoint of view used here was Lagrange himself in pointing out that the most convenientdefinition of ldquophase spacerdquo is the space of the physical motions [106] Many of the toolsused below are also used in hamiltonian treatments of generally covariant theories asconstrained systems although generally within a rather obscure interpretative cloud

31 Nonrelativistic mechanics mechanics is about timeevolution

I begin with a brief review of conventional mechanics This is useful tofix notations and introduce some notions that will play a role in the rela-tivistic formalism I give no derivations here they are standard and theycan be obtained as a special case of the derivations in the next section

Lagrangian A dynamical system with m degrees of freedom describes theevolution in time t of m lagrangian variables qi where i = 1 m Thespace in which the variables qi take value is the m-dimensional (nonrela-tivistic) configuration space C0 The dynamics of the system is determined

98


by a single function of 2m variables L(qi vi) the lagrangian Given twotimes t1 and t2 and two points qi1 and qi2 in C0 physical motions are suchthat the action

S[q] =int t2

t1

dt L(qi(t)

dqi(t)dt

)(31)

is an extremum in the space of the motions qi(t) such that qi(t1) = qi1and qi(t2) = qi2 A dynamical system is therefore specified by the couple(C0 L) Physical motions satisfy the Lagrange equations

ddt

pi

(qi(t)

dqi(t)dt

)= Fi

(qi(t)

dqi(t)dt

) (32)

where momenta and forces are defined by

pi(qi vi) =partL(qi vi)

partvi Fi(qi vi) =

partL(qi vi)partqi

(33)

Hamiltonian The Lagrange equations can be cast in first-order form byusing the lagrangian coordinates qi and the momenta pi as variablesInverting the function pi(qi vi) yields the function vi(qi pi) inserting thisin the function Fi(qi vi) defines the force fi(qi pi) equiv Fi(qi vi(qi pi)) asfunctions of coordinates and momenta The equations of motion (32)become

dqi(t)dt

= vi(qi(t) pi(t))dpi(t)

dt= fi(qi(t) pi(t)) (34)

These equations are determined by the function H0(qi pi) the nonrela-tivistic hamiltonian defined by H0(qi pi) = piv

i(qi pi) minus L(qi vi(qi pi))Indeed (34) is equivalent to (32) with

vi(qi pi) =partH0(qi pi)

partpi fi(qi pi) = minuspartH0(qi pi)

partqi (35)

Symplectic The Hamilton equations (34)ndash(35) can be written in a use-ful and compact geometric language The 2m-dimensional space coordi-natized by the coordinates qi and the momenta pi is the nonrelativisticphase space Γ0 (The reason for the subscript 0 will be clear below) Timeevolution is a flow (qi(t) pi(t)) in this space the vector field on Γ0 tangentto this flow is

X0 = vi(qi pi)part

partqi+ fi(qi pi)

part

partpi (36)

100 Mechanics

Therefore the dynamics is specified by assigning the vector field X0 on Γ0Now Γ0 can be interpreted as the cotangent space T lowastC0 Any cotangentspace carries a natural1 one-form θ0 = pidqi where dθ0 is nondegenerateA space equipped with such a one-form has the remarkable property thatevery function f determines a vector field Xf via the relation (dθ0)(Xf ) =minusdf A straightforward calculation shows that the flow defined by H0 isprecisely the time evolution vector field (36) Therefore the equations ofmotion (34)ndash(35) can be written simply2 as

(dθ0)(X0) = minusdH0 (37)

The two-form ω0 = dθ0 entering (37) is symplectic3 A dynamical systemis determined by a triple (Γ0 ω0 H0) where Γ0 is a manifold ω0 is asymplectic two-form and H0 is a function on Γ0

Presymplectic A very elegant formulation of mechanics and a crucialstep in the direction of the relativistic theory is provided by the presym-plectic formalism This formalism is based on the idea of describingmotions by using the graph of the function (qi(t) pi(t)) instead of thefunctions themselves The graph of the function (qi(t) pi(t)) is an un-parametrized curve γ in the (2m + 1)-dimensional space Σ = R times Γ0with coordinates (t qi pi) it is formed by all the points (t qi(t) pi(t)) inthis space The vector field

X =part

partt+ vi(qi pi)

part

partqi+ f i(qi pi)

part

partpi(38)

is tangent to all these curves (So is any other vector field obtained byscaling X namely any vector field X prime = fX where f is a scalar functionon Σ) Now consider the Poincare one-form

θ = pidqi minusH0(qi pi)dt (39)

on Σ The two-form ω = dθ is closed but it is degenerate (every two-formis degenerate in odd dimensions) that is there is a vector field X (calledthe null vector field of ω) satisfying

(dθ)(X) = 0 (310)

1It is defined intrinsically by θ0(X)(s) = s(πX) where X is a vector field on T lowastC0 s apoint in T lowastC0 and π the bundle projection

2The contraction between a two-form and a vector is defined by (αandβ)(X) = α(X)βminusβ(X)α

3That is closed and nondegenerate Closed means dω0 = 0 nondegenerate means thatω0(X) = 0 implies X = 0


The integral curves4 of the null vector field of a two-form ω are calledthe ldquoorbitsrdquo of ω It is easy to see that X given in (38) satisfies (310)Therefore the graphs of the motions are simply the orbits of dθ In otherwords (310) is a rewriting of the equations of motion

A space Σ equipped with a closed degenerate two-form ω is calledpresymplectic A dynamical system is thus completely defined by apresymplectic space (Σ ω) We use also the notation (Σ θ) where ω = dθ

Notice that (310) is homogeneous and therefore it determines X onlyup to scaling This is consistent with the fact that the vector field tangentto the motions is defined only up to scaling That is consistent with thefact that motions are represented by unparametrized curves in Σ

Finally it is easy to see that the action (31) is simply the line integral ofthe Poincare one-form (39) along the orbits if γ is an orbit (t qi(t) pi(t))of ω then the action of the motion qi(t) is

S[q] =int

γθ (311)

Extended Finally let me come to a formulation of dynamics that ex-tends naturally to general-relativistic systems In light of the presymplec-tic formulation described above it is natural to consider the relativisticconfiguration space

C = Rtimes C0 (312)

coordinatized by the m + 1 variables (t qi) and to describe motions withthe graphs of the functions qi(t) which are unparametrized curves in CConsider the cotangent space T lowastC with coordinates (t qi pt pi) and thefunction

H(t qi pt pi) = pt + H0(qi pi) (313)

on this space Let Σ be the surface in T lowastC defined by

H(qi t pi pt) = 0 (314)

We can coordinatize Σ with the coordinates (t qi pi) Since it is a cotan-gent space T lowastC carries a natural one-form which is

θ = pidqi + ptdt (315)

The restriction of this one-form to the surface (314) is precisely (39)Therefore the surface (314) is the presymplectic space that defines thedynamics

4An integral curve of a vector field is a curve everywhere tangent to the field

102 Mechanics

In other words the dynamics is completely defined by the couple (C H)a relativistic configuration space C and a function H on T lowastC The graphsof the motions are simply the orbits of dθ on the surface (314)5 I call Hthe relativistic hamiltonian

Remarkably the dynamics can be directly expressed in terms of a varia-tional principle based on (C H) An unparametrized curve γ in C describesa physical motion if γ extremizes the integral

S[γ] =int

γθ (316)

in the class of the curves γ in T lowastC satisfying (314) whose restriction γ toC connects two given points (t1 qi1) and (t2 qi2)

The relativistic configuration space C has the structure (312) and therelativistic hamiltonian H has the form (313) As we shall see the struc-ture (312)ndash(313) does not survive in the relativistic formulation of me-chanics

Relativistic phase space Denote Γ the space of the orbits of dθ in ΣThere is a natural projection π Σ rarr Γ that sends each point of Σ to thecurve to which it belongs It is not hard to show that there is one andonly one symplectic two-form ωph on Γ such that its pull-back to Σ is dθnamely πlowastωph = dθ Therefore Γ is a symplectic space Γ is the space ofthe physical motions I shall call it the relativistic phase space

The relation between the relativistic phase space Γ and the nonrela-tivistic phase space Γ0 = T lowastC0 is the following Γ0 is the space of theinstantaneous states the states that the system can have at a fixed timet = t0 On the other hand Γ is the space of all solutions of the equationsof motion Now fix a time say t = t0 If at t = t0 the system is in an ini-tial state in Γ0 it will then evolve in a well-defined motion The other wayaround each motion determines an instantaneous state at t = t0 There-fore there is a one-to-one mapping between Γ and Γ0 The identificationbetween Γ and Γ0 depends on the t0 chosen

HamiltonndashJacobi The HamiltonndashJacobi equation is

partS(qi t)partt

+ H0

(qi

partS(qi t)partqi

)= 0 (317)

If a family of solutions S(qi Qi t) depending on m parameters Qi is foundthen we can compute the function

Pi(qi Qi t) = minuspartS(qi Qi t)partQi

(318)

5More precisely the projections of these orbits on C


by simple derivation Inverting this function we obtain

qi(t) = qi(Qi Pi t) (319)

which are physical motions namely the general solution of the equationsof motion where the quantities (Qi Pi) are the 2m integration constants

Solutions of (317) can be found in the form S(qi Qi t) = EtminusW (qi Qi)where E is a constant and W satisfies

H0

(qi

partW (qi Qi)partqi

)= E (320)

S is called the principal HamiltonndashJacobi function W is called the char-acteristic HamiltonndashJacobi function

The HamiltonndashJacobi equation (317) can be obtained from the classicallimit of the Schrodinger equation

The Hamilton function Consider two points (t1 qi1) and (t2 qi2) in C Thefunction on G = C times C

S(t1 qi1 t2 qi2) =

int t2

t1

dt L(qi(t) qi(t)) (321)

where qi(t) is the physical motion from qi1(t1) to qi2(t2) (that minimizesthe action) is called the Hamilton function Equivalently

S(t1 qi1 t2 qi2) =

int

γθ (322)

where γ is the orbit into Σ that projects to qi(t) Notice the differencebetween the action (31) and the Hamilton function (321) the first is afunctional of the motion the second is a function of the end points It isnot hard to see that the Hamilton function solves the HamiltonndashJacobiequation (in both sets of variables) The Hamilton function is thereforea preferred solution of the HamiltonndashJacobi equation If we know theHamilton function we have solved the equations of motion because weobtain the general solution of the equations of motion in the form qi =qi(t Qi Pi T ) by simply inverting the function

Pi(t qi TQi) =partS(t qi TQi)

partQi(323)

with respect to qi The resulting function qi(t TQi Pi) is the generalsolution of the equations of motion where the integration constants arethe initial coordinate and momenta Qi Pi at time T

104 Mechanics

Thus the action defines a dynamical system the Hamilton function di-rectly gives all the motions6 The Hamilton function (321) is the classicallimit of the quantum mechanical propagator

Example a pendulum Let α be the lagrangian variable describing the elongation of asimple harmonic oscillator which I call ldquopendulumrdquo for simplicity The lagrangianis L(α v) = (mv22) minus (mω2α22) the nonrelativistic hamiltonian is H0(α p) =(p22m) + (mω2α22) The extended configuration space has coordinates (t α) andthe relativistic hamiltonian is

H(t α pt p) = pt +p2

2m+

mω2α2

2 (324)

Choose coordinates (t α p) on the constraint surface H = 0 which is therefore definedby pt = minusH0(α p) The restriction of the one-form θ = pt dt + pdα to this surface is

θ = pdαminus(

p2

2m+

mω2α2

2

)dt (325)

The presymplectic two-form is therefore

ω = dθ = dp and dαminus p

mdp and dtminusmω2α dα and dt (326)

The orbits are obtained by integrating the vector field

X = Xtpart

partt+ Xα

part

partα+ Xp

part

partp(327)

satisfying ω(X) = 0 Inserting (326) and (327) in ω(X) = 0 we get

ω(X) = Xt

(minus p

mdpminusmω2α dα

)+ Xα

(dp + mω2α dt

)+ Xp

(minusdα +

p

mdt

)

=(minus p

mXt + Xα

)dp +

(minusmω2αXt minusXp

)dα +

(mω2αXα +

p

mXp

)dt

= 0 (328)

Writing dt(τ)dτ = Xt dα(τ)dτ = Xα dp(τ)dτ = Xp equation (328) reads

dα(τ)

dτminus p

m

dt(τ)

dτ= 0 minusdp(τ)

dτminusmω2α

dt(τ)

dτ= 0 (329)

together with a third equation dependent on the first two Equation (329) can bewritten as

dα(t)

dt=

p

m

dp(t)

dt= minusmω2α (330)

which are the Hamilton equations of the pendulum We can write its general solutionin the form

α(t) = a eiωt + a eminusiωt (331)

The Hamilton function S(α1 t1 α2 t2) is the preferred solution of the HamiltonndashJacobiequation

partS(α t)

partt+

1

2m

(partS(α t)

partα

)2

+mω2α2

2= 0 (332)

6Hamilton (talking about himself in the third person) ldquoMr Lagrangersquos function statesthe problem Mr Hamiltonrsquos function solves itrdquo [107]


obtained by computing the action of the physical motion α(t) that goes from α(t1) = α1

to α(t2) = α2 This motion is given by (331) with

a =α1e

minusiωt2 + α2eminusiωt1

2i sin[(ω(t1 minus t2))] (333)

Inserting this in the action and integrating we obtain the Hamilton function

S(α1 t1 α2 t2) = mω2α1α2 minus (α2

1 + α22) cos[(ω(t1 minus t2))]

2 sin[(ω(t1 minus t2))] (334)

This concludes the short review of nonrelativistic mechanics I nowconsider the generalization of this formalism to relativistic systems



Is there a version of the notions of ldquostaterdquo and ldquoobservablerdquo broad enough to applynaturally to relativistic systems I begin by introducing the main notions and tools ofcovariant mechanics in the context of a simple system

The pendulum revisited Say we want to describe the small oscillations ofa pendulum To this aim we need two measuring devices a clock and adevice that reads the elongation of the pendulum Let t be the readingof the clock (in seconds) and α the reading of the device measuring theelongation of the pendulum (in centimeters) Call the variables t and α thepartial observables of the pendulum (I use also relativistic observables orsimply observables if there is no risk of confusion with the nonrelativisticnotion of observable which is different)

A useful observation is a reading of the time t and the elongation αtogether Thus an observation yields a pair (t α) Call a pair obtained inthis manner an event

Let C be the two-dimensional space with coordinates t and α CallC the event space of the pendulum (I use also relativistic configurationspace or simply configuration space if there is no risk of confusion withthe nonrelativistic configuration space C0 which is different)

Experience shows we can find mathematical laws characterizing se-quences of events This is the reason we can do science These laws havethe following form Call an unparametrized curve γ in C a motion of thesystem Perform a sequence of measurements of pairs (t α) and find thatthe points representing the measured pairs sit on a motion γ Then wesay that γ is a physical motion We express a motion as a relation in C

f(α t) = 0 (335)

Thus a motion γ is a relation or a correlation between partial observables

106 Mechanics

Then disturb the pendulum (push it with a finger) and repeat theentire experiment over At each repetition of the experiment a differentmotion γ is found That is a different mathematical relation of the form(335) is found Experience shows that the space of the physical motionsis very limited it is just a two-dimensional space Only a two-dimensionalspace of curves γ is realized in Nature

In the case of the small oscillations of a frictionless pendulum we cancoordinatize the physical motions by the two real numbers A ge 0 and0 le φ lt 2π and (335) is given by

f(α tA φ) = αminusA sin(ωt + φ) = 0 (336)

This equation gives a curve γ in C for each couple (A φ)Let Γ be the two-dimensional space of the physical motions coordina-

tized by A and φ Γ is the relativistic phase space of the pendulum (orthe space of the motions) A point in Γ is also called a relativistic state(Or a Heisenberg state or simply a state if there is no risk of confusionwith the nonrelativistic notion of state which is different)

Equation (336) is the mathematical law that captures the empiricalinformation we have on the pendulum This equation is the evolutionequation of the system The function f is the evolution function of thesystem

A relativistic state is determined by a couple (A φ) It determines acurve γ in the (t α) plane That is it determines a correlation betweenthe two partial observables t and α via (336) If we disturb the pendulumby interacting with it or if we start a new experiment over we have a newstate The state remains the same if we observe the pendulum and theclock without disturbing them (here we disregard quantum theory ofcourse)

Summarizing each state in the phase space Γ determines a correlationbetween the observables in the configuration space C The set of theserelations is captured by the evolution equation (336) namely by thevanishing of a function

f Γ times C rarr R (337)

The evolution equation f = 0 expresses all predictions that can be madeusing the theory Equivalently these predictions are captured by the sur-face f = 0 in the cartesian product of the phase space with the configu-ration space

General structure of the dynamical systems The (CΓ f) language de-scribed above is general It is sufficient to describe all predictions of con-ventional mechanics On the other hand it is broad enough to describe


general-relativistic systems All fundamental systems can be described (tothe accuracy at which quantum effects can be disregarded) by making useof these concepts

(i) The relativistic configuration space C of the partial observables

(ii) The relativistic phase space Γ of the relativistic states

(iii) The evolution equation f = 0 where f Γ times C rarr V

Here V is a linear space The state in the phase space Γ is fixed until thesystem is disturbed Each state in Γ determines (via f = 0) a motion γ ofthe system namely it describes a relation or a set of relations betweenthe observables in C

A motion is not necessarily a one-dimensional curve in C it can be asurface in C of any dimension k If k gt 1 we say that there is gaugeinvariance For a system with gauge invariance we call ldquomotionrdquo the mo-tion itself and any curve within it In this chapter we shall not deal muchwith systems with gauge invariance but we shall mention them whererelevant

Predictions are obtained as follows We first perform enough measure-ments to determine the state (In reality the state of a large system isoften ldquoguessedrdquo on the basis of incomplete observations and reasonableassumptions justified inductively) Once the state is so determined orguessed the evolution equation predicts all the possible events namelyall the allowed correlations between the observables in any subsequentmeasurement

In the example of the pendulum for instance the equation predicts thevalue of α that can be measured together with any given t or the valuesof t that can be measured together with any given α These predictionsare valid until the system is disturbed

The definitions of observable state configuration space and phase spacegiven here are different from the conventional definitions In particu-lar notions of instantaneous state evolution in time observable at afixed time play no role here These notions make no sense in a general-relativistic context For nonrelativistic systems the usual notions can berecovered from the definitions given The relation between the relativisticdefinitions considered here and the conventional nonrelativistic notions isdiscussed in Section 324

The task of mechanics is to find the (CΓ f) description for all phys-ical systems The first step kinematics consists in the specification ofthe observables that characterize the system Namely it consists in thespecification of the configuration space C and its physical interpretationPhysical interpretation means the association of coordinates on C with

108 Mechanics

measuring devices The second step dynamics consists in finding thephase space Γ and the function f that describe the physical motions ofthe system

In the next section I describe a relativistic hamiltonian formalism formechanics based on the relativistic notions of state and observable definedhere


Elementary physical systems can be described by hamiltonian mechanics7

Once the kinematics ndash that is the space C of the partial observables qa ndashis known the dynamics ndash that is Γ and f ndash is fully determined by givinga surface Σ in the space Ω of the observables qa and their momenta paThe surface Σ can be specified by giving a function H Ω rarr Rk Σ is thendefined by H = 08 Denote γ a curve in Ω (observables and momenta)and γ its restriction to C (observables alone) H determines the physicalmotions via the following

Variational principle A curve γ connecting the events qa1and qa2 is a physical motion if γ extremizes the action

S[γ] =int

γpa dqa (338)

in the class of the curves γ satisfying

H(qa pa) = 0 (339)

whose restriction γ to C connects qa1 and qa2

All (relativistic and nonrelativistic) hamiltonian systems can be formu-lated in this manner

If k = 1 H is a scalar function and is sometimes called the hamil-tonian constraint The case k gt 1 is the case in which there is gaugeinvariance In this case the system (339) is sometimes called the systemof the ldquoconstraint equationsrdquo I call H the relativistic hamiltonian or ifthere is no ambiguity simply the hamiltonian I denote the pair (C H)as a relativistic dynamical system The generalization to field theory isdiscussed in Section 33

The relativistic hamiltonian H is related to but should not be confusedwith the usual nonrelativistic hamiltonian denoted H0 in this book Halways exists while H0 exists only for nonrelativistic systems

7Perhaps because they are the classical limit of a quantum system8Different Hs that vanish on the same surface Σ define the same physical system


Indeed notice that this formulation of mechanics is similar to the ex-tended formulation of nonrelativistic mechanics defined in Section 31The novelty is that C and H do not have the structure (312)ndash(313) Thediscussion above shows that this structure is not necessary in order to havea well-defined physical interpretation of the formalism A nonrelativisticsystem is characterized by the fact that one of its partial observables qa

is singled out by having the special role of an independent variable tThis does not happen in a relativistic system The following simple ex-ample shows that the relativistic formulation of mechanics is a propergeneralization of standard mechanics

Timeless double pendulum I now introduce a genuinely timeless system which I willrepeatedly use as a simple model to illustrate the theory Consider a mechanical modelwith two partial observables say a and b whose dynamics is defined by the relativistichamiltonian

H(a b pa pb) = minus1

2

(p2a + p2

b + a2 + b2 minus 2E) (340)

where E is a constant The extended configuration space is C = R2 The constraintsurface has dimension 3 it is the sphere of radius

radic2E in T lowastC The phase space has

dimension 2 The motions are curves in the (a b) space For each state the theorypredicts the correlation between a and b

A straightforward calculation (see below) shows that the evolution equation deter-mined by H is an ellipse in the (a b) space

f(a bα β) =( a

sinα

)2

+( b

cosα

)2

+ 2a

sinα

b

cosαcosβ minus 2E2 sin2 β = 0 (341)

where α and β parametrize Γ Therefore motions are closed curves and in fact ellipsesin C The system does not admit a conventional hamiltonian formulation because fora nonrelativistic hamiltonian system motions in C = R times C0 are monotonic in t isin Rand therefore cannot be closed curves

The example is not artificial There exist cosmological models that have precisely thisstructure For instance we can identify a with the radius of a maximally symmetricuniverse and b with the spatially constant value of a field representing the mattercontent of that universe and adopt the approximation in which these are the only twovariables that govern the large-scale evolution of the universe Then the dynamics ofgeneral relativity reduces to a system with the structure (340)

The associated nonrelativistic system The system (340) can also be viewed as followsConsider a physical system which we denote the ldquoassociated nonrelativistic systemrdquoformed by two noninteracting harmonic oscillators Let me stress that the associatednonrelativistic system is a different physical system than the timeless double pendu-lum considered above The timeless double pendulum has one degree of freedom itsassociated nonrelativistic system has two degrees of freedom The partial observablesof the associated nonrelativistic system are the two elongations a and b and the timet The nonrelativistic hamiltonian that governs the evolution in t is

H0(a b pa pb) =1

2

(p2a + p2

b + a2 + b2 minus 2E) (342)

110 Mechanics

It it has the same form as the relativistic hamiltonian (340) of the timeless doublependulum9 The constant term 2E of course has no effect on the equations of motionit only redefines the energy Physically we can view the relation between the twosystems as follows Imagine that we take the associated nonrelativistic system but wedecide to ignore the clock that measures t we consider just measurements of the twoobservables a and b Furthermore assume that the energy of the double pendulum isconstrained to vanish namely

1

2

(p2a + p2

b + a2 + b2)

= E (344)

Then the observed relation between the measurements of a and b is described by therelativistic system (340)

Geometric formalism As for nonrelativistic hamiltonian mechanics theequations of motion can be expressed in an elegant geometric form Thevariables (qa pa) are coordinates on the cotangent space Ω = T lowastC Equa-tion (339) defines a surface Σ in this space The cotangent space carriesthe natural one-form

θ = padqa (345)

Denote θ the restriction of θ to the surface Σ The two-form ω = dθ onΣ is degenerate it has null directions The integral surfaces of these nulldirections are the orbits of ω on Σ Each such orbit projects from T lowastC toC to give a surface in C These surfaces are the motions

Consider the case k = 1 In this case Σ has dimension 2nminus1 the kernelof ω is generically one-dimensional and the motions are generically one-dimensional Let γ be a motion on Σ and X be a vector tangent to themotion then

ω(X) = 0 (346)

To find the motions we have just to integrate this equation Equation(346) is the equation of motion X is defined by the homogeneous equa-tion (346) only up to a multiplicative factor Therefore the tangent of theorbit is defined only up to a multiplicative factor and so the parametriza-tion of the orbit is not determined by (346)

The case k gt 1 is analogous In this case Σ has dimension 2n minus kthe kernel of ω is generically k-dimensional and the motions are generi-cally k-dimensional X is then a k-dimensional multi-tangent and it stillsatisfies (346)

Let π Σ rarr Γ be the projection map that associates with each pointof the constraint surface the motion to which the point belongs The

9The relativistic hamiltonian of the associated nonrelativistic system is

H(a b t pa pb pt) = pt +1

2

(p2a + p2

b + a2 + b2 minus 2E) (343)


projection π equips the phase space Γ with a symplectic two-form ωph

defined to be the two-form whose pull-back to Σ under π is ω Locally itexists and it is unique precisely because ω is degenerate along the orbits

Relation with the variational principle Let γ be an orbit of ω on Σ such that itsrestriction γ in C is bounded by the initial and final events q1 and q2 Let γprime be a curvein Σ infinitesimally close to γ such that its restriction γprime is also bounded by q1 and q2Let δs1 (and δs2) be the difference between the initial (and final) points of γ and γprimeThe four curves γ δs1 minusγprime and minusδs2 form a closed curve in Σ Consider the integral ofω over the infinitesimal surface bounded by this curve This integral vanishes becauseat every point of the surface one of the tangents is (to first order) a null directionof ω (the surface is a strip parallel to the motion γ) But ω = dθ and therefore byStokes theorem the integral of θ along the closed curve vanishes as well The integralof θ = padq

a along δs1 and δs2 is zero because qa is constant along these segmentsTherefore int

γ

θ +

int

minusγprimeθ = 0 (347)

or

δ

int

γ

θ = 0 (348)

for any variation in the class considered This is precisely the variational principlestated in Section 32

Hamilton equations Consider first the case k = 1 Motions are one-dimensional Parametrize the curve with an arbitrary parameter τ Thatis describe a motion (in Ω) with the functions (qa(τ) pa(τ)) These func-tions satisfy the Hamilton system

H(qa pa) = 0 (349)

dqa(τ)dτ

= N(τ) va(qa(τ) pa(τ))

dpa(τ)dτ

= N(τ) fa(qa(τ) pa(τ)) (350)

where

va(qa pa) =partH(qa pa)

partpa fa(qa pa) = minuspartH(qa pa)

partqa (351)

The function N(τ) is called the ldquoLapse functionrdquo It is arbitrary Differ-ent choices of N(τ) determine different parameters τ along the motionTo obtain a monotonic parametrization we need N(τ) gt 0 A preferredparametrization can be obtained by taking N(τ) = 1 that is replacing(350)ndash(351) by the equations (written in the usual compact form)

qa =partH

partpa pa = minuspartH

partqa (352)

112 Mechanics

where the dot indicates derivative with respect to τ This choice is calledthe Lapse = 1 gauge It is not preferred in a physical sense In particu-lar different but physically equivalent hamiltonians H defining the samesurface Σ determine different preferred parametrizations Nevertheless itis often the easiest gauge to compute with

If k gt 1 the function H has components Hj with j = 1 k and motions arek-dimensional surfaces We can parametrize a motion with k arbitrary parameters τ =τj Namely we can represent it using the 2n functions qa(τ) pa(τ) of k parametersτj These equations satisfy the system given by (349) and

partqa(τ)

partτj= Nj(τ)

partHj(qa pa)

partpa

partpa(τ)

partτj= minusNj(τ)

partHj(qa pa)

partqa (353)

A motion is determined by the full k-dimensional surface in C we can choose a particularcurve τ(τ) on this surface where τ is an arbitrary parameter and represent the motionby the one-dimensional curve qa(τ) = qa(τ(τ)) in C This satisfies the system formedby (349) and

dqa(τ)

dτ= Nj(τ)

partHj(qa pa)

partpa

dpa(τ)

dτ= minusNj(τ)

partHj(qa pa)

partqa(354)

for k arbitrary functions of one variable Nj(τ) Different choices of the functions Nj(τ)determine different curves on the single surface that defines a motion These are gauge-equivalent representations of the same motion

It is important to stress that the parameters τ or τj are an artifact ofthis technique They have no physical significance They are absent in thegeometric formalism as well as in the HamiltonndashJacobi formalism as weshall see below The physical content of the theory is in the motion in Cnot in the way the motion is parametrized That is the physical informa-tion is not in the functions qa(τ) it is in the image of these functions inC

Relation with the variational principle Parametrize the curve γ with a parameter τ The action (338) reads

S =

intdτ pa(τ)

dqa(τ)

dτ (355)

The constraint (339) can be implemented in the action with lagrange multipliers Ni(τ)This defines the action

S =

intdτ

(pa

dqa

dτminusNi H

i(pa qa)

) (356)

Varying this action with respect to Ni(τ) qa(τ) and pa(τ) gives the Hamilton equation(349) (354)

Example double pendulum Consider the system defined by the hamiltonian (340)The Hamilton equations (349) (352) in the Lapse = 1 gauge give

a = pa b = pb pa = minusa pb = minusb a2 + b2 + p2a + p2

b = 2E (357)


The general solution is

a(τ) = Aa sin(τ) b(τ) = Ab sin(τ + β) (358)

where Aa =radic

2E sinα and Ab =radic

2E cosα The motions are given by the image in Cof these curves These are the ellipses (341) The parametrization of the curves (358)has no physical significance The physics is in the unparametrized ellipses in C and inthe relation between a and b they determine

HamiltonndashJacobi HamiltonndashJacobi formalism is elegant general andpowerful it has a direct connection with quantum theory and is con-ceptually clear The relativistic formulation of HamiltonndashJacobi theory issimpler than the conventional nonrelativistic version indicating that therelativistic formulation unveils a natural and general structure of mechan-ical systems

The relativistic HamiltonndashJacobi formalism is given by the system of kpartial differential equations

H

(qa

partS(qa)partqa

)= 0 (359)

for the function S(qa) defined on the extended configuration space C LetS(qa Qi) be a family of solutions parametrized by the nminus k constants ofintegration Qi Pose

f i(qa Pi Qi) equiv partS(qa Qi)

partQi+ Pi = 0 (360)

for n minus k arbitrary constants Pi This is the evolution equation Theconstants Qi Pi coordinatize a 2(nminus k)-dimensional space Γ This is thephase space

The form of the relativistic HamiltonndashJacobi equation (359) is simplerthan the usual nonrelativistic HamiltonndashJacobi equation (317) Further-more there is no equation to invert as in the nonrelativistic formalismNotice also that the function S(qa Qi) can be identified with the principalHamiltonndashJacobi function S(t qi Qi) = Et+W (qi Qi) of the nonrelativis-tic formalism as well as with the characteristic HamiltonndashJacobi functionW (qi Qi) since (359) is formally like (320) with vanishing energy Thetwo functions are in fact identified in the relativistic formalism

Example double pendulum The HamiltonndashJacobi equation of the timeless system(340) is

(partS(a b)

parta

)2

+

(partS(a b)

partb

)2

+ a2 + b2 minus 2E = 0 (361)

114 Mechanics

A one-parameter family of solutions is given by

S(a b A) =a

2

radicA2 minus a2 +

A2

2arctan

(aradic

A2 minus a2

)

+b

2

radic2E minusA2 minus b2 +

2E minusA2

2arctan

(bradic

2E minusA2 minus b2

) (362)

The general solution (341) of the system is directly obtained by writing

partS(a b A)

partAminus φ = 0 (363)

where φ is an integration constant

Derivation of the HamiltonndashJacobi formalism Since the phase space Γ is a symplecticspace we can locally choose canonical coordinates (Qi Pi) over it These coordinatescan be pulled back to Σ where they are constant along the orbits In fact they labelthe orbits Let θph = PidQ

i therefore dθph = ω But ω = dθ = d(padqa) so on Σ we

have

d(θph minus θ) = d(PidQi minus padq

a) = 0 (364)

This implies that there should locally exist a function S on Σ such that

PidQi minus padq

a = minusdS (365)

Let us choose qa and Qi as independent coordinates on Σ Then (365) reads

dS(qa Qi) = pa(qa Qi)dqa minus Pi(q

a Qi)dQi (366)

that is

partS(qa Qi)

partqa= pa(q

a Qi) (367)

partS(qa Qi)

partQi= minusPi(q

a Qi) (368)

By the definition of Σ we have H(qa pa) = 0 which using (367) gives the HamiltonndashJacobi equation (359) Equation (368) is then immediately the evolution equation(360)

In other words S(qa Qi) is the generating function of a canonical transformationthat relates the observables and their momenta (qa pa) to new canonical variables(Qi Pi) satisfying Qi = 0 Pi = 0 These new variables are constants of motion andtherefore define Γ The relation between C and Γ given by the canonical transformationequations (367)ndash(368) is the evolution equation


Here I discuss in more detail how the notions and the structures of conven-tional mechanics described in Section 31 are recovered from the relativis-tic formalism A nonrelativistic system is simply a relativistic dynamicalsystem in which one of the partial observables qa is denoted t and calledldquotimerdquo and the hamiltonian H has the form

H = pt + H0 (369)

partt+ X0 (374)


where H0 is independent from pt and is called the nonrelativistic hamil-tonian The quantity E = minuspt is called energy The device that measuresthe partial observable t is called a clock

The relativistic configuration space therefore has the structure

C = Rtimes C0 (370)

with coordinates qa = (t qi) where i = 1 nndash1 The space C0 is theusual nonrelativistic configuration space Accordingly the cotangent spaceΩ = T lowastC has coordinates (qa pa) = (t qi pt pi)

If H has the form (369) the relativistic HamiltonndashJacobi equation(359) becomes the conventional nonrelativistic HamiltonndashJacobi equation(317)

Given a state and a value t of the clock observable we can ask whatare the possible values of the observables qi such that (qi t) is a possibleevent That is we can ask what is the value of qi ldquowhenrdquo the time is t Thesolution is obtained by solving the evolution function f i(qi tQi Pi) = 0for the qi This gives

qi = qi(tQi Pi) (371)

which is interpreted as the evolution equation of the variables qi in thetime t The form (369) of the hamiltonian guarantees that we can solve fwith respect to the qi because the Hamilton equation for t (in the gaugeLapse = 1) is simply t = τ which can be inverted

In the parametrized hamiltonian formalism the evolution equationfor t(τ) is trivial and gives taking advantage of the freedom in rescal-ing τ just t = τ Using this equations (353) become the conventionalHamilton equations and (349) simply fixes the value of pt namely theenergy

In the presymplectic formalism the surface Σ turns out to be

Σ = Rtimes Γ0 (372)

where the coordinate on R is the time t and Γ0 = T lowastC0 is the nonrela-tivistic phase space The restriction of θ to this surface has the Cartanform

θ = pidqi minusH0dt = θ0 minusH0dt (373)

We can take the vector field X to have the form

X =part

116 Mechanics

where X0 is a vector field on Γ0 Then the equation of motion (346)reduces to the equation

(dθ0)(X0) = minusdH0 (375)

which is the geometric form of the conventional Hamilton equations ThusH determines how the variables in Γ0 are correlated to the variable t Thatis ldquohow the variables in Γ0 evolve in timerdquo In this sense the nonrelativis-tic hamiltonian H0 generates ldquoevolution in the time trdquo This evolution isgenerated in Γ0 by the hamiltonian flow X0 of H0 A point s = (qi pi) inΓ0 is taken to the point s(t) = (qi(t) pi(t)) where

ds(t)dt

= X0(s(t)) (376)

The evolution of an observable (not depending explicitly on time) de-fined by At(s) = A(s(t)) = A(s t) can be written introducing the Poissonbracket notation

AB = minusXA(B) = XB(A) =sum

i

(partA

partqipartB

partpiminus partA

partpi

partB

partqi

) (377)

asdAt

dt= At H0 (378)

Instantaneous states and relativistic states The nonrelativistic definitionof state refers to the properties of a system at a certain moment of timeDenote this conventional notion of state as the ldquoinstantaneous staterdquo Thespace of the instantaneous states is the conventional nonrelativistic phasespace Γ0 Letrsquos fix the value t = t0 of the time variable and characterizethe instantaneous state in terms of the initial data For the pendulumthese are position and momentum (α0 p0) at t = t0 Thus (α0 p0) arecoordinates on Γ0

On the other hand a relativistic state is a solution of the equations ofmotion (If there is gauge invariance a state is a gauge equivalence classof solutions of the equations of motion) The relativistic phase space Γ isthe space of the solutions of the equations of motion

Given a value t0 of the time there is a one-to-one correspondence be-tween initial data and solutions of the equations of motion each solutionof the equation of motion determines initial data at t = t0 and eachchoice of initial data at t0 determines uniquely a solution of the equationsof motion Therefore there is a one-to-one correspondence between in-stantaneous states and relativistic states Therefore the relativistic phasespace Γ is isomorphic to the nonrelativistic phase space Γ sim Γ0 How-ever the isomorphism depends on the time t0 chosen and the physical


interpretation of the two spaces is quite different One is a space of statesat a given time the other a space of motions

In the case of the pendulum the nonrelativistic phase space Γ0 can becoordinatized with (α0 p0) the relativistic phase space Γ can be coordina-tized with (A φ) The identification map (A φ) rarr (α0 p0) is given by

α0(A φ) = A sin(ωt0 + φ) (379)p0(A φ) = ωmA cos(ωt0 + φ) (380)

The nonrelativistic phase space Γ0 plays a double role in nonrelativistichamiltonian mechanics it is the space of the instantaneous states but itis also the arena of nonrelativistic hamiltonian mechanics over which H0

is defined In the relativistic context this double role is lost one mustdistinguish the cotangent space Ω = T lowastC over which H is defined fromthe phase space Γ which is the space of the motions This distinction willbecome important in field theory where Ω is finite-dimensional while Γ isinfinite-dimensional

In a nonrelativistic system X0 generates a one-parameter group oftransformations in Γ0 the hamiltonian flow of H0 on Γ0 Instead of havingthe observables in C0 depending on t one can shift perspective and viewthe observables in C0 as time-independent objects and the states in Γ0

as time-dependent objects This is a classical analog of the shift fromthe Heisenberg to the Schrodinger picture in quantum theory and can becalled the ldquoclassical Schrodinger picturerdquo

In the relativistic theory there is no special ldquotimerdquo variable C doesnot split naturally as C = R times C0 the constraints do not have the formH = pt +H0 and the description of the correlations in terms of ldquohow thevariables in C0 evolve in timerdquo is not available in general In a system thatdoes not admit a nonrelativistic formulation the classical Schrodingerpicture in which states evolve in time is not available only the relativisticnotions of state and observable make sense

Special-relativistic systems There are relativistic systems that do not ad-mit a nonrelativistic formulation such as the example of the double pen-dulum discussed above There are also systems that can be given a nonrel-ativistic formulation but their structure is far more clean in the relativis-tic formalism Lorentz-invariant systems are typical examples They canbe formulated in the conventional hamiltonian picture only at the priceof breaking Lorentz invariance The choice of a preferred Lorentz framespecifies a preferred Lorentz time variable t = x0 The predictions of thetheory are Lorentz invariant but the formalism is not This way of deal-ing with the mechanics of special-relativistic systems hides the simplicityand symmetry of its hamiltonian structure The relativistic hamiltonian

118 Mechanics

formalism exemplified below for the case of a free particle is manifestlyLorentz invariant

Example relativistic particle The configuration space C is a Minkowski space M withcoordinates xμ The dynamics is given by the hamiltonian H = pμpμ+m2 which definesthe mass-m Lorentz hyperboloid Km The constraint surface Σ is therefore given byΣ = T lowastM|H=0 = MtimesKm The null vectors of the restriction of dθ = dpμ and dxμ to Σare

X = pμpart

partxμ (381)

because ω(X) = pμdpμ = 2d(p2) = 0 on pμpμ = minusm2 The integral lines of X namelythe lines whose tangent is X are

xμ(τ) = Pμτ + Xμ pμ(τ) = Pμ (382)

which give the physical motions of the particle The space of these lines is six-dimensional (it is coordinatized by the eight numbers (Xμ Pμ) but PμPμ = minusm2

and (Pμ Xμ) defines the same line as (Pμ Xμ + Pμa) for any a) and represents thephase space The motions are thus the timelike straight lines in M

Notice that all notions used are completely Lorentz invariant A state is a time-like geodesic an observable is any Minkowski coordinate a correlation is a point inMinkowski space The theory is about correlations between Minkowski coordinatesthat is observations of the particle at certain spacetime points On the other handthe split M = RtimesR3 necessary to define the usual hamiltonian formalism is observerdependent

The relativistic formulation of mechanics is not only more general butalso more simple and elegant and better operationally founded than theconventional nonrelativistic formulation This is true whether one usesthe Hamilton equations the geometric language or the HamiltonndashJacobiformalism

324 Discussion mechanics is about relations between observables

The key difference between the relativistic formulation of mechanics dis-cussed in this chapter and the conventional one ndash and in particular be-tween the relativistic definitions of state and observable and the conven-tional ones ndash is the role played by time In the nonrelativistic context timeis a primary concept Mechanics is defined as the theory of the evolutionin time In the definition considered here on the other hand no specialpartial observable is singled out as the independent variable Mechanicsis defined as the theory of the correlations between partial observables

Technically C does not split naturally as C = R times C0 the constraintsdo not have the form H = pt + H0 and the Schrodinger-like descriptionof correlations in terms of ldquohow states and observables evolve in timerdquo isnot available in general


It is important to understand clearly the meaning of this shift of per-spective

The first point is that it is possible to formulate conventional mechanicsin this time-independent language In fact the formalism of mechanicsbecomes even more clean and symmetric (for instance Lorentz covariant)in this language This is a remarkable fact by itself What is remarkableis that the formal structure of mechanics doesnrsquot really treat the timevariable on a different footing than the other variables The structure ofmechanics is the formalization of what we have understood about thephysical structure of the world Therefore we can say that the physical(more precisely mechanical) structure of the world is quite blind to thefact that there is anything ldquospecialrdquo about the variable t

Historically the idea that in a relativistic context we need the time-independent notion of state has been advocated particularly by Dirac(see [148] in Chapter 5) and by Souriau [105] The advantages of therelativistic notion of state are multi-fold In special relativity for instancetime transforms with other variables and there is no covariant definitionof instantaneous state In a Lorentz-invariant field theory in particularthe notion of instantaneous state breaks explicit Lorentz covariance theinstantaneous state is the value of the field on a simultaneity surfacewhich is such for a certain observer only The relativistic notion of stateon the other hand is Lorentz invariant

The second point is that this shift in perspective is forced in gen-eral relativity where the notion of a special spacelike surface over whichinitial data are fixed conflicts with diffeomorphism invariance A gen-erally covariant notion of instantaneous state or a generally covariantnotion of observable ldquoat a given timerdquo makes little physical sense In-deed none of the various notions of time that appear in general rel-ativity (coordinate time proper time clock time) play the role that tplays in nonrelativistic mechanics A consistent definition of state andobservable in a generally covariant context cannot explicitly involvetime

The physical reason for this difference is discussed in Chapter 2 Innonrelativistic physics time and position are defined with respect to asystem of reference bodies and clocks that are implicitly assumed to ex-ist and not to interact with the physical system studied In gravitationalphysics one discovers that no body or clock exists which does not inter-act with the gravitational field the gravitational field affects directly themotion and the rate of any reference body or clock Therefore one cannotseparate reference bodies and clocks from the dynamical variables of thesystem General relativity ndash in fact any generally covariant theory ndash isalways a theory of interacting variables that necessarily include the phys-ical bodies and clocks used as references to characterize spacetime points

120 Mechanics

In the example of the pendulum discussed in Section 321 for instancewe can assume that the pendulum and the clock do not interact In ageneral-relativistic context the two always interact and C does not splitinto C0 and R

Summarizing it is only in the nonrelativistic limit that mechanics canbe seen as the theory of the evolution of the physical variables in time Ina fully relativistic context mechanics is a theory of correlations betweenpartial observables


I describe here the relativistic version of a structure that plays an impor-tant role in the quantum theory

Hamilton function Notice that the Hamilton function defined in (321)is naturally a function on (two copies of) the relativistic configurationspace C In fact its definition extends to the relativistic context giventwo events qa and qa0 in C the Hamilton function is defined as

S(qa qa0) =int

γθ (383)

where γ is the orbit in Σ of the motion that goes from qa0 to qa This is alsothe value of the action along this motion For instance for a nonrelativisticsystem we can write

S(qa qa0) =int

γθ =

int

γpadqa (384)

=int 1

0pa(τ)qa(τ)dτ =

int 1

0

(pi(τ)qi(τ) + pt(τ)t(τ)

)dτ

=int 1

0

(pi(τ)qi(τ) minusH0(τ)t(τ)

)dτ

=int t

t0

(pi(t)

dqi(t)dt

minusH0(t))

dt

=int t

t0

L

(qi

dqi(t)dt

)dt (385)

where L is the lagrangian From the definition we have

partS(qa qa0)partqa

= pa(qa qa0) (386)

where pa(qa qa0) is the value of the momentum at the final event Notice


that this value depends on qa as well as on qa0 The derivation of thisequation is less obvious than appears at first sight I leave the details tothe acute reader

It follows from (386) that S(qa qa0) satisfies the HamiltonndashJacobi equa-tion (359) The quantities qa0 can be seen as the HamiltonndashJacobi inte-gration constants Notice that they are n not nminus1 Equations (360) nowread

fa(qa qa0 pa0) =partS(qa qa0)

partqa0+ pa0 = 0 (387)

Therefore the phase space is directly (over-)coordinatized by initial co-ordinates and momenta (qa0 pa0) These are not independent for tworeasons First they satisfy the equation H = 0 Second different sets(qa0(τ) pa0(τ)) along the same motion determine the same motion Fur-thermore one of the equations (387) turns out to be dependent on theothers

S(qa qa0) satisfies the HamiltonndashJacobi equation in both sets of vari-ables namely it satisfies also

H

(qa0 minus

partS(qa qa0)partqa0

)= 0 (388)

where the minus sign comes from the fact that the second set of variablesis in the lower integration boundary in (383)

If there is more than one physical motion γ connecting the boundarydata the Hamilton function is multivalued If γ1 γn are distinct so-lutions with the same boundary values we denote its different branchesas

Si(qa1 qa2) =

int

γi

θ (389)

The Hamilton function is strictly related to the quantum theory It isthe phase of the propagator W (qa qa0) which as we shall see in Chapter5 is the main object of the quantum theory If S is single valued we have

W (qa qa0) sim A(qa qa0) eiS(qaqa0 ) (390)

up to higher terms in If S is multivalued

W (qa qa0) simsum

i

Ai(qa qa0) eiSi(q

aqa0 ) (391)

122 Mechanics

Example free particle In the case of the free particle the value of the classical actionalong the motion is

S(x t x0 t0) =

int 1

0

(pt t + px)dt = pt

int t

t0

dt + p

int x

x0

dx

= minusm(xminus x0)2

2(tminus t0)+ m

(xminus x0)2

tminus t0

=m(xminus x0)

2

2(tminus t0) (392)

It is easy to check that S solves the HamiltonndashJacobi equation of the free particle Thefirst of the two equations (387) gives the evolution equation

partS(x t x0 t0)

partx0+ p0 = minusm

xminus x0

tminus t0+ p0 = 0 (393)

The second equation constrains the pt integration constant

partS(x t x0 t0)

partt0+ pt0 = minus 1

2mp20 + pt0 = 0 (394)

Recall that the propagator of the Schrodinger equation of the free particle is

W (x t x0 t0) =1

radici(tminus t0)

ei

m(xminusx0)2

2(tminust0) =1

radici(tminus t0)

eiS(xtx0t0) (395)

Example double pendulum The Hamilton function of the timeless system (340) canbe computed directly from its definition This gives

S(a b aprime bprime) = S(a b aprime bprime A(a b aprime bprime)

) (396)

where

S(a b aprime bprime A) = S(a b A) minus S(aprime bprime A) (397)

S(a b A) is given in (362) and A(a b aprime bprime) is the value of A of the ellipse (341) thatcrosses (a b) and (aprime bprime) This value can be obtained by noticing that (358) imply withlittle algebra that

A2 =a2 + aprime2 minus 2aaprime cos τ

sin2 τ(398)

and

E =(a2 + b2 + aprime2 + bprime2) minus 2(aaprime + bbprime) cos τ

sin2 τ (399)

The second equation can be solved for τ(a b aprime bprime) and inserting this in the first givesA(a b aprime bprime) It is not complicated to check that the derivative of partS(a b aprime bprime A)partAvanishes when A = A(a b aprime bprime) Using this it is easy to see that (396) solves theHamiltonndashJacobi equation in both sets of variables

Notice that for given (a b aprime bprime) equation (398) gives A as a function of τ We cantherefore consider also the function

S(a b aprime bprime τ) = S(a b A(τ)) minus S(aprime bprime A(τ)) (3100)

which is the value of the action of the nonrelativistic system formed by two harmonicoscillators evolving in a physical time τ with a nonrelativistic hamiltonian H that is


it is the Hamilton function of this system With some algebra this can be written alsoas

S(a b aprime bprime τ) = Mτ +(a2 + b2 + aprime2 + bprime2) cos τ minus 2(aaprime + bbprime)

sin τ (3101)

As for A we have immediately

partS(a b aprime bprime τ)

partτ

∣∣∣∣τ=τ(abaprimebprime)

= 0 (3102)

This means that the Hamilton function of the timeless system is numerically equalto the Hamilton function of the two oscillators for the ldquocorrectrdquo time τ needed to gofrom (aprime bprime) to (a b) staying on a motion of total energy E And that this ldquocorrectrdquotime τ = τ(a b aprime bprime) is the one that minimizes the Hamilton function of the twooscillators

More precisely for given (a b aprime bprime) there are two paths connecting (aprime bprime) with (a b)these are the two paths in which the ellipse that goes through (aprime bprime) and (a b) is cutby these two points Denote S1 and S2 the two values of the action along these pathsTheir relation is easily obtained by noticing that the action along the entire ellipse iseasily computed as

S1 + S2 = 2πE (3103)

The space of the boundary data G The Hamilton function is a functionon the space G = C times C An element α isin G is an ordered pair of elementsof the extended configuration space C α = (qa qa0) Notice that α isthe ensemble of the boundary conditions for a physical motion For anonrelativistic system α = (t qi t0 qi0) the motion begins at qi0 at timet0 and ends at qi at time t

The space G carries a natural symplectic structure In fact let i G rarr Γbe the map that sends each pair to the orbit that the pair defines Thenwe can define the two-form ωG = ilowastωph where ωph is the symplecticform of the phase space defined in Section 322 In other words α =(qa qa0) can be taken as a natural over-coordinatization of the phase spaceInstead of coordinatizing a motion with initial positions and momentawe coordinatize it with initial and final positions In these coordinatesthe symplectic form is given by ωG

The two-form ωG can be computed without having first to computeΓ and ωph Denote γα the orbit in Σ with boundary data α and γα itsprojection to C Then α is the boundary of γα We write α = partγα Denotes and s0 the initial and final points of γα in Σ That is s = (qa pa) ands0 = (qa0 p0a) where in general both pa and p0a depend on qa and onqa0 Let δα = (δqa δqa0) be a vector (an infinitesimal displacement) at αThen the following is true

ωG(α)(δ1α δ2α) = ωG(qa qa0)((δ1qa δ1qa0) (δ2qa δ2qa0))= ω(s)(δ1s δ2s) minus ω(s0)(δ1s0 δ2s0) (3104)

124 Mechanics

Notice that δ1s the variation of s is determined by δ1q as well as byδ1q0 and so on This equation expresses ωG directly in terms of ω As weshall see this equation admits an immediate generalization in the fieldtheoretical framework where ω will be a five-form and ωG is a two-form

Now fix a pair α = (qa qa0) and consider a small variation of only oneof its elements say

δα = (δqa 0) (3105)

This defines a vector δα at α on G which can be pushed forward toΓ If the variation is along the direction of the motion then the pushforward vanishes that is ilowastδα = 0 because α and α + δα define the samemotion It follows that if the variation is along the direction of the motionωG(δα) = 0 Therefore the equation

ωG(X) = 0 (3106)

gives the solutions of the equations of motionThus the pair (G ωG) contains all the relevant information of the sys-

tem The null directions of ωG define the physical motions and if we divideG by these null directions the factor space is the physical phase spaceequipped with the physical symplectic structure

Example free particle The space G has coordinates α = (t x t0 x0) Given this pointin G there is one motion that goes from (t0 x0) to (t x) which is

t(τ) = t0 + (tminus t0)τ (3107)

x(τ) = x0 + (xminus x0)τ (3108)

Along this motion

p = mxminus x0

tminus t0 (3109)

pt = minusm(xminus x0)2

2(tminus t0)2 (3110)

The map i G rarr Γ is thus given by

P = p = mxminus x0

tminus t0 (3111)

Q = xminus p

mt = xminus xminus x0

tminus t0t (3112)

and therefore the two-form ωG is

ωG = ilowastωΓ = dP (t x t0 x0) and dQ(t x t0 x0)

= m dxminus x0

tminus t0and d

(xminus xminus x0

tminus t0t

)

=m

tminus t0

(dxminus xminus x0

tminus t0dt

)and

(dx0 minus xminus x0

tminus t0dt0

) (3113)


Immediately we see that a variation δα = (δt δx 0 0) (at constant (x0 t0)) such thatωG(δα) = 0 must satisfy

δx =xminus x0

tminus t0δt (3114)

This is precisely a variation of x and t along the physical motion (determined by(x0 t0)) Therefore ωG(δα) = 0 gives again the equations of motion The two nulldirections of ωG are thus given by the two vector fields

X =xminus x0

tminus t0partx + partt (3115)

X0 =xminus x0

tminus t0partx0 + partt0 (3116)

which are in involution (their Lie bracket vanishes) and therefore define a foliation ofG with two-dimensional surfaces These surfaces are parametrized by P and Q givenin (3111) (3112) and in fact

X(P ) = X(Q) = X0(P ) = X0(Q) = 0 (3117)

We have simply recovered in this way the physical phase space the space of thesesurfaces is the phase space Γ and the restriction of ωG to it is the physical symplecticform ωph

Physical predictions from S There are several different ways of derivingphysical predictions from the Hamilton function S(qa qa0)

bull Equation (387) gives the evolution function f in terms of the Hamil-ton function

bull If we can measure the partial observables qa as well as their momentapa then the Hamilton function can be used for making predictionsas follows Let

p1a(q

a1 q

a2) =

partS(qa1 qa2)

partqa1

p2a(q

a1 q

a2) =

partS(qa1 qa2)

partqa2 (3118)

The two equations

p1a = p1

a(qa1 q

a2)

p2a = p2

a(qa1 q

a2) (3119)

relate the four partial observables of the quadruplet (qa1 p1a q

a2 p

2a)

The theory predicts that it is possible to observe the quadruplet(qa1 p

1a q

a2 p

2a) only if this satisfies (3119) In this way the classical

theory determines which combinations of values of partial observ-ables can be observed

126 Mechanics

bull Alternatively we can fix two points qai and qaf in C and ask whethera third point qa is on the motion determined by qai and qaf That isask whether or not we could observe the correlation qa given thatthe correlations qai and qaf are observed A moment of reflection willconvince the reader that if the answer to this question is positivethen

S(qaf qa) + S(qa qai ) = S(qaf q

ai ) (3120)

because the action is additive along the motion Furthermore theincoming momentum at qa and the outgoing one must be equaltherefore

partS(qaf qa)

partqa= minus

partS(qa qaf )partqa

(3121)


A physical system is often defined by an action which is the integral of alagrangian in an evolution parameter But there are two different physicalmeanings that the evolution parameter may have

We have seen that the variational principle governing any hamiltoniansystem can be written in the form (here k = 1)

S =int

dτ(pa

dqa

dτminusNH(pa qa)

) (3122)

The action is invariant under reparametrizations of the evolution param-eter τ The evolution parameter τ has no physical meaning there is nomeasuring device associated with it

On the other hand consider a nonrelativistic system where qa = (t qi)and H = pt + H0 The action (3122) becomes

S =int

dτ(pt

dtdτ

+ pidqi

dτminusN(pt + H0(pi qi))

) (3123)

Varying N we obtain the equation of motion

pt = minusH0 (3124)

Inserting this relation back into the action we obtain

S =int

dτ(minusH0

dtdτ

+ pidqi

dτ

) (3125)


We can now change the integration variable from τ to t(τ) Defining (inbad physicistsrsquo notation) qi(t) equiv qi(τ(t)) and so on we can write

S =int

dτdtdτ

(minusH0 + pi

dqi

dt

)=

intdt

(pi

dqi(t)dt

minusH0

) (3126)

The evolution parameter in the action is no longer an arbitrary unphysicalparameter τ It is one of the partial observables the time observable t

If we are given an action we must understand whether the evolutionparameter in the action is a partial observable such as t or an unphysicalparameter such as τ If the action is invariant under reparametrizationsof its evolution parameter then the evolution parameter is unphysical Ifit is not then the evolution parameter is a partial observable

The same is true if the action is given in lagrangian form In performingthe Legendre transform from the lagrangian to the hamiltonian formalismthe consequence of the invariance of the action under reparametrizationsis doublefold First the relation between velocities and momenta cannotbe inverted The map from the space of the coordinates and velocities(qa qa) to the space of coordinates and momenta (qa pa) is not invertibleThe image of this map is a subspace Σ of Ω and we can characterize Σby means of an equation H = 0 for a suitable hamiltonian H Secondthe canonical hamiltonian computed via the Legendre transform vanisheson Σ In the language of constrained system theory this is because thecanonical hamiltonian generates evolution in the parameter of the actionsince this is unphysical this evolution is gauge the generator of a gaugeis a constraint and therefore vanishes on Σ

The evolution parameter in the action is often denoted t whether it isa partial observable or an unphysical parameter One should not confusethe t in the first case with the t in the second case They have verydifferent physical interpretations The time coordinate t in Maxwell theoryis a partial observable The time coordinate t in GR is an unphysicalparameter The fact that the two are generally denoted with the sameletter and with the same name is a very unfortunate historical accident

Example relativistic particle As we have seen the hamiltonian dynamics of a relativis-tic particle is defined by the relativistic hamiltonian H = pμp

μ + m2 namely by theaction principle

S =

intdτ

(pμx

μ minus N

2(pμp

μ + m2)

) (3127)

The relation between velocities and momenta obtained by varying pμ is xμ = NpμThe inverse Legendre transform therefore gives

S =1

2

intdτ

(xμx

μ

NminusNm2

) (3128)

128 Mechanics

We can also get rid of the Lagrange multiplier N from this action by writing its equationof motion

minus xμxμ

N2minusm2 = 0 (3129)

which is solved by

N =

radicminusxμxμ

m (3130)

and inserting this relation back into the action This gives

S = m

intdτ

radicminusxμxμ (3131)

which is the best known reparametrization invariant action for the relativistic particle


In GR it is often convenient to use complex dynamical variables sincethese simplify the form of the dynamical equations A particularly conve-nient choice is a mixture of complex and real variables where one canoni-cal variable is complex while the conjugate one is real As we shall see theselfdual connection (219) which is complex naturally leads to canonicalvariables of this type To exemplify how the use of such variables affectsdynamics consider a free particle with coordinate x momentum p andhamiltonian H0(x p) = p22m and assume we want to describe its dy-namics in terms of the variables (x z) where

z = xminus ip (3132)

In terms of these variables the nonrelativistic hamiltonian reads

H0(x z) = minus 12m

(xminus z)2 (3133)

Consider z as a configuration variable and ix as its momentum variableThe HamiltonndashJacobi equation becomes

partS(z t)partt

= minusH0

(minusi

partS(z t)partz

z

)=

12m

(ipartS(z t)

partz+ z

)2

(3134)

This is solved by

S(z t k) = kz +i2z2 minus k2

2mt (3135)

Equating the derivative of S with respect to the parameter k to a constantwe obtain the solution

C =partS(z t k)

partk= z minus k

mt (3136)

33 Field theory 129

that is

z(t) =k

mt + C (3137)

This is not the end of the story since so far k and C can be arbitrarycomplex constants To find the good solutions corresponding to real x andp we have to remind ourselves that z and x are not truly independentsince x is the real part of z

z + z = 2x (3138)

that is

z + z = minus2ipartS

partz (3139)

Inserting the solutions (3137) in the lhs we get

Im [k]t

m+ Im[C] = minusk (3140)

Therefore k is real and the imaginary part of C is minusk This immediatelygives the correct solution

Equation (3138) is called the reality condition The example illustratesthat in the HamiltonndashJacobi formalism the reality condition restricts thevalues of the HamiltonndashJacobi constants once the solutions of the evolu-tion equations are inserted

33 Field theory

There are several ways in which a field theory can be cast in hamiltonian form Onepossibility is to take the space of the fields at fixed time as the nonrelativistic configu-ration space Q This strategy badly breaks special- and general-relativistic invarianceLorentz covariance is broken by the fact that one has to choose a Lorentz frame for thet variable Far more disturbing is the conflict with general covariance The very founda-tion of generally covariant physics is the idea that the notion of a simultaneity surfaceover all the Universe is devoid of physical meaning It is better to found hamiltonianmechanics on a notion not devoid of physical significance

A second alternative is to formulate mechanics on the space of the solutions of theequations of motion The idea goes back to Lagrange In the generally covariant con-text a symplectic structure can be defined over this space using a spacelike surface butone can show that the definition is surface independent and therefore it is well definedThis strategy has been explored by several authors [108] The structure is viable inprinciple and has the merit of showing that the hamiltonian formalism is intrinsicallycovariant In practice it is difficult to work with the space of solutions to the field equa-tions in the case of an interacting theory Therefore we must either work over a spacethat we canrsquot even coordinatize or coordinatize the space with initial data on someinstantaneity surface and therefore effectively go back to the conventional fixed-timeformulation

130 Mechanics

The third possibility which I consider here is to use a covariant finite-dimensionalspace for formulating hamiltonian mechanics I noted above that in the relativisticcontext the double role of the phase space as the arena of mechanics and the spaceof the states is lost The space of the states namely the phase space Γ is infinite-dimensional in field theory essentially by definition of field theory But this does notimply that the arena of hamiltonian mechanics has to be infinite-dimensional as wellThe natural arena for relativistic mechanics is the extended configuration space C ofthe partial observables Is the space of the partial observables of a field theory finite-or infinite-dimensional


Consider a field theory for a field φ(x) with N components The fieldis defined over spacetime M with coordinates x and takes values in anN -dimensional target space T

φ M minusrarr T

x minusrarr φ(x) (3141)

For instance this could be Maxwell theory for the electric and magneticfields φ = ( E B) where N = 6 In order to make physical measurementson the field described by this theory we need N measuring devices to mea-sure the components of the field φ and four devices (one clock and threedevices giving us the distance from three reference objects) to determinethe spacetime position x Field values φ and positions x are therefore thepartial observables of a field theory Therefore the operationally motivatedrelativistic configuration space for a field theory is the finite-dimensionalspace

C = M times T (3142)

which has dimension 4 + N A correlation is a point (x φ) in C It repre-sents a certain value (φ) of the fields at a certain spacetime point (x) Thisis the obvious generalization of the (t α) correlations of the pendulum ofthe example in Section 321

A physical motion γ is a physically realizable ensemble of correlationsA motion is determined by a solution φ(x) of the field equations Such asolution determines a 4-dimensional surface in the ((4 +N)-dimensional)space C the surface is the graph of the function (3141) Namely theensemble of the points (x φ(x)) The space of the solutions of the fieldequations namely the phase space Γ is therefore an (infinite-dimensional)space of 4d surfaces γ in the (4 + N)-dimensional configuration space CEach state in Γ determines a surface γ in C

Hamiltonian formulations of field theory defined directly on C = MtimesTare possible and have been studied The main reason is that in a local field

33 Field theory 131

theory the equations of motion are local and therefore what happens at apoint depends only on the neighborhood of that point There is no needtherefore to consider full spacetime to find the hamiltonian structure ofthe field equations I refer the reader to the beautiful and detailed paper[109] and the ample references therein for a discussion of this kind ofapproach I give a simple and self-contained illustration of the formalismbelow with the emphasis on its general covariance


Consider a field theory on Minkowski space M Call φA(xμ) the fieldwhere A = 1 N The field is a function φ M rarr T where T = RN

is the target space namely it is the space in which the field takes valuesThe extended configuration space of this theory is the finite-dimensionalspace C = M times T with coordinates qa = (xμ φA) The coordinates qa

are the (4 + N) partial observables whose relations are described by thetheory A solution of the equations of motion defines a four-dimensionalsurface γ in C If we coordinatize this surface using the coordinates xμthen this surface is given by [xμ φA(xμ)] where φA(xμ) is a solution of thefield equations If alternatively we use an arbitrary parametrization withparameters τρ ρ = 0 1 2 3 then the surface is given by [xμ(τρ) φA(τρ)]where φA(xμ(τρ)) = φA(τρ)

In the case of a finite number of degrees of freedom (and no gauges)motions are given by one-dimensional curves At each point of the curvethere is one tangent vector and momenta coordinatize the one-forms Infield theory motions are four-dimensional surfaces and have four inde-pendent tangents Xμ or a ldquoquadritangentrdquo X = εμνρσXμotimesXνotimesXρotimesXσ

at each point Accordingly momenta coordinatize the four-forms LetΩ = Λ4T lowastC be the bundle of the four-forms pabcddqa and dqb and dqc and dqd

over C A point in Ω is thus a pair (qa pabcd) The space Ω carries thecanonical four-form

θ = pabcd dqa and dqb and dqc and dqd (3143)

In general given the finite-dimensional space C of the partial observ-ables qa dynamics is defined by a relativistic hamiltonian H Ω rarr V where Ω = Λ4T lowastC and V is a vector space Denote γ a four-dimensionalsurface in Ω and γ the projection of this surface on C The physical mo-tions γ are determined by the following

Variational principle A surface γ with a boundary α is aphysical motion if γ extremizes the integral

S[γ] =int

γpabcd dqa and dqb and dqc and dqd (3144)

132 Mechanics

in the class of the surfaces γ satisfying

H(qa pabcd) = 0 (3145)

and whose restriction γ to C is bounded by α

This is a completely straightforward generalization of the variational prin-ciple of Section 32 Equation (3145) defines a surface Σ in Ω As beforewe denote θ the restriction of θ to Σ and ω = dθ

For a field theory on Minkowski space without gauges the system(3145) is given by

pABCD = pABCμ = pABμν = 0 (3146)

H = π + H0(xμ φA pμA) = 0 (3147)

where H0 is DeDonderrsquos covariant hamiltonian [110] (see below for anexample) It is convenient to use the notation pμνρσ = πεμνρσ and pAνρσ =pμAεμνρσ for the nonvanishing momenta and to use coordinates (xμ φA pμA)on Σ On the surface defined by (3146)

θ = π d4x + pμA dφA and d3xμ (3148)

where we have introduced the notation d4x = dx0 and dx1 and dx2 and dx3 andd3xμ = d4x(partμ) = 1

3εμνρσdxν and dxρ and dxσ On Σ defined by (3146) and(3147)

θ = θ|Σ = minusH0(xμ φA pμA)d4x + pμA dφA and d3xμ (3149)

and ω is the five-form

ω = minusdH0(xμ φA pμA) and d4x + dpμA and dφA and d3xμ (3150)

An orbit of ω is a four-dimensional surface m immersed in Σ such thatat each of its points a quadruplet X of tangents to the surface satisfies

ω(X) = 0 (3151)

I leave to the reader the exercise of showing that the projection of anorbit on C is a physical motion

In more detail let (partμ partA partAμ ) be the basis in the tangent space of

Σ determined by the coordinates (xμ φA pμA) Parametrize the surfacewith arbitrary parameters τρ The surface is then given by the points[xμ(τρ) φA(τρ) pμA(τρ)] Let partρ = partpartτρ Then let

Xρ = partρxμ(τρ) partμ + partρφ

A(τρ) partA + partρpμA(τρ) partA

μ (3152)

33 Field theory 133

Then X = X0 otimesX1 otimesX2 otimesX3 is a rank four tensor on Σ If ω(X) = 0then φA(xμ) determined by φA(xμ(τρ)) = φA(τρ) is a physical motion

Summarizing the canonical formalism of field theory is completely de-fined by the couple (C H) where C is the finite-dimensional space ofthe partial observables (field values and spacetime coordinates) and H ahamiltonian on the finite-dimensional space Ω = Λ4T lowastC Equivalently itis completely defined by the finite-dimensional presymplectic space (Σ θ)The formalism as well as its interpretation make sense even in the case inwhich the coordinates of C do not split into xμ and φA and the relativistichamiltonian does not have the particular form (3146)ndash(3147)

Example scalar field As an example consider a scalar field φ(xμ) on Minkowski spacesatisfying the field equations

partμpartμφ(xμ) + m2φ(xμ) + V prime(φ(xμ)) = 0 (3153)

Here the Minkowski metric has signature [+minusminusminus] and V prime(φ) = dV (φ)dφ The fieldis a function φ M rarr T where here T = R The relativistic configuration space of thistheory is the five-dimensional space C = M times T with coordinates (xμ φ) The spaceΩ has coordinates (xμ φ π pμ) (equation (3146) is trivially satisfied) and carries thecanonical four-form

θ = π d4x + pμ dφ and d3xμ (3154)

The dynamics is defined on this space by the DeDonder relativistic hamiltonian

H = π + H0 = 0 (3155)

H0 =1

2

(pμpμ + m2φ2 + 2V (φ)

) (3156)

Therefore we can use coordinates (xμ φ pμ) on the surface Σ defined by these equationsand (3149) gives

θ = minus1

2

(pμpμ + m2φ2 + 2V (φ)

)d4x + pμ dφ and d3xμ (3157)

The couple (Σ θ) defines the presymplectic formulation of the system ω is the five-form

ω = dθ = minus(pμdpμ + m2φdφ + V prime(φ)dφ

)and d4x + dpμ and dφ and d3xμ (3158)

A tangent vector has the form

V = Xμpartxμ + Xφpartφ + Y μpartpμ (3159)

If we coordinatize the orbits of ω with the coordinates xμ at every point we have thefour independent tangent vectors

Xμ = partxμ + (partμφ)partφ + (partμpρ)partpρ (3160)

and the quadritangent X = εμνρσXμ otimesXν otimesXρ otimesXσ Inserting (3160) and (3158) inω(X) = 0 a straightforward calculation yields

partμφ(x) = pμ(x) (3161)

partμpμ(x) = minusm2φ(x) minus V prime(φ(x)) (3162)

and therefore precisely the field equations (3153) Notice that the canonical formalismis manifestly Lorentz covariant and no equal-time initial data surface has to be chosen

134 Mechanics

A state is a 4d surface (x φ(x)) in the extended configurations space C It representsa set of combinations of measurements of partial observables that can be realized inNature The phase space Γ is the infinite-dimensional space of these states A statedetermines whether or not a certain correlation (x φ) or a certain set of correlations(x1 φ1) (xn φn) can be observed They can be observed if the points (xi φi) lie onthe 4d surface that represents the state Conversely the observation of a certain setof correlations gives us information on the state the surface has to pass through theobserved points


The space of boundary data G described in Section 325 plays a key role inquantum theory In the finite-dimensional case G is the cartesian productof the extended configuration space with itself but the same is not true inthe field theoretical context where we need an infinite number of bound-ary data to characterize solutions Recall that in the finite-dimensionalcase G is the space of the possible boundaries of a motion in C In fieldtheory a motion is a 4d surface in C Its boundary is a three-dimensionalsurface α without boundaries in C Let us therefore define G in field theoryas a space of oriented three-dimensional surfaces α without boundaries inC As C = M times T the boundary data α includes a 3d boundary surfaceσ in spacetime as well as the value ϕ of the field on this surface

More precisely let xμ be spacetime coordinates in M and φA coordi-nates in the target space Coordinatize the 3d surface α with 3d coordi-nates τ = (τ1 τ2 τ3) Then α is given by the functions

α = [σ ϕ] (3163)σ τ rarr xμ(τ) (3164)ϕ τ rarr ϕA(τ) (3165)

The functions xμ(τ) define the 3d surface σ without boundaries in space-time The functions ϕA(τ) define the value of the field φ(x) on this surface

φA(x(τ)) equiv ϕA(τ) (3166)

Say σ is the boundary of a connected region R of M Then genericallyϕ determines a solution φ(x) of the equations of motion in the interior Rsuch that φ|σ = ϕ Imagine that σ is a cylinder in Minkowski space Todetermine a solution in the interior we need the initial value of the fieldon the bottom of the cylinder its final value on the top of the cylinder aswell as spatial boundary conditions on the side of the cylinder The dataα determine all these field values as well as the spacetime location of thecylinder itself These data form the field theoretical generalization of theset (t qi tprime qprimei) which form the argument of the Hamilton function and ofthe quantum propagator in finite-dimensional mechanics Alternatively

33 Field theory 135

the surface α need not be connected For instance it can be formed bytwo components which we can view as initial and final configurations

The Hamilton function S[α] = S[σ ϕ] is defined as the action of thesolution of the equations of motion φ(x) such that φ|σ = ϕ in R We shallsee below that S[α] satisfies a functional HamiltonndashJacobi equation andcan be seen as the classical limit of a quantum mechanical propagator10

We can give a more formal definition of S[α] analogous to the definition(383) Let γ be the motion in C bounded by α Let γ be the lift of γ toΣ That is let γ be the orbit of ω that projects down to γ Then

S[α] =int

γθ (3167)

Example scalar field For a scalar field for instance

S[α] =

int

γ

θ =

int

γ

(πd4x + pμdφ and d3xμ) =

int

R(π + pμpartμφ) d4x

=

int

R

(minus1

2pμpμ minus 1

2m2φ2 minus V (φ) + pμpartμφ

)d4x (3168)

=

int

R

(1

2partμφpart

μφminus 1

2m2φ2 minus V (φ)

)d4x

=

int

RL(φ partμφ) d4x (3169)

where L is the lagrangian density and we have used the equation of motion pμ = partμφIt is not hard to compute the Hamilton function for a free scalar field in the special

case in which α is formed by the two spacelike parallel hypersurfaces xμ(τ) = (t1 τ)and xμ(τ) = (t2 τ) and by the values φ1(x) and φ2(x) of the field on these surfacesThe calculation is simplified by the fact that a free field is essentially a collection of

oscillator with modes of wavelength k and frequency ω(k) =

radic|k|2 + m2 Using this

fact and (334) it is straightforward to compute the field for given boundary values andits action This gives

S(φ1 t1 φ2 t2) =

intd3k ω(k)

2φ1(k)φ2(k) minus (|φ1|2(k) + |φ2|2(k)) cos[ω(k)(t1 minus t2)]

2 sin[ω(t1 minus t2)]

(3170)

where φ(k) are the Fourier components of φ(x)

The symplectic structure on G As in the finite-dimensional case wecan define a symplectic structure on G Let s be the 3d surface in Σ thatbounds γ That is s = [xμ(τ) ϕA(τ) pμA(τ)] where the momenta pμA(τ)are determined by the solution of the field equations determined by theentire α

10S[α] is only defined on the regions of G where this solution exists and it is multivaluedwhere there is more than one solution

136 Mechanics

Define a two-form on G as follows

ωG [α] =int

sω (3171)

The form ωG is a two-form it is the integral of a five-form over a 3dsurface More precisely let δα be a small variation of α This variationcan be seen as a vector field δα(τ) defined on α This variation determinesa corresponding small variation δs which in turn is a vector field δs(τ)over s Then

ωG [α](δ1α δ2α) =int

sα

ω(δ1s δ2s) (3172)

Thus the five-form ω on the finite-dimensional space Σ defines the two-form ωG on the infinite-dimensional space G

Consider a small local variation δα of α This means varying the surfaceαM in Minkowski space as well as varying the value of the field over itAssume that this variation satisfies the field equations that is the vari-ation of the field is the correct one for the solution of the field equationsdetermined by α We have

ωG [α](δα) =int

sα

ω(δs) (3173)

But the variation δs is by construction along the orbit namely in the nulldirection of ω and therefore the right-hand side of this equation vanishesIt follows that if δα is an infinitesimal physical motion then

ωG(δα) = 0 (3174)

The pair (G ωG) contains all the relevant information on the systemThe null directions of ωG determine the variations of the 3-surface α alongthe physical motions The space G divided by these null directions namelythe space of the orbits of these variations is the physical phase space Γand the ωG restricted to this space is the physical symplectic two-formof the system

Example scalar field Letrsquos compute ωG in a slightly more explicit form for the exampleof the scalar field From the definition (3171)

ωG [α] =

int

s

ω =

int

s

dπ and d4x + dpμ and dφ and d3xμ

=

int

s

(pνdpν + m2φdφ + V primedφ) and d4x + dpμ and dφ and d3xμ

=

int

αM

d3xν

[(pμminuspartμφ)dpμ and dxν + (m2φ + V prime + partμp

μ)dφ and dxν + dpν and dφ]

=

int

αM

d3xν dpν and dφ (3175)

33 Field theory 137

where we have used the xμ coordinates themselves as integration variables and there-fore the integrand fields are the functions of the xμ Notice that since the integral is ons the pμ in the integrand is the one given by the solution of the field equation deter-mined by the data on α Therefore it satisfies the equations of motion (3161)ndash(3162)which we have used above Using (3161) again we have

ωG [α] =

int

αM

d3x nν d(nablaνφ) and dφ (3176)

In particular if we consider variations δα that do not move the surface and such thatthe change of the field on the surface is δφ(x) we have

ωG [α](δ1α δ2α) =

int

αM

d3x nν

(δ1φnablaνδ2φminus δ2φnablaνδ1φ

) (3177)

This formula can be directly compared with the expression of the symplectic two-formgiven on the space of the solutions of the field equations in [108] The expression is thesame but with a nuance in the interpretation ωG is not defined on the space of thesolutions of the field equations it is defined on the space of the lagrangian data G andthe normal derivative nνnablaνφ of these data is determined by the data themselves viathe field equations


A HamiltonndashJacobi equation for the field theory can be written as a localequation on the boundary satisfied by the Hamilton function I illustratehere the derivation of the HamiltonndashJacobi equation in the case of thescalar field leaving the generalization to the interested reader From thedefinition

S[α] =int

γθ =

int

γ(πd4x + pμdφ and d3xμ) (3178)

we can write

δS[α]δxμ(τ)

= π(τ) nμ(τ) + εμνρσ pν(τ) partiφ(τ) partjxρ(τ) partkx

σ(τ) εijk (3179)

where

nμ(τ) =13εμνρσpart1x

ν(τ)part2xρ(τ)part3x

σ(τ) (3180)

is the normal to the 3-surface σ The momentum π depends on the fullα Contracting this equation with nμ we obtain


+ pi(τ) partiφ(τ) (3181)

Using the equation of motion pμ = partμφ this becomes


+ partiφ(τ)partiφ(τ) (3182)

138 Mechanics

Also

δS[α]δϕ(τ)

= pμ(τ)nμ(τ) (3183)

The derivation of these two equations requires steps analogous to the oneswe used to derive (386)

Now from (3155) and (3156) we have that the scalar field dynamicsis governed by the equation

π +12

(pμpμ + m2φ2 + 2V (φ)

)= 0 (3184)

We split pμ into its normal (p = pμnμ) and tangential (pi) components(so that pμ = pipartix

μ + pnμ) obtaining

π +12

(p2 minus pipi + m2φ2 + 2V (φ)

)= 0 (3185)

Inserting (3182) and (3183) we obtain

δS[α]δxμ(τ)

nμ(τ) +12

[(δS[α]δϕ(τ)

)2

+partjϕ(τ)partjϕ(τ) + m2ϕ2(τ) + 2V (ϕ(τ))

]

= 0

(3186)

This is the HamiltonndashJacobi equation Notice that the function

S[xμ(τ) ϕ(τ)] = S[σ ϕ] = S[α] (3187)

is a function of the surface not the way the surface is parametrizedTherefore it is invariant under a change of parametrization It followsthat

δS[α]δxμ(τ)

partjxμ(τ) +

δS[α]δϕ(τ)

partjϕ(τ) = 0 (3188)

(This equation can be obtained also from the tangential component of(3179)) The two equations (3186) and (3188) govern the HamiltonndashJacobi function S[α]

The connection with the nonrelativistic field theoretical HamiltonndashJacobi formalism is the following We can restrict the formalism to apreferred choice of parameters τ Choosing τ j = xj we obtain S in theform S[t(x) φ(x)] and the HamiltonndashJacobi equation (3186) becomes

δS

δt(x)+

12

[(δS[α]δφ(x)

)2

+ partjφpartjφ + m2φ2 + 2V (φ)

]

= 0 (3189)

33 Field theory 139

Further restricting the surfaces to the ones of constant t gives the func-tional S[t φ(x)] satisfying the HamiltonndashJacobi equation

partS

partt+

12

intd3x

[(δS

δφ(x)

)2

+ |nablaφ|2 + m2φ2 + 2V (φ)

]

= 0 (3190)

which is the usual nonrelativistic HamiltonndashJacobi equation

partS

partt+ H

(φ nablaφ

δS[α]δφ(x)

)= 0 (3191)

where H(φ nablaφ parttφ) is the nonrelativistic hamiltonian

Canonical formulation on G We can write a hamiltonian density functionH(τ) directly for the infinite-dimensional space G H(τ) is a function onthe cotangent space T lowastG We coordinatize this cotangent space with thefunctions (xμ(τ) ϕ(τ)) and their momenta (πμ(τ) p(τ)) The hamiltonianis then

H[xμ ϕ πμ p](τ) = πμ(τ)nμ(τ) +12

[p2(τ) + partjϕ(τ)partjϕ(τ)

+m2ϕ2(τ) + 2V (ϕ(τ))] (3192)

and the HamiltonndashJacobi equation (3186) reads

H

[xμ ϕ

δS[α]δxμ

δS[α]δϕ

](τ) = 0 (3193)

If we restrict the surface xμ(τ) to the case xμ(τ) = (t τ) then H(τ)becomes

H[xμ ϕ πμ p](x) = π0(x) + H0(x) (3194)

where H0(x) is the conventional nonrelativistic hamiltonian density

H0[φ p] =12

[p2+partjϕpart

jϕ + m2ϕ2 + 2V (ϕ)] (3195)

Physical predictions from S The complete physical predictions of thetheory can be obtained directly from the Hamilton function S[α] = S[σ ϕ]as follows Let p(τ) be a function on the surface σ Define

F [σ ϕ p](τ) =δS[σ ϕ]δϕ(τ)

minus p(τ) (3196)

140 Mechanics

Given a closed surface σ in spacetime we can observe field boundaryvalues φ(x(τ)) = ϕ(τ) together with momenta nμpartμφ(x(τ)) = p(τ) if andonly if

F [σ ϕ p](τ) = 0 (3197)

This equation is equivalent to the equations of motion and expresses di-rectly the physical content of the theory as a restriction on the partialobservables that can be observed on a boundary surface

As in the case of finite-dimensional systems the general solution ofthe equations of motion can be obtained by derivations For instance letα be formed by two connected components that we denote α = [σ ϕ]and α0 = [σ0 ϕ0] parametrized by τ and τ0 respectively Consider theequation for α

f [α](τ) =δS[α cup α0]δϕ0(τ0)

minus p0(τ0) = 0 (3198)

where p0(τ) is an arbitrary initial value momentum This is the evolutionequation that determines all surfaces α compatible with the initial dataϕ0 p0 on σ0


Earth lay with Sky and after them was born TimeThe wily youngest and most terrible of her children

Hesiod Theogony [111]

In the macroscopic world the physical variable t measured by a clockhas peculiar properties It is not easy to pinpoint these properties withprecision without referring to a presupposed notion of time but it is alsodifficult to deny that they exist From the point of view developed inthis book at the fundamental level the variable t measured by a clockis on the same footing as any other partial observable If we accept thisidea we have then to reconcile the fact that time is not a special variableat the fundamental level with its peculiar properties at the macroscopiclevel What is so special about time An interesting possibility is that itis statistical mechanics and therefore thermodynamics that singles outt and gives it its special properties I briefly illustrate this idea in thissection

The world around us is made up of systems with a large number ofdegrees of freedom such as fields We never measure the totality of thesedegrees of freedom Rather we measure certain macroscopic parametersand make predictions on the basis of assumptions on the state of the other


degrees of freedom The viability of our choice of macroscopic parametersand our assumptions about the state of the others is justified a posterioriif the system of prediction works We represent our incomplete knowledgeand assumptions in terms of a statistical state ρ The state ρ can berepresented as a normalized positive function on the phase space Γ

ρ Γ rarr R+ (3199)int

Γds ρ(s) = 1 (3200)

ρ(s) represents the assumed probability density of the state s in Γ Thenthe expectation value of any observable A Γ rarr R in the state ρ is

ρ[A] =int

Γds A(s) ρ(s) (3201)

The fundamental postulate of statistical mechanics is that a system leftfree to thermalize reaches a time-independent equilibrium state that canbe represented by means of the Gibbs statistical state

ρ0(s) = NeminusβH0(s) (3202)

where β = 1T is a constant ndash the inverse temperature ndash and H0 isthe nonrelativistic hamiltonian Classical thermodynamics follows fromthis postulate Time evolution At = αt(A) of A is determined by (378)Equivalently At(s) = A(t(s)) where s(t) is the hamiltonian flow of H0 onΓ The correlation probability between At and B is given by

WAB(t) = ρ0[αt(A)B] =int

Σds A(s(t)) B(s) eminusβH0(s) (3203)

In this chapter we have seen that the formulas of mechanics do notsingle out a preferred variable because all mechanical predictions canbe obtained using the relativistic hamiltonian H which treats all vari-ables on an equal footing instead of using the nonrelativistic hamiltonianH0 which singles t out Is this true also for statistical mechanics andthermodynamics Equations (3200)ndash(3201) are meaningful also in therelativistic context where Γ is the space of the solutions of the equationsof motion But this is not true for (3202) and (3203) These dependon the nonrelativistic hamiltonian They depend on the fact that tis a variable different from the others Equations (3202) and (3203) defi-nitely single out t as a special variable This observation indicates that thepeculiar properties of the t variable have to do with statistical mechanicsand thermodynamics rather than with mechanics With purely mechani-cal measurements we cannot recognize the time variable With statisticalor thermal measurements we can

142 Mechanics

Indeed notice that if we try to pinpoint what is special about thevariable t we generally find features connected to thermodynamics irre-versibility convergence to equilibrium memory feeling of ldquoflowrdquo and soon

Indeed there is an intriguing fact about (3202) and (3203) Imaginethat we study a system which is in equilibrium at inverse temperature βand we do not know its nonrelativistic hamiltonian H0 In principle wecan figure out H0 simply by repeated microscopic measurements on copiesof the system without any need of observing time evolution Indeed ifwe find out the distribution of microstates ρ0 then up to an irrelevantadditive constant we have

H0 = minus 1β

ln ρ0 (3204)

Therefore in a statistical context we have in principle an operationalprocedure for determining which one is the time variable First measureρ0 second compute H0 from (3204) third compute the hamiltonianflow s(t) of H0 on Σ The time variable t is the parameter of this flowA ldquoclockrdquo is any measuring apparatus whose reading grows linearly withthis flow The multiplicative constant in front of H0 just sets the unit inwhich time is measured Up to this unit we can find out which one is thetime variable just by measuring ρ0 This is in strident contrast with thepurely mechanical context where no operational procedure for singlingout the time variable is available

Now let me come to the main observation Imagine that we have atruly relativistic system where no partial observable is singled out as thetime variable Imagine that we make measurements on many copies of thesystem and find that the statistical state describing the system is givenby a certain arbitrary11 state ρ Define the quantity

Hρ = minus ln ρ (3205)

Let s(tρ) be the hamiltonian flow of Hρ Call tρ ldquothermal timerdquo Callldquothermal clockrdquo any measuring device whose reading grows linearly withthis flow Given an observable A consider the one-parameter family ofobservables Atρ defined by Atρ(s) = A(tρ(s)) Then it follows that thecorrelation probability between the observables Atρ and B is given by

WAB(tρ) =int

Σds A(tρ(s)) B(s) eminusHρ(s) (3206)

What is the difference between the physics described by (3202)ndash(3203)and that described by (3205)ndash(3206) None That is whatever the

11For (3205) to make sense assume that ρ nowhere vanishes on Σ


statistical state ρ there exists always a variable tρ measured by the ther-mal clock with respect to which the system is in equilibrium and whosephysics is the same as in the conventional nonrelativistic statistical caseThis key observation naturally leads us to the following hypothesis

The thermal time hypothesis In Nature there is no preferredphysical time variable t There are no equilibrium states ρ0 preferreda priori Rather all variables are equivalent we can find the systemin an arbitrary state ρ if the system is in a state ρ then a preferredvariable is singled out by the state of the system This variable iswhat we call time

In other words it is the statistical state that determines which variable isphysical time and not any a priori hypothetical ldquoflowrdquo that drives the sys-tem to a preferred statistical state All variables are physically equivalentat the mechanical level But if we restrict our observations to macroscopicparameters and assume the other dynamical variables are distributed ac-cording to a statistical state ρ then a preferred variable is singled outby this procedure This variable has the property that correlations withrespect to it are described precisely by ordinary statistical mechanics Inother words it has precisely the properties that characterize our macro-scopic time parameter

In other words when we say that a certain variable is ldquothe timerdquo weare not making a statement concerning the fundamental mechanical struc-ture of reality12 Rather we are making a statement about the statisticaldistribution we use to describe the macroscopic properties of the systemthat we describe macroscopically

The hamiltonian Hρ determined by a state ρ is called the thermalhamiltonian The ldquothermal time hypothesisrdquo is the idea that what wecall ldquotimerdquo is simply the thermal time of the statistical state in whichthe world happens to be when described in terms of the macroscopicparameters we have chosen

Let the system be in the mechanical microstate s Describe it with macroscopicobservables Ai In general (but not always) there exists a statistical state ρ whose meanvalues give the correct predictions for the Ai that is Ai(s) sim ρ[Ai] Assuming it exitsρ codes in a sense our ignorance of the microscopic details of the state Intuitivelywe can therefore say that the existence of time is the result of this ignorance of oursTime is the expression of our ignorance of the microstate

The thermal time hypothesis works surprisingly well in a number ofcases For example if we start from a radiation-filled covariant cosmo-logical model having no preferred time variable and write a statistical

12Time Kρoνoς comes after matter (Earth Γαια and Sky Ovρανoς) also in Greek

mythology See Hesiodrsquos quote [111] at the beginning of this section

144 Mechanics

state representing the cosmological background radiation then the ther-mal time of this state turns out to be precisely the Friedmann time [112]Furthermore we will see in Section 551 that this hypothesis extends inan extremely natural way to the quantum context and even more nat-urally to the quantum field theoretical context where it leads also to ageneral abstract state-independent notion of time flow

mdashmdash


The hamiltonian theory of systems with constraints is one of Diracrsquos manymasterpieces The theory is not just a technical complication of standardhamiltonian mechanics it is a powerful generalization of mechanics whichremains valid in the general-relativistic context The title of Diracrsquos initialwork on the subject was ldquoGeneralized Hamiltonian dynamicsrdquo [113] Thetheory is synthesized in [114] For modern accounts and developments see[115] For the notion of partial observable I have followed [116] On thegeneral structure of mechanics I have followed [117 118] For a nontrivialexample of relational evolution treated in detail see [119]

The canonical treatment of field theory on finite-dimensional spaces de-rives from the Weyl and DeDonderrsquos calculus of variations [110 120] Abeautiful comprehensive and mathematically precise discussion of covari-ant hamiltonian field theory is in [109] which contains complete referencesto the literature on the subject See also [121]

The idea of the thermal origin of time was introduced in [112 122] inthe context of classical field theory and was independently suggested byAlain Connes It is developed in quantum field theory (see Section 551)in [125] see also [124] For a related Boltzmann-like approach see [123]

4Hamiltonian general relativity

I begin this chapter by presenting the HamiltonndashJacobi formulation of GR This is thebasis of the quantum theory

In the remainder of the chapter I present formulations of hamiltonian GR on afinite-dimensional configuration space along the lines illustrated at the end of theprevious chapter

This order of presentation is inverse to the logical order which should start from thefinite-dimensional configuration space of the partial observables But I do not want toforce the hurried reader to navigate through the entire chapter before finding the fewsimple equations that are the basis of the quantum theory

I take the cosmological constant to be zero and ignore matter fields leaving to thereader the generally easy exercise of adding the cosmological and matter terms to therelevant equations


GR can be expressed in terms of a complex field Aia(τ) and a 3d real

momentum field Eai (τ) defined on a three-dimensional space σ without

boundaries satisfying the reality conditions

Aia + Ai

a = Γia[E] (41)

where Γ is defined below in (423)ndash(424) The theory is defined by thehamiltonian system

DaEai = 0 (42)

Eai F

iab = 0 (43)

F ijabE

ai E

bj = 0 (44)

145


where F ijab = εijk F

kab (see pg xxii) and Da and F i

ab are the covariantderivative and the curvature of Ai

a defined by

Davi = partavi + εijkAjav

k (45)

F iab = partaA

ib minus partbA

ia + εijkA

jaA

kb (46)

I sketch the derivation of these equations from the lagrangian formalismbelow An indirect derivation via a finite-dimensional canonical formula-tion is given at the end of this chapter

The HamiltonndashJacobi system is given in terms of the functional S[A]by writing

Eai (τ) =

δS[A]δAi

a(τ)(47)

in the hamiltonian system The first two equations that we obtain

DaδS[A]δAi

a(τ)= 0 F i

ab(τ)δS[A]δAi

a(τ)= 0 (48)

require that S[A] is invariant under 3d diffeomorphisms (diffs) and localSO(3) transformations as I will show in a moment The last reads

F ijab(τ)

δS[A]δAi

a(τ)δS[A]

δAjb(τ)

= 0 (49)

This is the HamiltonndashJacobi equation of GR It defines the dynamics ofGR

Smeared form Equivalently we can integrate equations (42)ndash(44)against suitable ldquotestrdquo functions and demand the integral to vanish forany such function For the first two we get

G[λ] = minusint

d3τλi DaEai =

intd3τDaλ

iEai = 0 (410)

C[f ] = minusint

d3τfaF iabE

bi = 0 (411)

The quantities Daλi and faF i

ab that appear in these equations are theinfinitesimal transformations of the connection under an internal gaugetransformation with generator λi(τ) and under (the combination of aninternal gauge transformation and) an infinitesimal diffeomorphism gen-erated by the vector field fa(τ)

δλAia = Daλ

i δfAia = f bF i

ab (412)


Therefore the smeared form of (48) readsint

d3τ δλAia(τ)

δS[A]δAi

a(τ)= 0

intd3τ δfA

ia(τ)

δS[A]δAi

a(τ)= 0 (413)

which is the requirement that S[A] is invariant under gauge and diffeo-morphisms

The quantity (44) on the other hand is a density of weight two Tobe able to integrate it against a scalar quantity and get a well-definedresult we need a density of weight one This can be obtained by dividingthe hamiltonian by the square roote of the determinant of E exploitingthe freedom in the definition of the hamiltonian The Poisson bracketderived below in (425) between the volume

V =int

d3xradic| detE(x)| (414)

and the connection is

V Aia(x) = (8πiG)

Ebj (x)Ec

k(x)εabcεijk

4radic

| detE(x)| (415)

Using this we can write (44) in the form

H[N ] =int

N tr(F and V A) = 0 (416)

This form of the hamiltonian will prove convenient in the quantum theoryEquations (410) (411) and (416) define GR


What is the relation between the 4d fields used in Chapter 2 and the3d fields used above Consider a solution (eIμ(x) Ai

μ(x)) of the Einsteinequations (221) Choose a 3d surface σ τ = (τa) rarr xμ(τ) withoutboundaries in the coordinate space The four-dimensional forms Ai (theselfdual connection defined in (219)) Σi (the 4d Plebanski two-formdefined in (223)) and eI (the gravitational field introduced in (21))induce the three-dimensional forms

Ai(τ) = Aia(τ) dτa (417)

Σi(τ) = Σiab(τ) dτa and dτ b (418)

eI(τ) = eIa(τ) dτa (419)

on σ The 3d field E is defined as the vector density associated to Σi thatis

Eai(τ) = εabc Σibc(τ) (420)


Letrsquos write eI(τ) = (e0(τ) ei(τ)) Choose a gauge in which

e0(τ) = 0 (421)

(The extension of the formalism to a more general gauge deserves to beinvestigated See [127]) It is easy to see that in this gauge Ea

i (τ) is realand

Eai (τ) =

12εijk εabc ejb(τ) ekc (τ) (422)

The connection Γi[E](τ) = εijkΓj

k[E](τ) used in (41) is defined by

dei + Γij [E] and ej = 0 (423)

(this is the first Cartan structure equation for σ) which is solved by

Γjak =

12ebk(partae

jb minus partbe

ja + ecjealpartbe

lc) (424)

That is it is the spin connection of the triad eia It is also easy to verifythat in this gauge the two quantities Ai

a(τ) and Eai (τ) defined by (417)

and (420) satisfy the ldquoreality conditionrdquo (41)The quantity Ea

i (τ) is (8πiG times) the momentum conjugate to Aia(τ)

Hence we can write immediately the Poisson brackets

Aia(τ) Eb

j (τprime) = (8πiG) δbaδ

ijδ

3(τ τ prime) (425)

In Maxwell and YangndashMills theories the momentum conjugate to thethree-dimensional connection A is called electric field The field E istherefore called the gravitational electric field In the gauge (421) weare considering E is determined just by eia(τ) the triad field of σEquation (422) shows that E is the inverse matrix of the triad eia(τ)multiplied by its determinant

Eai = (det e)eai (426)

I sketch here the derivation of the basic equations of the hamiltonianformalism namely the Poisson brackets (425) and the constraint system(42)ndash(44) For a detailed discussion of this derivation see for instanceI [2 9 20 126] An indirect derivation via a finite-dimensional canonicalformulation is given at the end of this chapter We can start for instancefrom the action (227) without the cosmological constant and write it as


follows

S[Σ A] =minusi

16πG

intΣi and F i =

minusi16πG

intΣiμνF

iρσε

μνρσd4x

=minusi

8πG

int (ΣiabF

ic0 + Σi0aF

ibc

)εabcd4x

=minusi

8πG

int (Ec

i

(part0A

ic minus partcA

i0 + εijkA

j0A

kc

)+ PiIJe

Jae

J0F

ibcε

abc)d4x

=minusi

8πG

int (Ec

i Aic + Ai

0DcEci +

12(εijke

jae

k0 + ie0

0eia

)Fibcε

abc

)d4x

=minusi

8πG

int (Ec

i Aic + λi

0

(DcE

ci

)+ λb

(Ea

i Fiab

)+ λ

(Ea

jEbkF

jkab

))d4x

(427)

The dot over A indicates time derivative I have used the gauge conditione0i = 0 and the Lagrange multipliers are multiples of the nondynamical

variables Ai0 e

00 e

i0 The first term shows that Ec

i 8iπG is the momentumconjugate to Ai

c varying with respect to the Lagrange multipliers yieldsthe constraint system (42)ndash(44)

The geometry of the three surface In Section 214 we saw that the grav-itational field has a metric interpretation The metric structure inheritedby σ depends on the gravitational electric field E In particular considera two-dimensional surface S σ = (σ1 σ2) rarr τ(σi) embedded in thethree-dimensional surface σ What is the area of S From the definitionof the area equation (270) we have in a few steps

A(S) =int

Sd2σ |E| (428)

Here the norm is defined by |v| =radicδijvivj and

Ei(σ) = Eai (τ(σ)) na(σ) (429)

the normal to the surface being defined by

na(σ) = εabcpartτ b(σ)partσ1

partτ c(σ)partσ2

(430)

Equation (428) can be interpreted as the surface integral of the norm ofthe two-form

Ei = Eai εabc dxb and dxc (431)

and writtenA(S) =

int

S|E| (432)


Thus Eai or more precisely its norm or ldquolengthrdquo |E| defines the area

element We could therefore say that in gravity ldquothe length of the electricfield is the areardquo or more precisely the area of a surface is the flux of(the norm of) the gravitational electric field across the surface

Using (273) a similar calculation gives the volume of a 3d region R

V(R) =int

Rd3τ

radic| detE| (433)

If we know the area of any surface and the volume of any region we knowthe geometry

These expressions for area and volume in terms of the gravitationalelectric field E play a major role in quantum gravity The correspondingquantum operators have a discrete spectrum their eigenstates are knownand determine a convenient basis in the quantum state space

For later use notice that detE = det ((det e)eminus1)=(det e)3(det e)minus1 =(det e)2 Hence

radicn middot n = | det e| =

radic| detE| where n is defined in (430)

Phase space states and relation with the Einstein equations A state ofGR is an equivalence class of 4d field configurations eIμ(x) solving the Ein-stein equations under the two gauge transformations (2123) and (2124)The space Γ of these equivalence classes is the phase space of GR

Given one solution of the Einstein equations consider a 3d surface σwithout boundaries in coordinate space Let Ai

a(τ) be the connection on σinduced by the 4d selfdual connection Ai

μ(x) A state determines a familyof possible 3d fields Ai

a(τ) called compatible with the state obtainedby changing the representative in the equivalence class of solutions orequivalently changing σ A family of 3d fields Ai

a(τ) compatible with astate can be obtained in principle from a solution of the HamiltonndashJacobisystem as follows

In general to solve the system we need a solution S[Aα] of theHamiltonndashJacobi system depending on a sufficiently large number αn ofparameters A state is then determined by constants αn and βn as followsThe equation

F [Aα] =partS[Aα]partαn

minus βn = 0 (434)

determines the Aia(τ) compatible with the state In this sense a solution

S[Aα] of the HJ equation (49) contains the solution of the Einstein equa-tions In what follows I focus on the particular solution of the HamiltonndashJacobi system provided by the Hamilton function



A preferred solution of the HamiltonndashJacobi equation is the Hamiltonfunction S[A] This is defined as the value of the action of the regionR bounded by σ = partR computed on a solution of the field equationsdetermined by the boundary value A

A boundary value A on σ determines a solution (eIμ(x) Aiμ(x)) of the

Einstein equations in the region R In turn this solution induces on σ the3d field E[A] The Hamilton function satisfies

δS[A]δAi

a(τ)= Ea

i (τ)[A] (435)

Notice that Eai (τ)[A] is the value of Ea

i (τ) which is determined via theEinstein equations by the value of A on the entire surface σ Define thefunctional

F [AE](τ) =δS[A]δAi

a(τ)minus Ea

i (τ) (436)

then the equation

F [AE] = 0 (437)

is equivalent to the Einstein equations It expresses the conditions thatthe Einstein equations put on the possibility of having fields A and E ona 3d surface

This can be viewed as an evolution problem in the special case in whichwe take σ to be formed by two connected components σin and σoutbounding a single connected region R For instance σin and σout could betwo spacelike surfaces of a spatially closed universe In this case a solutionis determined by the components [Ain Aout] of the connection on σin andσout and also by the components [Ain Ein] of the connection and electricfield on σin alone We can write

F [Aout Ain Ein](τ) =δS[Ain Aout]δAin

ia(τ)

minus Einai (τ) = 0 (438)

Taking Aout as the unknown and Ain and Ein as data this equation givesthe general solution of the Einstein equations for fixed Ain and Ein itis solved by all 3d connections Aout on σ that are compatible with asolution bounded by a 3d surface with ldquoinitial conditionsrdquo Ain and EinThat is (438) determines all fields Aout that can ldquoevolverdquo from the ldquoinitialconditionsrdquo Ain and Ein Therefore the Hamilton function S[A] containsthe full solution of the Einstein equations S[A] expresses the full dynamicsof GR


As we shall see the full dynamics of quantum GR is contained in thecorresponding quantum propagator W [A] To the first relevant order in W [A] will be related to eminus

iS[A]

I have made no request that the 3d surface σ be spacelike In particular I haveavoided the issue of whether arbitrary boundary values A admit or determine an inter-polating solution In general the function S[A] will be defined only on a region andcan be multivalued in some other region These issues are important and I refer theinterested reader to [109] and references therein for literature on the topic On theother hand I think that the insistence on spacelike surfaces might be more tied to ourprerelativistic thinking habits than to their relations with Cauchy problems In viewof the construction of the quantum theory these problems can perhaps be postponedIf needed the requirement that the 3d surface is spacelike can be implemented as arestriction on the momentum E

Experiments Suppose we knew explicitly the Hamilton function S[A]How could we compare the theory with experience The answer is simpleWe should measure the 3d fields A and E on a closed 3-surface σ Thetheory predicts that the only fields (AE) we could measure are the onesthat satisfy (436) and (437) Therefore the theory determines which 3dfields could be measured and which could not In turn this determinesrestrictions (namely predictions) on any other quantity depending onthese fields Several important observations are in order

First the prediction is local in the sense that it regards a finite regionof spacetime Observables that require the full spacetime or the full spaceto be observed are not realistic

Second and most important where is the surface σ located Whichsurface σ should we consider The remarkable answer is it doesnrsquot matterThis is a key point in the interpretation of GR and should be understoodin detail

Consider a concrete experimental situation Consider for instance ascattering experiment in a particle accelerator or the propagation andreception of waves (electromagnetic or gravitational) In a nonrelativisticsituation say on Minkowski spacetime we can view the situation as fol-lows We have a certain number of objects and detectors located in certainknown positions of spacetime We measure the initial or incoming dataWe measure the final or outgoing data Furthermore we specify spatialboundary values (that forbid for instance spurious incoming radiation)The initial final and boundary values of the fields can be represented bythe value of the fields on a compact 3d surface σ These data howeverare not sufficient to make theoretical predictions we also need to knowthe location of σ in spacetime To fix the ideas say that σ is a cylinderin Minkowski space The height of the cylinder for instance is the timelapse between the beginning and the end of the experiment


Notice that the only relevant aspects of the location of objects appara-tus and detectors are their relative distances and time lapses Thereforethe only relevant aspect of the location of σ is the value of the metricon the surface and in its interior Indeed if we displace the surface (thatis the full experiment) in such a way that the geometry of the experimentremains the same we expect that the outcome will not change Since thegeometry of the interior is dictated (on Minkowski) by the geometry of σwe actually need to know only the geometry of σ It is this geometry thatdetermines the relative distances and time lapses between emissions anddetections Thus the full data that we need in a prerelativistic situationare

(i) the value of the dynamical fields on σ and(ii) the geometry of σ

Consider now the general-relativistic situation The same data as aboveare needed but now the geometry of σ is determined by the value ofthe dynamical fields on σ because the geometry is determined by thegravitational field Therefore the data that we need is

(i) the value of the dynamical fields on σ

and nothing elseThe ldquolocationrdquo of σ in the coordinate manifold is irrelevant because it

only reflects the arbitrary choice of coordinatization of spacetime In otherwords the distances and the time lapses among the detectors are preciselypart of the boundary data (AE) on σ For instance if σ is a cylinder thetime lapse between the initial and final measurement is precisely codedin the value of the gravitational field on the vertical (timelike) side ofthe cylinder Asking what happens after a longer time means nothing butasking what happens for larger values of E on the side of the cylinder


421 Euclidean GR

In this section I describe a field theory different from GR but which playsan important role in quantum gravity This is often called ldquoeuclideanGRrdquo Usual physical GR is then denoted ldquolorentzianrdquo to emphasize itsdistinction from euclidean GR Euclidean GR can be defined by the sameequations as GR for instance the action (213) with the only differencethat indices I J in the internal space are raised and lowered with theeuclidean metric δIJ instead of the Minkowski metric ηIJ Accordinglythe euclidean spin connection ω is an SO(4) connection instead of anSO(3 1) connection


It is still convenient to define the selfdual connection A as in (219) butthe appropriate selfdual projector P is now defined without the imaginaryfactor that is

Ai = ωi + ω0i (439)

Therefore the selfdual connection A is real in the euclidean case Theabsence of the imaginary factor gives immediately the Poisson brackets

Aia(τ) Eb

j (τprime) = (8πG) δbaδ

ijδ

3(τ τ prime) (440)

instead of (425)There is an important difference between the lorentzian and euclidean

cases In the euclidean case the connection lives in the so(4) algebraThis algebra decomposes as so(4) = so(3) oplus so(3) The real connection(439) is simply one of the two components Therefore (439) has half theinformation of ω

In the lorentzian case on the other hand the Lorentz algebra so(3 1)does not decompose at all However its complexification so(3 1C) de-composes as so(3 1C) = so(3C) oplus so(3C) A real ω determines twocomplex components which are complex conjugate to each other andeach component contains the same information as ω itself In this caseindeed the connection (219) has three complex components which isprecisely the same information as the six real components of ω

Remarkably the canonical formalism for the euclidean theory parallelscompletely the one for the lorentzian theory The theory is defined bythe same HamiltonndashJacobi equations (42)ndash(44) with the only differencethat (in the gauge (421)) the reality conditions (41) are replaced by

Eai minus Eai = 0 Aia minusAi

a = 0 (441)

The world is described by lorentzian GR not by euclidean GR Why then is euclideanGR useful at all Because euclidean GR plays a role in the search for a physical quantumtheory of gravity in several ways These will be discussed in more detail in the secondpart of the book but it is appropriate to anticipate some of these reasons here

First the key difficulty of quantum gravity is to understand how to formulate anontrivial generally covariant quantum field theory Euclidean GR is an example of anontrivial generally covariant field theory which is simpler than lorentzian GR becausethe reality conditions are simpler Therefore a complete and consistent formulation ofeuclidean quantum GR is not yet a quantum theory of gravity but is probably a majorstep in that direction Euclidean GR is a highly nontrivial model of the true theory

Second it is well known that the euclidean version of flat-space quantum field the-ories is strictly connected to the physical lorentzian version Under wide assumptionsone can prove that physical n-point functions are analytical continuations of the onesof the euclidean theory Naively one can simply Wick-rotate the time coordinate in theimaginary plane More precisely solid theorems from axiomatic quantum field theory


assure us that Wightman distributions are indeed the analytic continuation of the mo-ments of an euclidean process (the Schwinger functions) under very general hypothe-ses Defining the euclidean quantum field theory is therefore equivalent to defining thephysical theory In fact calculations are routinely performed in the euclidean region instandard quantum field theory We cannot assume naively that the same remains truein quantum gravity There is no Wick rotation to consider (recall the coordinate t isirrelevant for observable amplitudes anyway) and we are outside the hypotheses ofthe axiomatic approach Therefore we cannot as we do on flat space content ourself todefine the euclidean quantum field theory and lazily be sure that a consistent physicaltheory will follow

Still the very strict connection between the euclidean and the lorentzian theorythat exists on flat space strongly suggests that some connection between euclideanand lorentzian quantum GR is likely to exist Stephen Hawking in particular has ex-plored the hypothesis that physical quantum gravity could be directly defined in termsof the quantization of the euclidean theory There are various indications for that Firstthe formal functional path integral of the euclidean theory solves the WheelerndashDeWittequation for the lorentzian theory as well Second there is a standard technique forobtaining the vacuum of a quantum theory by propagating for an infinite euclideantime the adaptation of this idea to gravity led Jim Hartle and Stephen Hawking to theidea that a quantum gravitational ldquovacuumrdquo is obtained from propagation in imaginarytime or equivalently from the quantum euclidean theory

Finally as I show in the next section the lorentzian theory admits a formulationthat has the same kinematics as the euclidean theory It is therefore reasonable toexpect that the kinematical features of the two theories are the same and thereforekinematical aspects of the physical theory can be studied in the euclidean context


Let us return to lorentzian GR In this context define the quantity

Ai = ωi + ω0i (442)

precisely as in (439) This quantity does not transform as a connectionunder a local Lorentz transformation (as it does in the euclidean case)but it is still a well-defined field If we fix the gauge (421) then thereduced local internal gauge invariance is SO(3) and A defined in (442)transforms as a connection under SO(3) transformations For this reasonit is denoted the ldquoreal connectionrdquo of the lorentzian theory

Remarkably we can take the real connection or more precisely itsthree-dimensional restriction to the boundary surface as a canonical co-ordinate Lorentzian GR in other words can be expressed in terms of areal SO(3) connection The reality conditions are trivial The only differ-ence with respect to the euclidean theory is the form of the hamiltonianwhich acquires another more complicated term with respect to (44)

H = (F ijab + 2Ki

[aKjb]) Ea

i Ebj (443)


where Kia = Ai

a minus Γia[E] (See for instance [20]) The connection (442)

and the hamiltonian (443) provide a second hamiltonian formalism forGR alternative to the one described at the beginning of this chapter


Finally there is a third possible formalism for lorentzian GR It consistsin using the connection

Ai = ωi + γω0i (444)

where γ is an arbitrary complex parameter This is called the Barberoconnection it derives naturally from the use of the Holst action (see Sec211) The case γ = i gives the selfdual connection When γ is real it iscalled the Immirzi parameter In this case the reality conditions are stilltrivial (that is A = A) and the hamiltonian is a small modification of(443) (see [20])

H = (F ijab + (γ2 + 1)Ki

[aKjb]) Ea

i Ebj (445)

We will use this formalism in the quantum theorySince γ scales the term ω0i which is the one that has nonvanishing

Poisson brackets with E it is easy to see that the Poisson brackets betweenthe Barbero connection and the electric field are

Aia(x) Eb

j (y) = (8πγG) δbaδijδ

3(x y) (446)

The fact that γ can be arbitrary is important because as we shall seethe quantum theories obtained starting with different values of γ leadto different physical predictions That is in pure gravity γ has no effectin the classical theory but has an effect in the quantum theory (In thepresence of minimally coupled fermions γ appears in the equation ofmotion [128]) Presumably the presence of this parameter reflects a one-parameter quantization ambiguity of the theory γ is a parameter of thequantum theory that is absent in the classical theory such as for instancethe θ parameter of the QCD θ-vacua In fact γ can also be introduced asthe constant in front of a topological term added to the action precisely asthe θ parameter in QCD Such terms do not affect the classical equationsof motion but affect the quantum theory

As we shall see in Chapter 8 γ enters in several key predictions of thequantum theory In particular it enters in the computation of the black-hole entropy Comparing the black-hole entropy with the one determinedthermodynamically then determines γ A calculation along these linessketched in Chapter 8 suggests the value

γ asymp 02375 (447)


It has also been repeatedly suggested that γ may determine the relationbetween the bare and renormalized Newton constant Nevertheless thephysical interpretation of this parameter is not yet clear

43 Hamiltonian GR

I give here a formulation of canonical GR on a finite-dimensional config-uration space along the lines described in Section 332


Let T be the space on which the fields e and ω take value This is a (16 +24)-dimensional space with coordinates (eIμ ω

IJμ ) Let Σ = M times T be the

(4 + 16 + 24)-dimensional space with coordinates (xμ eIμ ωIJμ ) Consider

the four-form

θ = εIJKL eIμ eJν DωKLρ and dxμ and dxν and dxρ (448)

defined on this space Here the covariant differential D is defined by

DωKLρ = dωKL

ρ + ωKσI ω

ILρ dxσ (449)

This structure defines GR as follows Consider a four-dimensional surfaceγ in Σ Recall from Section 332 that we say that γ is an orbit of ω if thequadritangent X to the orbit is in the kernel of the five-form ω = dθ

dθ(X) = 0 (450)

The orbits of ω are the solutions of the Einstein equations If we use thex as coordinates on the γ then γ is represented by

γ = (xμ eIμ(x) ωIJμ (x)) (451)

If γ is an orbit of ω then the functions eIμ(x) ωIJμ (x) solve the Einstein

equations The demonstration is a straightforward calculation along thelines sketched for the scalar field example in Section 332


Consider the space Σ with coordinates (xμ Aiμ e

Iμ) where Ai

μ is complexand eIμ is real Define the gauge-covariant differential acting on all quan-tities with internal i indices as

Dvi = dvi + εijkAjμv

kdxμ (452)


andDAi

μ = dAiμ + εijkA

jνA

kμdxν (453)

GR is defined by the four-form

θ = PIJi eI and eJ and DAi (454)

where P iIJ is the selfdual projector defined in (217) Indeed the orbits

(xμ Aiμ(xμ) eIμ(xμ)) of ω = dθ satisfy the Einstein equations in the form

eI and (deJ + PJKi Ai and eK) = 0 (455)

PIJi and eI and F i = 0 (456)

where F i is the curvature of Ai The calculation is straightforward


Above I have defined canonical GR directly as a presymplectic (Σ θ)system This form can be derived from a configuration space and a hamil-tonian namely from the (C H) formalism described in Section 332 asfollows

Consider the finite-dimensional space C with coordinates (xμ Aiμ) Here

Aiμ is a complex matrix Assuming immediately (3146) the corresponding

space Ω has coordinates (xμ Aiμ π p

μνi ) and carries the canonical four-

formθ = πd4x + pμνi dAi

ν and d3xμ (457)

Using D the canonical form (457) reads

θ = pd4x + pμνi DAiμ and d3xν (458)

where p = π minus pμνi AjνAk

μεijk Also define

Eiμν = εμνρσ δijpρσj (459)

and the forms Ai = AiμdxμDAi = dAi

μ and dxμ + AjνAk

μεijkdx

ν and dxν Ei =Ei

μνdxμ and dxν and so on on Ω

GR is defined by the hamiltonian system

p = 0 (460)

pμνi + pνμi = 0 (461)

Ei and Ej = 0 (462)

(δikδjl minus13δijδkl)Ei and Ej = 0 (463)


The key point is that the constraints (462) (463) imply that thereexists a real four by four matrix eIμ where I = 0 1 2 3 such that Ei

μν isthe selfdual part of eIμe

Jν In fact it is easy to check that (462) and (463)

are solved by

Ei = P iIJ eI and eJ (464)

and the counting of degrees of freedom indicates that this is the uniquesolution Therefore we can use the coordinates (xμ Ai

μ eIμ) on the con-

straint surface Σ (where Aiμ is complex and eIμ is real) and the induced

canonical four-form is (454) Thus we recover the above (Σ θ) structure


Let α be a three-dimensional surface in C Thus α = [xμ(τ) Aiμ(τ)] where

τ = (τ1 τ2 τ3) = (τa) Define the functional

S[α] =int

γθ (465)

as in (3167) That is γ is the four-dimensional surface in Σ which is anorbit of dθ and therefore a solution of the field equations and is suchthat the projection of its boundary to C is α From the definition (454)

δS[α]δAi

μ(τ)= PiIJ εμνρσeJρ (τ)eIσ(τ)nν(τ) (466)

where nν is defined in (3180) Since from this equation we have immedi-ately

nμ(τ)δS[α]δAi

μ(τ)= 0 (467)

it follows that the dependence of S[α] on Aiμ(τ) is only through the restric-

tion of Ai(τ) to the 3-surface αM that is only through the components

Aia(τ) = partax

μ(τ)Aiμ(τ) (468)

Thus S[α] = S[xμ(τ) Aia(τ)] and

δS[α]δAi

a(τ)= PiJK εaνbc partbx

ρ(τ)partcxσ(τ)eJρ (τ)eKσ (τ)nν(τ) equiv Eai (τ) (469)

Therefore Eai is the conjugate momentum to the connection Ai

a Noticethat Ea

i is the dual of the restriction to the boundary surface σ of thePlebanski two-form Σi = Σi

μνdxμ and dxν defined in (223) Assume for


simplicity that the boundary surface is given by x0 = 0 and coordinatizedby x(τ) = τ and that we have chosen the gauge e0

b(τ) = 0 Then nμ =(1 0 0 0) and

Eai = εabc Σibc (470)

Its real part is the densitized inverse triad

ReEai = minusεijk εabc ejbe

kc = det(e) eai (471)

where eai is the matrix inverse to the ldquotriadrdquo one-form eia Its imaginarypart is

ImEai = εabc eibe

0c (472)

The projection of the field equations (456) on σ written in terms ofEa

i read DaEai = 0 F i

abEai = 0 and F i

abEaiEbkεijk = 0 where Da and F i

abare the covariant derivative and the curvature of Ai

a Using (469) thesegive the three HamiltonndashJacobi equations of GR

DaδS[α]δAi

a(τ)= 0 (473)

δS[α]δAi

a(τ)F iab = 0 (474)

F ijab(τ)

δS[α]δAi

a(τ)δS[α]

δAjb(τ)

= 0 (475)

Kinematical gauges Equation (473) could have been obtained by simplyobserving that S[α] is invariant under local SU(2) gauge transformationson the 3-surface Under one such transformation generated by a functionf i(τ) the variation of the connection is δfAi

a = Dafi Therefore S satisfies

0 = δfS =int

d3τ δfAia(τ)

δS[α]δAi

a(τ)=

intd3τ Daf

i(τ)δS[α]δAi

a(τ)

= minusint

d3τ f i(τ) DaδS[α]δAi

a(τ) (476)

This gives (473) Next the action is invariant under a change of coordi-nates on the 3-surface αM Under one such transformation generated by afunction fa(τ) the variation of the connection is δfAi

a = f bpartbAia+Ai

bpartafb

Integrating by parts as in (476) this gives

partbAia

δS[α]δAi

a(τ)+ (partbAi

a)δS[α]δAi

a(τ)= 0 (477)

which combined with (473) gives (474) Thus (473) and (474) aresimply the requirement that S[α] is invariant under internal gauge andchanges of coordinates on the 3-surface The three equations (473) (474)and (475) govern the dependence of S on Ai

a(τ)


Dropping the coordinates It is easy to see that S is independent fromxμ(τ) A change of coordinates xμ(τ) tangential to the surface cannotaffect the action which is independent of the coordinates used Moreformally the invariance under change of parameter τ implies

δS[α]δxμ(τ)

partjxμ(τ) =

δS[α]δAi

a(τ)δjA

ia(τ) (478)

and we have already seen that the right-hand side vanishes The variationof S under a change of xμ(τ) normal to the surface is governed by theHamiltonndashJacobi equation proper equation (3186) In the present casefollowing the same steps as for the scalar field we obtain

δS[α]δxμ(τ)

nμ(τ) + εijkFiab

δS[α]

δAja(τ)

δS[α]δAk

b (τ)= 0 (479)

But the second term vanishes because of (475) Therefore S[α] is inde-pendent of tangential as well as normal parts of xμ(τ) S depends onlyon [Ai

a(τ)]We can thus drop altogether the spacetime coordinates xμ from the

extended configuration space Define a smaller extended configurationspace C as the 9d complex space of the variables Ai

a Geometrically thiscan be viewed as the space of the linear mappings A D rarr sl(2C) whereD = R3 is a ldquospace of directionsrdquo and we have chosen the complex selfdualbasis in the sl(2C) algebra We then identify the space G as a space ofparametrized 3d surfaces A with components [Ai

a(τ)] and without bound-aries in C GR is defined on this space by the HamiltonndashJacobi system

DaδS[A]δAi

a(τ)= 0 (480)

δS[A]δAi

a(τ)F iab = 0 (481)

F ijab(τ)

δS[A]δAi

a(τ)δS[α]

δAjb(τ)

= 0 (482)

These are the equations presented at the beginning of this chapter onwhich we will base quantum gravity

Equivalently we can solve immediately (480) and (481) by definingthe space G0 of the equivalence classes of 3d SU(2) connections A undergauge and 3d diffeomorphisms (Aa

i(τ) = partτ primebpartτa A

primebi(τ prime(τ))) transformation

Then GR is defined by the sole equation (482) on this space (wherefunctions S[Ai

a(τ)] overcoordinatize G0) Accordingly we can interpretGR as the dynamical system defined by the extended configuration space


G0 and the relativistic hamiltonian

H(τ) = F ijab(τ) Ea

i (τ) Ebj (τ) (483)


The two variables on which we have based the canonical formulation ofGR described above are a complex 3d connection Ai

a and its complex con-jugate momentum Eai They have 9 complex components each On theother hand the degrees of freedom of GR have (9 + 9) real componentsof which (2 + 2) are physical degrees of freedom 7 are constrained and 7are gauges The explanation of the apparent doubling of the componentsis that A and E are like the coordinates z = x + ip and z = x minus ip overthe phase space of a one-dimensional system That is they are not inde-pendent of each other

To find out these relations let us write the real and imaginary parts ofA and E From their definition we have

ReAia = ωi

a (484)ImAi

a = ω0ia (485)

ReEai = det(e) eai (486)

ImEai = εabc e0

b eic (487)

We have chosen a gauge in which e0a = 0 Then (487) implies that E

is real Recall that the tetrad and the connection ω are related by theequation deI = ωI

J and eJ Projecting this equation on the 3-surface weobtain

dei = ωij and ej + ωi

0 and e0 (488)

In the gauge chosen the last term vanishes and ωij is the spin connection

of the triad ei Hence (484) implies that the real part of A satisfies (41)Without fixing the gauge e0

a = 0 the reality conditions are a bit morecumbersome

mdashmdash


The hamiltonian formulation of GR was developed independently byPeter Bergmann and his group [129] and by Dirac [130] The long-termgoal of both was quantum gravity The main tool for this the hamiltoniantheory of constrained systems was developed for this purpose The greatalgebraic complexity of the hamiltonian formalism was dramatically re-duced by the introduction of the ADM variables by Arnowitt Deser and


Misner [131] and then by the selfdual connection variables systematizedby Ashtekar [132]

The conventional derivation of the fundamental equations (42ndash44)from the lagrangian formalism can be found in many books and arti-cles see for instance I [2 9 20 126] See also the original articles [132]The expression (416) of the hamiltonian which plays an important rolein the quantum theory was introduced by Thomas Thiemann [133] Theusefulness of the Barbero connection was pointed out in [134] on itsgeometrical interpretation see [136] The importance of the Immirzi pa-rameter for the quantum theory in [135] An (inconclusive) discussion onthe Immirzi parameter and its physical interpretation is in [137]

For the finite-dimensional formulation I have followed here [138 139]On other versions of this formalism see [140] For the covariant HamiltonndashJacobi formalism for GR see also [141]

5Quantum mechanics

Quantum mechanics (QM) is not just a theory of micro-objects it is our currentfundamental theory of motion It expresses a deeper understanding of Nature thanclassical mechanics Precisely as classical mechanics the conventional formulationof QM describes evolution of states and observables in time Precisely as classicalmechanics this is not sufficient to deal with general relativistic systems because thesesystems do not describe evolution in time they describe correlations between observ-ables Therefore a formulation of QM slightly more general than the conventional onendash or a quantum version of the relativistic classical mechanics discussed in the previouschapter ndash is needed In this chapter I discuss the possibility of such a formulationIn the last section I discuss the general physical interpretation of QM

QM can be formulated in a number of more or less equivalent formalisms canonical(Hilbert spaces and self-adjoint operators) covariant (Feynmanrsquos sum-over-histories)algebraic (states as linear functionals over an abstract algebra of observables) andothers Generally but not always we are able to translate these formalisms into oneanother but often what is easy in one formulation is difficult in another A general-relativistic sum-over-histories formalism has been developed by Jim Hartle [26] HereI focus on the canonical formalism because the canonical formalism has provided themathematical completeness and precision needed to explicitly construct the mathemat-ical apparatus of quantum gravity Later I will consider alternative formalisms


Conventional QM can be formulated as follows

States The states of a system are represented by vectors ψ in a complexseparable Hilbert space H0

Observables Each observable quantity A is represented by a self-adjointoperator A on H0 The possible values that A can take are thenumbers in the spectrum of A

164


Probability The average of the values that A takes over many equal statesrepresented by ψ is a = 〈ψ|A|ψ〉〈ψ|ψ〉

Projection If the observable A takes values in the spectral interval Ithe state ψ becomes then the state PIψ where PI is the spectralprojector on the interval I

Evolution States evolve in time according to the Schrodinger equation

iparttψ(t) = H0ψ(t) (51)

where H0 is the hamiltonian operator corresponding to the energyEquivalently states do not evolve in time but observables do andtheir evolution is governed by the Heisenberg equation

ddt

A(t) = minus i

[A(t) H0] (52)

A given quantum system is defined by a family (generally an algebra) ofoperators Ai including H0 defined over an Hilbert space H0

This scheme for describing Nature differs substantially from the newto-nian one Here are the main features of the physical content of the abovescheme

Probability Predictions are only probabilistic

Quantization Some physical quantities can take certain discrete valuesonly (are ldquoquantizedrdquo)

Superposition principle If a system can be in a state A where a physicalquantity q has value a as well as in state B where q has value bthen the system can also be in states (denoted ψ = caA+ cbB with|ca|2 + |cb|2 = 1) where q has value a with probability |ca|2 andvalue b with probability |cb|2

Uncertainty principle There are couples of (conjugate) variables thatcannot have determined values at the same time

Effect of observations on predictions The properties we expect the sys-tem to have at some time t2 are determined not only by the proper-ties we know the system had at time t0 but also by the propertieswe know the system has at the time t1 where t0 lt t1 lt t21

1Bohr expressed this fact by saying that observation affects the observed system Butformulations such as Bohmrsquos or consistent histories force us to express this physicalfact using more careful wording


In Section 56 I discuss the physical content of QM in more depthIn general a quantum system (H0 Ai H0) has a classical limit which

is a mechanical system describing the results of observations made onthe system at scales and with accuracy larger than the Planck constantIn the classical limit Heisenberg uncertainty can be neglected and theobservables Ai can be taken as coordinates of a commutative phase spaceΓ0 Quantum commutators define classical Poisson brackets and (52)reduces to Hamilton equation (378)

If the classical limit is known the search for a quantum system fromwhich this limit may derive is called the quantization problem There isno reason for the quantization problem to have a unique solution Theexistence of distinct solutions is denoted ldquoquantization ambiguityrdquo Ex-perience shows that the simplest quantization of a given classical systemis very often the physically correct one If we are given a classical systemdefined by a nonrelativistic configuration space C0 with coordinates qi andby a nonrelativistic hamiltonian H0(qi pi) then a solution of the quan-tization problem can be obtained by interpreting the HamiltonndashJacobiequation (317) as the eikonal approximation of the wave function (51)that governs the quantum dynamics [142] This can be achieved by defin-ing multiplicative operators qi derivative operators pi = minusi part

partqiand the

hamiltonian operator

H0 = H0

(qiminusi

part

partqi

)(53)

on the Hilbert space H0 = L2[C0] the space of the square integrablefunctions on the nonrelativistic configuration space [143]

In a special-relativistic context this structure remains the same but theEvolution postulate above is extended to the requirement that H0 carriesa unitary representation of the Poincare group and H0 is the generator ofthe time translations of this representation

This structure is not generally relativistic In particular the notions ofldquostaterdquo and ldquoobservablerdquo used above are the nonrelativistic ones Can thestructure of QM be extended to the relativistic framework In Section52 I discuss such an extension As a preliminary step however in therest of this section I introduce and illustrate some tools needed for thisreformulation in the context of a very simple system ndash as I did for classicalmechanics


Nonrelativistic formulation The quantum theory of the pendulum canbe written on the Hilbert space H0 = L2[R] of wave functions ψ0(α) in


terms of the multiplicative position operator α the momentum operatorpα = minusi part

partα and the hamiltonian

H0 = minus 2

2mpart2

partα2+

mω2

2α2 (54)

More precisely the theory is defined on a rigged Hilbert space or Gelfand triple AGelfand triple S sub H sub S prime is formed by a Hilbert space H a proper subset S densein H and equipped with a weak topology and the dual S prime of S with their naturalidentifications A manifold M with a measure dx determines a rigged Hilbert spaceSM sub HM sub S prime

M where SM is the space of smooth functions on M with fast decrease(Schwarz space) HM = L2[M dx] and S prime

M is the space of the tempered distributionson M This setting allows us in particular to deal with eigenstates of observables withcontinuous spectrum and Fourier transforms

The operators (here h = 1)

α(t) = eitH0αeminusitH0 (55)

which solve (52) are the Heisenberg position operators that give the posi-tion at any time t Denote |α t〉 the generalized eigenstate of the operatorα(t) with eigenvalue α (which are in S prime)

α(t)|α t〉 = α|α t〉 (56)

and |α〉 = |α 0〉 Clearly |α t〉 = eitH0 |α〉 Given a state |ψ〉 theSchrodinger wave function

ψ(α t) = 〈α t|ψ〉 = 〈α|eminusitH0 |ψ〉 (57)

satisfies the Schrodinger equation (51) Conversely each solution of theSchrodinger equation restricted to t = 0 defines a state in H0 Thereforethere is a one-to-one correspondence between states at fixed time ψ0(α)and solutions of the Schrodinger equation ψ(α t) I call H the space ofthe solutions of the Schrodinger equation Thanks to the identificationjust mentioned H is a Hilbert space isomorphic to the Hilbert space H0

of the states at fixed time I call

R0 H rarr H0 (58)ψ(α t) rarr ψ0(α) = ψ(α 0) (59)

the identification map The relation between H and H0 is analogous tothe relation between the spaces Γ and Γ0 in classical mechanics discussedin Chapter 3

The propagator is defined as

W (α t αprime tprime) = 〈α t|αprime tprime〉 = 〈α|eminusi(tminustprime)H0 |αprime〉=

sum

n

Hn(α) eminusiEn(tminustprime) Hn(αprime) (510)


where Hn(α) is the eigenfunction of H0 with eigenvalue En Explicitly astraightforward calculation that can be found in many books gives

W (α t αprime tprime) =radic

mω

ih sin[ω(tminus tprime)]e

iωm2h

[(α2+αprime2) cos[ω(tminustprime)]minus2ααprime

sin2[ω(tminustprime)]

]

(511)

where h = 2πh The propagator satisfies the Schrodinger equation in thevariables (α t) (and the conjugate equation in the variables (αprime tprime))

Spacetime states It is convenient to consider the following states Givenany compact support complex function f(α t) the state

|f〉 =int

dα dt f(α t) |α t〉 (512)

is in H0 and is called the ldquospacetime smeared staterdquo or simply the ldquospace-time staterdquo of the function f(α t) Since standard normalizable statesare dimensionless (for 〈ψ|ψ〉 = 1 to make sense) and the states |α t〉have dimension Lminus12 the function f must have dimensions Tminus1Lminus12These states generalize the conventional wave packets for which f(α t) =f(α)δ(t) Conventional wave packets can be thought of as being associatedwith results of instantaneous position measurements with finite resolutionin space as I will illustrate later on spacetime states can be associatedwith realistic measurements where the measuring apparatus has finiteresolution in space as well as in time The Schrodinger wave function of|f〉 is

ψf (α t) = 〈α t|f〉

= 〈α t|int

dαprimedtprime f(αprime tprime) |αprime tprime〉

=int


and satisfies the Schrodinger equation The scalar product of two space-time states is

〈f |f prime〉 =int


In particular we can associate a normalized state |R〉 to each spacetimeregion R

|R〉 = CR

int

Rdα dt |α t〉 (515)


where the factor

Cminus2R =

intdα dt dαprimedtprime W (α t αprime tprime) (516)

fixes the normalization 〈R|R〉 = 1 as well as giving the state the rightdimensions


As discussed in Chapter 3 the kinematics of a pendulum is describedby two partial observables time t and elongation α These coordinatizethe relativistic configuration space C The classical relativistic formalismtreats α and t on an equal footing The quantum relativistic formalismas well treats α and t on an equal footing and therefore it is based onfunctions f(α t) on C

To be precise let S sub K sub S prime be the Gelfand triple defined by C and the measuredαdt That is S is the space of the smooth functions f(α t) on C with fast decreaseK = L2[C dαdt] and S prime is formed by the tempered distributions over C

I call S the ldquokinematical state spacerdquo and its elements f(α t) ldquokine-matical statesrdquo

In the relativistic formalism the dynamics of the system is defined bythe relativistic hamiltonian H(α t p pt) given in (324) The quantumdynamics is defined by the ldquoWheelerndashDeWittrdquo (WdW) equation

H ψ(α t) = 0 (517)

where

H = H

(α tminusih

part

partαminusih

part

partt

)

= minusihpart

partt+ H0

= minusihpart

parttminus h2

2mpart2

partα2+

mω2

2α2 (518)

and H0 is given in (54) In the case of the pendulum (517) reducesto the Schrodinger equation (51) but (517) is more general than theSchrodinger equation because in general H does not have the nonrela-tivistic form H = pt +H0 Solutions ψ(α t) of this equation form a linearspace H which carries a natural scalar product that I will construct in amoment The key object for the relativistic quantum theory is the oper-ator

P =int



defined on S prime This operator maps arbitrary functions f(α t) into solu-tions of the WdW equation (517) namely into H

To see this expand a function f(α t) as

f(α t) =sum

n

intdE fn(E) Hn(α) eminusiEt (520)

Acting with P on this function we obtain

[Pf ](α t) =

intdτ eminusiτH

sum

n

intdE fn(E) Hn(α) eminusiEt

=

intdτ

sum

n

intdE eminusiτ(minusE+En) fn(E) Hn(α) eminusiEt

=sum

n

intdE δ(E minus En) fn(E) Hn(α) eminusiEt

=sum

n

ψn Hn(α) eminusiEnt (521)

where ψn = fn(En) which is the general solution of (517) Therefore P sends arbitraryfunctions into solutions of the WdW equation Intuitively P sim δ(H)

The integral kernel of P is the propagator (510) Indeed the inverseof (520) gives

ψn = fn(En) =int

dαdt Hn(α) eiEnt f(α t) (522)

Inserting this in (521) we have

[Pf ](α t) =sum

n

intdαprimedtprime Hn(αprime) eiEntprimeHn(α) eminusiEnt f(αprime tprime)

=int


P is often called ldquothe projectorrdquo although improperly so Intuitively it ldquoprojectsrdquoon the space of the solutions of the WdW equation In some systems (when 0 is aneigenvalue in the discrete spectrum of H) P is indeed a projector But generically andin particular for the nonrelativistic systems (where 0 is in the continuum spectrum ofH) P is not a projector because its domain is smaller than the full S prime In particularit does not contain the solutions of the WdW equation namely P rsquos codomain Thedomain of P contains on the other hand S

The matrix elements of P

〈f |P |f prime〉K =int



define a degenerate inner product in S Dividing S by the kernel of thisinner product that is identifying f and f prime if Pf = Pf prime and completingin norm we obtain a Hilbert space But if Pf = Pf prime then f and f prime definethe same solution of the WdW equation In fact they define the solutionthat corresponds to the spacetime state |f〉 defined above Therefore anelement of this Hilbert space corresponds to a solution of the WdW equa-tion the Hilbert space can be identified with the space of the solutions ofthe WdW equation H Therefore

P S rarr Hf rarr |f〉 (525)

It follows that P directly equips the space H of the solutions with aHilbert space structure if ψ = Pf and ψprime = Pf prime are two solutions of theWdW equation (517) their scalar product is defined by

〈ψ|ψprime〉 equiv 〈f |P |f prime〉K (526)

where the right-hand side is the scalar product in K and is explicitlygiven in (524)

Notice that the scalar product on the space of the solutions of the WdWequation can be defined just by using the relativistic operator P withoutany need of picking out t as a preferred variable

For all nonrelativistic systems the configuration space has the structureC = C0timesR where t isin R and a function ψ(α t) in H is uniquely determinedby its restriction ψt = Rtψ on C0 for a fixed t

ψt(α) equiv ψ(α t) (527)

For each t denote Ht the space of the L2[C0] functions ψt(α) so thatRt H rarr Ht The spaces H and Ht are in one-to-one correspondence theinverse map Rminus1

t is the evolution determined by equation (517) In par-ticular H0 is the Hilbert space used in the nonrelativistic formulation ofthe quantum theory Under the identification between H and H0 given byR0 the scalar product defined above is precisely the usual scalar productof the nonrelativistic Hilbert space

This can be directly seen by noticing that the right-hand side of (524) is precisely(514) More explicitly let ψ(α t) =

sumn ψnHn(α)eminusiEnt be a function in H namely a

solution of the WdW equation Its restriction to t = 0 is ψ0(α) =sum

n ψnHn(α) and itsnorm in H0 is ||ψ0||2 =

intdα |ψ0(α)|2 =

sumn |ψn|2 A function f such that Pf = ψ is

for instance simply f(α t) = ψ0(α)δ(t) =sum

n ψnHn(α)int

dEeminusiEt (This is actuallynot in S0 but we could take a sequence of functions in S0 converging to f But f isin the domain of P and such a procedure would not give anything new) The norm


of ψ is

||ψ||2 = 〈f |f〉H = 〈f |P |f〉K

=

intdτ

intdα

intdt f(α t) eminusiτH f(α t)

=

intdτ

intdα

intdt

sum

n

ψn Hn(α)

intdEeiEteminusiτH

sum

m

ψm Hm(α)

intdEprimeeminusiEprimet

=sum

n

intdτ

intdt

intdE

intdEprime |ψn|2eiEteminusiτ(EprimeminusEn) eminusiEprimet

=sum

n

|ψn|2 = ||ψ0||2 (528)


Consider two events (α t) and (αprime tprime) in the extended configuration spaceSuppose we have observed the event (αprime tprime) What is the probability ofobserving the event (α t)

To measure this probability we need measuring apparata for α and fort In general these apparata will have a certain resolution say Δα andΔt The proper question is therefore what is the probability of observingan event included in the region R = (αplusmn Δα tplusmn Δt) It is important toremark that no realistic measuring device or detector can have Δα = 0nor Δt = 0 Most QM textbooks put much emphasis on the fact thatΔα gt 0 and completely ignore the fact that Δt gt 0 Consider thus tworegions R and Rprime If a detector at Rprime has detected the pendulum what isthe probability PRRprime that a detector at R detects the pendulum

If the regions R and Rprime are much smaller than any other physical quan-tity in the problem including the spatial and temporal separation of Rand Rprime a direct application of perturbation theory shows that

PRRprime = γ2 |〈R|Rprime〉|2 (529)

where γ2 is a dimensionless constant related to the efficiency of the de-tector (We may assume that a ldquoperfectrdquo detector is defined by γ = 1)The reader can repeat the calculation himself or find it for instance in[144] Explicitly we can write this probability as the modulus square ofthe amplitude PRRprime = |ARRprime |2

ARRprime = γ〈R|Rprime〉

radic〈R|R〉

radic〈Rprime|Rprime〉

(530)

〈R|Rprime〉 =int

Rdαdt

int

Rprimedαprimedtprime W (α t αprime tprime) (531)

Therefore the propagator has all the information about transition proba-bilities


Assume that R is sufficiently small so that the wave function ψ(α t) =〈α t|Rprime〉 is constant within R and has the value ψ(α t) Then we can writethe probability of the pendulum being detected in R as

PR = γ (VRCR)2 |ψ(α t)|2 (532)

where VR is the volume of the region R Now assume the region R hassides ΔαΔt A direct calculation (see [144]) shows that if Δt mΔα2hthen (VRCR)2 is proportional to Δα therefore

PR sim Δα |ψ(α t)|2 (533)

So for small regions we have the two important results that (i) the tem-poral resolution of the detector drops out from the detection probabilityand (ii) the probability is proportional to the spacial resolution of thedetector Because of (i) we can forget the temporal resolution of the de-tector and take the idealized limit of an instantaneous detector Becauseof (ii) we can associate a probability density in α to each infinitesimalinterval dα in α Fixing the overall normalization by requiring that anidealized perfect detector covering all values of α detects with certaintythis yields the results that |ψ(α t)|2 is the probability density in α todetect the system at (α t) with an instantaneous detector That is werecover the conventional probabilistic interpretation of the wave functionfrom (529)

In the opposite limit when Δt mΔα2h (VRCR)2 is proportionalto (Δt)minus12 Therefore

PR sim (Δt)minus12 |ψ(α t)|2 (534)

and we cannot associate a probability density in t with this detectorbecause the detection probability does not scale linearly with Δt Thedifferent behavior of the probability in α and t is a consequence of thespecific form of the dynamics

Partial observables in quantum theory Recall that α and t are partial ob-servables They determine commuting self-adjoint operators in K Theseact simply by multiplication Their common generalized eigenstates |α t〉are in S The states |α t〉 satisfy

〈α t|P |αprime tprime〉 = W (α t αprime tprime) (535)

We can view the states |α t〉 as ldquokinematical statesrdquo that do not know any-thing about dynamics They correspond to a single quantum event Theldquokinematicalrdquo scalar product of these states in K given below in (536)expresses only their independence while the ldquophysicalrdquo scalar product of


these states in H given in (535) expresses the physical relation betweenthe two events it determines the probability that one event happens giventhat the other happened

Do not confuse |α t〉 with |α t〉 The first is an eigenstate of α and t the secondis an eigenstate of α(t) They both determine (generalized) functions on C The state|α t〉 determines a delta distribution at the point (α t)

〈αprime tprime|α t〉 = δ(αprime α)δ(tprime t) (536)

while the state |α t〉 determines a solution of the Schrodinger equation This solutionhas support all over C and is such that on the line t = constant it is a delta functionin α

〈αprime t|α t〉 = δ(αprime α) (537)

while for different trsquos〈α t|αprime tprime〉 = W (α t αprime tprime) (538)

The relation between the two is simply

|α t〉 = P |α t〉 (539)

Notice that (538) and (539) give

W (α t αprime tprime) = 〈α t|P daggerP |α t〉H (540)

which is consistent with (535) because the definition of the scalar product in H (indi-

cated in (540) by 〈middot|middot〉H) is (526)

514 Boundary state space K and covariant vacuum |0〉In this subsection I introduce some notions that play an important role inthe field theoretical context Fix two times t = 0 and t Let H0 = L2[Rdα]be the space of the instantaneous quantum states ψ0 at t = 0 Let Ht sim H0

be the space of the instantaneous states ψt at t The probability amplitudeof measuring a state ψt at t if the state ψ0 was measured at t = 0 is

A = 〈ψt|eminusiH0t|ψ0〉 (541)

Consider the boundary state space

Kt = Hlowastt otimesH0 = L2[R2 dαdαprime] (542)

The linear functional ρt defined by

ρt(ψt otimes ψ0) = 〈ψt|eminusiH0t|ψ0〉 (543)

is well defined on Kt This functional captures the entire dynamical infor-mation about the system A linear functional on a Hilbert space definesa state I denote |0t〉 the state defined by ρt

ρt(ψ) = 〈0t|ψ〉Kt (544)

and call it the ldquodynamical vacuumrdquo state in boundary state space Kt


These definitions can be given the following physical interpretationWe make a measurement on the system at t = 0 and a measurement att We can measure the positions (α αprime) or the momenta or other com-binations The outcomes of the two measurements are not independentbecause of the dynamics but to start with letrsquos ignore the dynamicsAll possible outcomes of measurements at t = 0 (with their kinematicalrelations) are described by instantaneous states at t = 0 namely by thenonrelativistic Hilbert space H0 Similarly for t If we ignore the dynam-ical correlations we can view the two measurements as if they were doneon two independent systems and therefore we can describe the outcomesof the two measurements using the Hilbert space Kt Dynamics is a cor-relation between the two measurements These correlations are describedby a probability amplitude associated with any given couple of statesNamely to any state in Kt

It is a simple exercise that I leave to the reader to show that in therepresentation Kt = L2[R2dαdαprime] the state |0t〉 is precisely the propaga-tor

〈0t |α αprime〉 = W (α t αprime 0) (545)

Dynamical vacuum versus Minkowski vacuum Denote |0M〉 the lowesteigenstate of H0 in H0

〈α|0M〉 = H0(α) =1radic2π

eminus12α2

(546)

and call it the ldquoMinkowskirdquo vacuum because of its analogy with thevacuum state of the quantum field theories on Minkowski space Considerthe analytic continuation in imaginary time of the propagator (510)

W (αminusit αprime 0) = 〈α|eminusH0t|αprime〉 =sum

n

Hn(α) eminusEnt Hn(αprime) (547)

For large t only the lowest-energy state survives in the sum and we have

W (αminusit αprime 0) minusrarrtrarrinfin H0(α) eminusE0t H0(αprime) (548)

Using the definitions of the previous section this can be written as

limtrarrinfin

eE0t |0minusit〉 = |0M〉 otimes 〈0M| (549)

(The ket and bra in the right-hand side are in H0 while the ket in theleft-hand side is in K = Hlowast

0 otimesH0) This expression relates the dynamicalvacuum |0t〉 and the Minkowski vacuum |0M〉 We will use this equationto find the quantum states corresponding to Minkowski spacetime fromthe spinfoam formulation of quantum gravity


The boundary state space K and covariant vacuum |0〉 The constructionabove can be given a more covariant formulation as follows Consider theHilbert space

K = Klowast otimesK = L2[R4dα dt dαprimedtprime] = L2[G] (550)

I call this space the ldquototalrdquo quantum space The propagator defines apreferred state |0〉 in K

〈α t αprime tprime|0〉 = W (α t αprime tprime) (551)

I call this state the covariant vacuum stateTo run a complete experiment in a one-dimensional quantum system

we need to measure two events a ldquopreparationrdquo and a ldquomeasurementrdquoThe space K describes all possible (a priori equal) outcomes of the mea-surements of these two events Any couple of measurements is representedby operators on K and any outcome is represented by a state ψ isin K whichis an eigenstate of these operators The dynamics is given by the bra 〈0|The probability amplitude of the given outcome is determined by

A = 〈0|ψ〉 (552)

This is a compact and fully covariant formulation of quantum dynamics


The interpretation of the theory is already entirely contained in (529)Still to make the connection with the nonrelativistic formalism moredirect we can also consider operators related to observable quantitieswhose probability distribution can be predicted by the theory

In the classical theory if we know the (relativistic) state of the pen-dulum we can predict the value of α when t has value say t = T Inthe quantum theory there is an operator that corresponds to this physi-cal prediction It is of course the Heisenberg position operator (55) fort = T that is α(T ) (For clarity it is convenient to distinguish the par-ticular numerical value T from the argument of the wave function t) Inow define and characterize this operator in a relativistic language

First of all notice that the operator α(T ) defined on H0 in (55) is infact well defined on H as

α(T ) = Rminus10 α(T )R0 = Rminus1

0 eiTH0 α eminusiTH0R0 = Rminus1T α RT (553)

The operator α(T ) can be directly defined on H without referring to H0as follows Consider the operator

a(T ) = eminusiω(Tminust)(α + i

pαmω

)(554)


and its real part

α(T ) = Re [a(T )] =a(T ) + adagger(T )

2 (555)

defined on S These operators commute with H for any T Therefore theyare well defined on the space of the solutions of (517) namely on H Therestriction of the operator (555) to H is precisely the operator (553)

The operator α(T ) is characterized by two properties First the factthat it commutes with the hamiltonian

[α(T ) H] = 0 (556)

Second if we put T = t in the expressions (554) (555) we obtain αThat is α(T ) is defined as an operator function α(T )(α pα t) such that

α(T )(α pα T ) = α (557)

Intuitively these two equations determine α(T ) since the second fixes itat t = T and the first evolves it for all t Operators of this kind arecalled ldquoevolving constants of motionrdquo They are ldquoevolvingrdquo because theydescribe the evolution (here the evolution of α with respect to t) theyare ldquoconstants of motionrdquo because they commute with the hamiltonianIn GR the operators of this kind are independent from the temporalcoordinate

52 Relativistic QM

In the previous section I used the example of a pendulum to introducea certain number of notions on which a relativistic hamiltonian formula-tion of QM can be based It is now time to attempt a general theory ofrelativistic QM


Kinematical states Kinematical states form a space S in a rigged Hilbertspace S sub K sub S prime

Partial observables A partial observable is represented by a self-adjointoperator in K Common eigenstates |s〉 of a complete set of com-muting partial observables are denoted quantum events

Dynamics Dynamics is defined by a self-adjoint operator H in K the(relativistic) hamiltonian The operator from S to S prime

P =int



is (sometimes improperly) called the projector (The integrationrange in this integral depends on the system) Its matrix elements


are called transition amplitudes

Probability Discrete spectrum the probability of the quantum event sgiven the quantum event sprime is

Pssprime = |W (s sprime)|2 (560)

where |s〉 is normalized by 〈s|P |s〉 = 1 Continuous spectrum theprobability of a quantum event in a small spectral region R given aquantum event in a small spectral region Rprime is

PRRprime =

∣∣∣∣∣

W (RRprime)radic

W (RR)radicW (Rprime Rprime)

∣∣∣∣∣

2

(561)

whereW (RRprime) =

int

Rds

int

Rprimedsprime W (s sprime) (562)

To this we may add

Boundary quantum space and covariant vacuum For a finite number ofdegrees of freedom the boundary Hilbert space K = Klowast otimes K rep-resents any observations of pairs of quantum events The covariantvacuum state |0〉 isin K defined by

〈0|(ψ otimes ψprime)〉K = 〈ψ|P |ψprime〉K (563)

expresses the dynamics It determines the correlation probabilityamplitude of any such observation The extension to QFT is con-sidered in Section 535

States A physical state is a solution of the WheelerndashDeWitt equation

Hψ = 0 (564)

Equivalently it is an element of the Hilbert space H defined by thequadratic form 〈 middot |P | middot 〉 on S (Elements of K are called kinematicalstates and elements of K are called boundary states)

Complete observables A complete observable A is represented by a self-adjoint operator on H A self-adjoint operator A in K defines acomplete observable if

[AH] = 0 (565)


Projection If the value of the observable A is restricted to the spectral in-terval I the state ψ becomes the state PIψ where PI is the spectralprojector on the interval I If an event corresponding to a sufficientlysmall region R is detected the state becomes |R〉

A relativistic quantum system is defined by a rigged Hilbert space ofkinematical states K and a set of partial observables Ai including a rela-tivistic hamiltonian operator H Alternatively it is defined by giving theprojector P

Axiomatizations are meant to be clarifying not prescriptive The struc-ture defined above is still tentative and perhaps incomplete There are as-pects of this structure that deserve to be better understood clarified andspecified Among these is the precise meaning of the ldquosmallnessrdquo of the re-gion R in the case of the continuum spectrum and the correct treatmentof repeated measurements On the other hand the conventional structureof QM is certainly physically incomplete in the light of GR The aboveis an attempt to complete it making it general relativistic


In general a quantum system (K Ai H) has a classical limit which isa relativistic mechanical system (C H) describing the results of observa-tions on the system at scales and with accuracy larger than the Planckconstant In the classical limit Heisenberg uncertainty can be neglectedand a commuting set of partial observables Ai can be taken as coordinatesof a commutative relativistic configuration space C

If we are given a classical system defined by a nonrelativistic config-uration space C with coordinates qa and by a relativistic hamiltonianH(qa pa) a solution of the quantization problem is provided by the mul-tiplicative operators qa the derivative operators

pa = minusihpart

partqa (566)

and the hamiltonian operator

H = H

(qaminusih

part

partqa

)(567)

on the Hilbert space K = L2[Cdqa] or more precisely the Gelfand tripledetermined by C and the measure dqa The physics is entirely containedin the transition amplitudes

W (qa qprimea) = 〈qa|P |qprimea〉 (568)


where the states |qa〉 are the eigenstates of the multiplicative operatorsqa

In turn the space K has the structure

K = L2[G] (569)

As we shall see this remains true in field theory and in quantum gravityThe space G was defined in Section 325 for finite-dimensional systems inSection 333 for field theories and in Section 434 in the case of gravity

In the limit h rarr 0 the WheelerndashDeWitt equation becomes the rela-tivistic HamiltonndashJacobi equation (359) and the propagator has the form(writing q equiv (qa))

W (q qprime) simsum

i

Ai(q qprime) eihSi(qq

prime) (570)

where Si(q qprime) are the different branches of the Hamilton function as in(389) Now the reverse of each path is still a path The Hamilton functionand the amplitude of a reversed path acquires a minus giving

W (q qprime) simsum

i

Ai(q qprime) sin[

1h Si(q qprime)

] (571)

and W is real Assuming only one path matters

W (q qprime) sim A(q qprime) sin[

1h S(q qprime)

](572)

and we can write for instance

limhrarr0

1W

ihpart

partqaih

part

partqbW (q qprime) =

partS(q qprime)partqa

partS(q qprime)partqb

(573)

This equation provides a precise relation between a quantum theory (en-tirely defined by the propagator W (q qprime)) and a classical theory (entirelydefined by the Hamilton function S(q qprime)) Using (386) and (566) thisequation can be written in the suggestive form

limhrarr0

1W

papbW (q qprime) = pa(q qprime) pb(q qprime) (574)


Pendulum An example of relativistic formalism is provided by the quan-tization of the pendulum described in the previous section the kinematicalstate space is K = L2[R2dαdt] The partial observable operators are the


multiplicative operators α and t acting on the functions ψ(α t) in K Dy-namics is defined by the operator H given in (518) The WheelerndashDeWittequation is therefore

(minusih

part

parttminus h2

2mpart2

partα2+

mω2

2α2

)Ψ(α t) = 0 (575)

H is a space of solutions of this equation The ldquoprojectorrdquo operator P K rarr H defined by H is given in (523) and defines the scalar product inH Its matrix elements W (α t αprime tprime) between the common eigenstates ofα and t are given by the propagator (511) They express all predictionsof the theory Because of the specific form of H these define a probabilitydensity in α but not in t as explained in Section 513

Equivalently the quantum theory can be defined by the boundarystate space K = L2[G] where G is the boundary space of the classi-cal theory with coordinates (α t αprime tprime) and the covariant vacuum state〈α t αprime tprime|0〉 = W (α t αprime tprime) which determines the amplitude A = 〈0|ψ〉of any possible outcome ψ isin K of a preparationmeasurement experiment

Timeless double pendulum An example of a relativistic quantum sys-tem which cannot be expressed in terms of conventional relativistic quan-tum mechanics is provided by the quantum theory of the timeless system(340) The kinematical Hilbert space K is L2[R2 dadb] and the WheelerndashDeWitt equation is

12

(minush2 part2

parta2minus h2 part2

partb2+ a2 + b2 minus 2E

)Ψ(a b) = 0 (576)

Below I describe this system in some detail

States Since H = Ha + Hb minus E where Ha (resp Hb) is the harmonicoscillator hamiltonian in the variable a (resp b) this equation is easy tosolve by using the basis that diagonalizes the harmonic oscillator Let

ψn(a) = 〈a|n〉 =1radicn

Hn(a) eminusa22h (577)

be the normalized nth eigenfunction of the harmonic oscillator with eigen-value En = h(n+12) Here Hn(a) is the nth Hermite polynomial Thenclearly

Ψnanb(a b) = ψna(a)ψnb

(b) equiv 〈a b|na nb〉 (578)

solves (576) ifh(na + nb + 1) = E (579)


Therefore the quantum theory exists (with this ordering) only if Eh =N + 1 is an integer which we assume from now on The general solutionof (576) is

Ψ(a b) =sum

n= 0N

cn ψn(a) ψNminusn(b) (580)

Therefore H is an (N+1)-dimensional proper subspace of K An orthonor-mal basis is formed by the N + 1 states |nN minus n〉 with n = 0 N

Projector The projector P S rarr H is in fact a true projector and canbe written explicitly as

P =sum

n= 0N

|nN minus n〉〈nN minus n| (581)

This can be obtained from (558) by taking the integration range to be2π determined by the range of τ in the classical hamiltonian evolutionor by the fact that H is the generator of an U(1) unitary action on Kwith period 2π Indeed

int 2π

0dτ eminus

ihτH =

int 2π

0dτ

sum

nanb

|na nb〉eminusihτ(h(na+nb+1)minusE)〈na nb|

=sum

nanb

|na nb〉δ(na + nb + 1 minus Eh)〈na nb|

= P (582)

Transition amplitudes The transition amplitudes are the matrix elementsof P In the basis that diagonalizes a and b

W (a b aprime bprime) = 〈a b|P |aprime bprime〉 =sum

n=0N

〈a b|nN minus n〉〈nN minus n|aprime bprime〉

(583)Explicitly this is

W (a b aprime bprime) =sum

n=0N

1radic

n(N minus n)Hn(a)HNminusn(b)

timesHn(aprime)HNminusn(bprime) eminus(a2+b2+aprime2+bprime2)2h (584)

This function codes all the properties of the quantum system Roughlyit determines the probability density of measuring (a b) if (aprime bprime) wasmeasured Let us study its properties


Semiclassical limit of the projector Notice that by inserting (582) into(583) we can write the projector as


0dτ 〈a b|eminus i

hHτ |aprime bprime〉

=int 2π

0dτ e

ihEτ 〈a|eminus i

hHaτ |aprime〉〈b|eminus i

hHbτ |bprime〉 (585)


0dτ e

ihEτW (a aprime τ) W (b bprime τ) (586)

where W (a aprime τ) is the propagator of the harmonic oscillator in a physicaltime τ given in (511) Inserting (511) in (586) we obtain


0dτ

1sin τ

eminusihS(abaprimebprimeτ) (587)

where S(a b aprime bprime τ) is given in (3101) We can evaluate this integral ina saddle-point approximation This gives

W (a b aprime bprime) simsum

i

1sin τi

eminusihS(aaprimebbprimeτi) (588)

where the τi are determined by

partS(a b aprime bprime τ)partτ

∣∣∣∣τ=τi(abaprimebprime)

= 0 (589)

But this is precisely (3102) that defines the value of τ giving the Hamil-ton function of the timeless system This equation has two solutions cor-responding to the two portions into which the ellipse is cut The relationbetween the two actions is given in (3103) Recalling that Eh is aninteger this gives


(eminus

ihS(aaprimebbprime) minus e

ihS(aaprimebbprime)

) (590)

that is


sin[

1hS(a aprime b bprime)

] (591)

as in (572) Here sim indicates equality in the lowest order in h Thisequation expresses the precise relation between the quantum theory andthe classical theory


Propagation ldquoforward and backward in timerdquo Notice that the two termsin (590) have two natural interpretations One is that they representthe two classical paths going from (aprime bprime) to (a b) in C The other moreinteresting interpretation is that they correspond to a trajectory goingfrom (aprime bprime) to (a b) and a ldquotime reversedrdquo trajectory going from (a b)to (aprime bprime) In fact the projector (which recall is real) can be naturallyinterpreted as the sum of two propagators one going forward and onegoing backward in the parameter time τ

The distinction between forward and backward in the parameter timeτ has no physical significance in the classical theory because the physicsis only in the ellipses in C not in the orientation of the ellipses

However in the quantum theory we can identify in H ldquoclockwise-movingrdquo and ldquoanticlockwise-movingrdquo components These components arethe eigenspaces of the positive and negative eigenvalues of the angular mo-mentum operator L = apartbminusbparta (or L = partφ where a = r sinφ b = r cosφ)Thus we can write wave packets ldquotraveling along the ellipses purely for-ward or purely backward in the parameter timerdquo If we consider only alocal evolution in a small region of C and we interpret say b as the in-dependent time variable and a as the dynamical variable then these twocomponents have respectively positive and negative energy In a sensethey can be viewed as particles and antiparticles


I assume the reader is familiar with standard quantum field theory (QFT) Here Iillustrate the connection between QFT and the relativistic formalism developed aboveand I recall a few techniques that will be used in Part II and are not widely knownOf particular importance are the distinction between Minkowski vacuum and covariantvacuum the functional representation of a field theory and the construction of thephysical Hilbert space of lattice YangndashMills theory

In Chapter 3 we have seen that a classical field theory can be definedcovariantly by the boundary space G of closed surfaces α in a finite-dimensional space C and a relativistic hamiltonian H on T lowastG For in-stance in a scalar field theory C = M timesR has coordinates (xμ φ) wherexμ is a point in Minkowski space and φ a field value A surface α isdetermined by the two functions

α = [xμ(τ) ϕ(τ)] (592)

and determines a boundary 3-surface xμ(τ) in Minkowski space M andboundary values φ(x(τ)) = ϕ(τ) of the field on this surface

A quantization of the theory can be obtained precisely as in the finite-dimensional case in terms of a boundary state space K of functionalsΨ[α] on G Notice however that the difference between the kinematical


state space K and the boundary state space K is far less significant infield theory than for finite-dimensional systems In the finite-dimensionalcase the states ψ(qa) in K are functions on the extended configurationspace C while the states ψ(qa qaprime) in K are functions on the boundaryspace G = C times C In the field theoretical case both states have the formΨ[α] The difference is that the states in K are functions of an ldquoinitialrdquosurface α where xμ(τ) can be for instance the spacelike surface x0 = 0in this case α contains only one-half of the data needed to determine asolution of the field equations On the other hand the states Ψ[α] in Kare functions of a closed surface α In fact the only difference betweenK and K is in the global topology of α If we disregard this and considerlocal equations we can confuse K and K (see Section 535)

The relativistic hamiltonian is given in (3192) The complete solutionof the classical dynamics is known if we know the Hamilton function S[α]which is the value of the action

S[α] = S[R φ] =int

RL(φ(x) partμφ(x))d4x (593)

where R is the four-dimensional region bounded by x(τ) and φ(x) isthe solution of the equations of motion in this region determined bythe boundary data φ(x(τ)) = ϕ(τ) If there is more than one of thesesolutions we write them as φi(x) and the Hamilton function is multivalued

Si[α] = S[R φi] =int

RL(φi(x) partμφi(x))d4x (594)

The relativistic Hamiltonian gives rise to the WheelerndashDeWitt equation

H

[xμ φminusih

δ

δxμminusih

δ

δϕ

](τ) Ψ[α] = 0 (595)

precisely as in the finite-dimensional case The HamiltonndashJacobi equation(3193) can be interpreted as the eikonal approximation for this waveequation

The complete solution of the dynamics is known if we know the propa-gator W [α] which is a solution of this equation Formally the field prop-agator can be written as a functional integral

W [α] =int

φ(x(τ))=ϕ(τ)[Dφ] eminus

ihS[Rφ] (596)

Of course one should not confuse the field propagator W [α] with theFeynman propagator The first propagates field the second the particlesof a QFT The first is a functional of a surface and the value of the field


on this surface the second is a function of two spacetime points To thelowest order in h the saddle-point approximation gives

W [α] simsum

i

Ai[α] eminusihSi[α] (597)

There are two characteristic difficulties in the field theoretical contextthat are absent in finite dimensions the definition of the scalar productand the need to regularize operator products

First in finite dimensions a measure dqa on C is sufficient to definean associated L2 Hilbert space of wave functions In the field theoreticalcase we have to define the scalar product in some other way The scalarproduct must respect the invariances of the theory and must be such thatreal classical variables be represented by self-adjoint operators This isbecause self-adjoint operators have a real spectrum and the spectrumdetermines the values that a quantity can take in a measurement Givena set of linear operators on a linear space the requirement that they areself-adjoint puts stringent conditions on the scalar product As we shallsee in all cases of interest these requirements are sufficient to determinethe scalar product

Second local operators are in general distributions and their productsare ill defined Operator products arise in physical observable quantities aswell as in the dynamical equation namely in the WheelerndashDeWitt equa-tion In particular functional derivatives are distributions In the classicalHamiltonndashJacobi equation we have products of functional derivatives ofthe HamiltonndashJacobi functional which are well-defined products of func-tions In the corresponding quantum WheelerndashDeWitt equation thesebecome products of functional-derivative operators which are ill definedwithout an appropriate renormalization procedure The definition of gen-erally covariant regularization techniques will be a major concern in thesecond part of the book


Consider a simple free scalar theory where V = 0 I describe this well-known QFTin some detail in order to illustrate certain techniques that play a role in quantumgravity In particular I illustrate the functional representation of quantum field the-ory a simple form of the WheelerndashDeWitt equation the general form of W [α] and itsphysical interpretation The functional representation is the representation in whichthe field operator is diagonal The quantum states will be represented as functionalsΨ[φ] = 〈φ|Ψ〉 where |φ〉 is the (generalized) eigenstate of the field operator with eigen-value φ(x) The relation between this representation and the conventional one on the

Fock basis |k1 k1〉 is precisely the same as the relation between the Schrodingerrepresentation ψ(x) and the one on the energy basis |n〉 for a simple harmonic oscilla-tor I also illustrate the way in which the scalar product on the space of the solutions


of the WheelerndashDeWitt equation is determined by the reality properties of the fieldoperators

To start with and to connect the generally covariant formalism de-scribed above with conventional QFT letrsquos restrict the surface x(τ) in α toa spacelike surface xμ(τ) = (t τ) in Minkowski space Then α = [t φ(x)]and Ψ[α] = Ψ[t φ(x)] The HamiltonndashJacobi equation (3186) reduces to(3190) The corresponding quantum WheelerndashDeWitt equation becomes

ihpart

parttΨ = H0Ψ (598)

where the nonrelativistic hamiltonian operator H0 is

H0 =int

d3x H0

[φminusih

δ

δφ

](x) (599)

and H0[φ p](x) is given in (3195) The factor ordering of this operatorcan be chosen in order to avoid the divergence that would result from thenaive factor ordering

H0 naive =12

intd3x

[minush2 δ

δφ(x)δ

δφ(x)+|nablaφ|2(x) + m2φ2(x)

] (5100)

The Fourier modes

φ(k) = (2π)minus32

intd3x e+ikmiddotxφ(x) (5101)

decouple

H0 =12

intd3k

[p2(k) + ω2(k)φ2(k)

] (5102)

where ω =radic

|k|2 + m2 The dangerous divergence is produced by the

vacuum energy of the quantum oscillators associated with each mode kand can be avoided by normal ordering In terms of the positive andnegative frequency fields

a(k) =iradic2ω

p(k) +radic

ω

2φ(k) (5103)

adagger(k) = minus iradic2ω

p(minusk) +radic

ω

2φ(minusk) (5104)

the hamiltonian reads

H0 =int

d3k ω(k) adagger(k) a(k) (5105)


We define the quantum hamiltonian by this equation where

a(k) =hradic2ω

δ

δφ(k)+

radicω

2φ(k) (5106)

adagger(k) = minus hradic2ω

δ

δφ(minusk)+

radicω

2φ(minusk) (5107)

The lowest-energy eigenvector of the hamiltonian has vanishing eigen-value and is called the Minkowski vacuum state This state is usuallydenoted |0〉 I denote it here as |0M〉 where M stands for Minkowski inorder to distinguish it from other vacuum states that will be introducedlater on The Minkowski vacuum state is determined by a(k)|0M〉 = 0 Inthe functional representation this state reads

Ψ0M [φ] equiv 〈φ|0M〉 (5108)

and is determined by

a(k)Ψ0M [φ] =hradic2ω

δ

δφ(k)Ψ0M [φ] +

radicω

2φ(k)Ψ0M [φ] = 0 (5109)

The solution of this equation gives the functional form of the vacuumstate

Ψ0M [φ] = Neminus12h

intd3k ω(k)φ(k)φ(k) (5110)

The one-particle state with momentum k is created by adagger(k)

Ψk[φ] equiv 〈φ|k〉 = adagger(k)Ψ0M [φ] =

radic2ω φ(k) Ψ0M [φ] (5111)

It has energy hω(k) Therefore the time-dependent state

Ψk[t φ] equiv

radic2ω eminusiω(k)t φ(k) Ψ0M [φ] (5112)

is a solution of the WheelerndashDeWitt equation (598)A generic one-particle state with wave function f(k) is defined by

|f〉 equivint

d3kradic2ω

f(k) |k〉 (5113)

and its functional representation is therefore

Ψf [φ] equiv 〈φ|f〉 =int

d3k f(k) φ(k) Ψ0[φ] (5114)

or

Ψf [φ] = φ[f ] Ψ0[φ] (5115)


where

φ[f ] =int

d3k f(k) φ(k) (5116)

The corresponding solution of the WheelerndashDeWitt equation (598) is

Ψf [t φ] =int

d3k f(k) eminusiω(k)t φ(k) Ψ0[φ] (5117)

or in Fourier transform

Ψf [t φ] =int

d3x F (t x) φ(x) Ψ0[φ] (5118)

where

F (x) = F (t x) = (2π)minus32

intd3k ei(kmiddotxminusω(k)t) f(k) (5119)

is a positive-energy solution of the KleinndashGordon equationThe n-particle states |k1 kn〉 can be obtained using again the cre-

ation operator adagger(k) in the well-known way They have energy h(ω1 +middot middot middot + ωn) where ωi = ω(ki) The general solution of the WheelerndashDeWittequation is therefore

Ψ[t φ] =sum

n

intd3k1 d3knradic

2ω1 2ωnf(k1 kn) eminusi(ω1+middotmiddotmiddot+ωn)t

times adagger(k1) adagger(kn)Ψ0[φ] (5120)

The space F of these solutions labeled by the functions f(k1 kn) isthe physical state space H of the theory Since Ψ[t φ] is determined byΨ[φ] = Ψ[0 φ] we can also represent the quantum states by their valueon the t = 0 surface namely as functionals Ψ[φ]

Scalar product The scalar product can be determined on the space ofthe solutions of the WheelerndashDeWitt equation from the requirement thatreal quantities are represented by self-adjoint operators The scalar fieldφ(x) and its momentum p(x) are real Therefore we must demand thatthe corresponding operators are self-adjoint It follows that the operatoradagger(k) is the adjoint of the operator a(k) Using this we obtain easily

〈k|kprime〉 = 〈adagger(k)0|adagger(kprime)0〉 = 〈0|a(k)adagger(kprime)0〉 = hδ(k minus kprime) (5121)

It follows from (5113) that

〈f |f prime〉 = h

intd3k

2ωf(k) f prime(k) (5122)


(Recall that d3k2ω is the Lorentz-invariant measure) Therefore the one-particle state space is H1 = L2[R3 d3k2ω] Let us write

f(x) = (2π)minus32

intd3k

2ωeikx f(k) (5123)

The spacetime function

f(x) = f(t x) = (2π)minus32

intd3k

2ωei(kmiddotxminusω(k)t) f(k) (5124)

is a positive-frequency solution of the KleinndashGordon equation with initialvalue f(0 x) = f(x) (not to be confused with the one defined in (5119)which is F = iparttf) Then easily

〈f |f prime〉 = ihint

d3x[f(x)part0f

prime(x) minus f prime(x)part0f(x)]t=0

(5125)

This is the well-known KleinndashGordon scalar product which is positivedefinite on the positive-frequency solutions

Notice that the one-particle Hilbert space can be represented in various equivalentmanners It is

bull the space of the positive-frequency solutions f(x) of the KleinndashGordon equationwith the scalar product (5125)

bull the space H = L2[R3 d3k2ω] of the functions f(k)

bull the space H = L2[R4 δ(k2 + m2)θ(k0)d4k] of the functions f(k)

bull the space H = L2[R3 d3x] of the functions

f(x) =

intd3kradic2ω

eikmiddotx f(k) (5126)

(the position operator x in this representation is obviously self-adjoint it is thewell-known NewtonndashWigner operator which has a far more complicated form inother representations)

bull and so on

Using the same technique the entire space F can be equipped with ascalar product The resulting Hilbert space is of course the well-knownFock space over this one-particle Hilbert space


Consider now a surface Σt formed by two parallel spacelike planes inMinkowski space say xμ1 (τ) = (t1 τ) and xμ2 (τ) = (t2 τ) Consider twoscalar fields ϕ1(τ) ϕ2(τ) on these planes Let α be the union of thesetwo surfaces with their fields that is α is formed by two disconnected


components α = α1 cup α2 = [xμ1 (τ) ϕ1(τ)] cup [xμ2 (τ) ϕ2(τ)] Consider thefield propagator (596) for this value of α Thus W [α] = W [t1 ϕ1 t2 ϕ2]In this case we can simply write

W [t1 ϕ1 t2 ϕ2] = 〈t1 ϕ1|t2 ϕ2〉 = 〈ϕ1|eminusihH0(t1minust2)|ϕ2〉 (5127)

The calculation of the propagator is simplified by the fact that the quan-tum field theory is essentially a collection of one harmonic oscillator foreach mode k Using the propagator of the harmonic oscillator given in(511) one obtains with some algebra

W [t1 ϕ1 t2 ϕ2]

= N exp

minus i2h

intd3k

(2π)3ω

[(|ϕ1|2 + |ϕ2|2

)cos[ω(t1 minus t2)] minus 2ϕ1ϕ2

sin[ω(t1 minus t2)]

]

(5128)

where N is the formal divergent normalization factor

N simprod

k

radicmω(k)

hexp

minusV

2

intd3k

(2π)3ln

[sin[ω(k)(t1 minus t2)]

] (5129)

This has the form (597) see the classical Hamilton function given in(3170)

Minkowski vacuum from the euclidean field propagator The state spaceat time zero Ht=0 is Fock space where the field operators ϕ(x) = φ(x t)and the hamiltonian H0 are defined Fock space is separable and thereforeadmits countable bases Choose a basis |n〉 of eigenstates of H0 witheigenvalues En and consider the operator

W (T ) =sum

n

eminusThEn |n〉〈n| (5130)

In the large-T limit this becomes the projection on the only eigenstatewith vanishing energy namely the Minkowski vacuum

limTrarrinfin

W (T ) = |0M〉〈0M| (5131)

In the functional Schrodinger representation the operator (5130) reads

W [ϕ1 ϕ2 T ] = 〈ϕ1|eminusihH0(minusiT )|ϕ2〉 = W [0 ϕ1 iT ϕ2] (5132)


Therefore it is the analytical continuation of the field propagator (5127)and satisfies the euclidean Schrodinger equation

minushpart

partTW [ϕ1 ϕ2 T ] = Hϕ1 W [ϕ1 ϕ2 T ] (5133)

We can obtain the vacuum (up to normalization) as

Ψ0M [ϕ] = 〈ϕ|0M〉 = limTrarrinfin

W [ϕ 0 T ] (5134)

We can derive all particle scattering amplitudes from the functionalW [ϕ1 ϕ2 T ] For instance the 2-point function can be obtained as theanalytic continuation of the Schwinger function

S(x1 x2) = limTrarrinfin

intDϕ1Dϕ2 W [0 ϕ1 T ]ϕ1(x1)

timesW [ϕ1 ϕ2 (t1 minus t2)]ϕ2(x2) W [ϕ2 0 T ] (5135)

This can be generalized to any n-point function where the times t1 tnare on the t = 0 and the t = T surfaces these in turn are sufficient tocompute all scattering amplitudes since time dependence of asymptoticstates is trivial

W [ϕ1 ϕ2 T ] admits the well-defined functional integral representation

W [ϕ1 ϕ2 T ] =int

φ|t=T =ϕ1

φ|t=0=ϕ2

Dφ eminus1hSET [φ] (5136)

Here the integral is over all fields φ on the strip R bounded by the twosurfaces t = 0 and t = T with fixed boundary value The action SE

T [φ]is the euclidean action Notice that using this functional integral repre-sentation the expression (5135) for the Schwinger function becomes thewell-known expression

S(x1 x2) =int

Dφ φ(x1) φ(x2) eminus1hSE [φ] (5137)

obtained by joining at the two boundaries the three functional integralsin the regions tltt2 t2lttltt1 and t1ltt The functional W [ϕ1 ϕ2 T ] canbe computed explicitly in the free field theory Its expression in terms ofthe Fourier transform ϕ of ϕ is the analytic continuation of (5128)

W [ϕ1 ϕ2 T ] = N expminus 1

2h

intd3k

(2π)3ω

( |ϕ1|2 + |ϕ2|2tanh (ωT )

minus 2ϕ1ϕ2

sinh (ωT )

)

(5138)


The dynamical vacuum |0ΣT〉 Consider the boundary state space KΣt

associated with the entire surface Σt as in Section 514 That is defineKΣt = Ht otimes Hlowast

0 Denote ϕ = (ϕ1 ϕ2) a field on Σt The field basis ofthe Fock space induces the basis |ϕ〉 = |ϕ1 ϕ2〉 equiv |ϕ1〉t otimes 〈ϕ2|0 in KΣt the vectors |Ψ〉 of KΣt are written in this basis as functionals Ψ[ϕ] =Ψ[ϕ1 ϕ2] equiv 〈ϕ1 ϕ2|Ψ〉 This is the field theoretical generalization of theboundary state space defined in (542)

The functional W defines a preferred state in this Hilbert space as in(543)ndash(544) Denote this state |0Σt〉 and call it the dynamical vacuum Itis defined by 〈ϕ|0Σt〉 equiv W [t ϕ1 0 ϕ2] This state expresses the dynamicsfrom t = 0 to t A state in the tensor product of two Hilbert spaces definesa linear mapping between the two spaces The linear mapping from Ht=0

to Ht=T defined by |0ΣT〉 is precisely the time evolution eminusiHt

The interpretation of this state is the same as in the finite-dimensionalcase The tensor product of two quantum state spaces describes the en-semble of the measurements described by the two factors Therefore KΣt

is the space of the possible results of all measurements performed at time0 and at time t Observations at two different times are correlated by thedynamics Hence KΣt is a ldquokinematicalrdquo state space in the sense that itdescribes more outcomes than the physically realizable ones Dynamics isthen a restriction on the possible outcomes of observations It expressesthe fact that measurement outcomes are correlated The linear functional〈0Σt | on KΣt assigns an amplitude to any outcome of observations Thisamplitude gives us the correlation between outcomes at time 0 and out-comes at time t

Therefore the theory can be represented as follows The Hilbert spaceKΣt describes all possible outcomes of measurements made on Σt The dy-namics is given by a single bra state 〈0Σt | Kt rarr C For a given collectionof measurement outcomes described by a state |Ψ〉 the quantity 〈0Σt |Ψ〉gives the correlation probability amplitude between these measurements

Using (5131) we have then the relation between the dynamical vacuumand the Minkowski vacuum (the braket mismatch is apparent only asthe three states are in different spaces)

limtrarrinfin

|0Σminusit〉 = |0M〉 otimes 〈0M| (5139)


So far I have considered only boundary surfaces formed by two parallelspacelike planes This restriction is sufficient and convenient in ordinaryQFT on Minkowski space but it has no meaning in a generally covariantcontext It is therefore necessary to consider arbitrary boundary surfacesso let us study the extension of the formalism to the case where the surface


Σ instead of being formed by two parallel planes is the boundary of a(sufficiently regular) arbitrary finite region of spacetime R

Let Σ be a closed connected 3d surface in Minkowski spacetime withthe topology (but in general not the geometry) of a 3-sphere and Σ =partR Let ϕ be a scalar field on Σ and consider the functional

W [ϕΣ] =int

φ|Σ=ϕDφ eminusSE

R[φ] (5140)

The integral is over all 4d fields on R that take the value ϕ on Σ and theaction in the exponent is the euclidean action where the 4d integral is overR In the free theory the integral is a well-defined gaussian integral andcan be evaluated The classical equations of motion with boundary valueϕ on Σ form an elliptic system which in general has a solution φcl[ϕ] thatcan be obtained by integration from the Green function for the shape R Achange of variable in the integral reduces it to a trivial gaussian integrationtimes eminusSE

R[ϕcl] Here SER[ϕ] is the field theoretical Hamilton function the

action of the bulk field determined by the boundary condition ϕ

W [ϕΣ] can be defined in the Minkowski regime as well If Σ is a rectangular boxin Minkowski space let ϕ = (ϕout ϕin ϕside) be the components of the field on thespacelike bases and timelike side Consider the field theory defined in the box withtime-dependent boundary conditions ϕside and let U [ϕside] be the evolution operatorfrom t = 0 to t = T generated by the (time-dependent) hamiltonian of the theoryThen we can write

W [ϕΣ] equiv 〈ϕout|U [ϕside]|ϕin〉 (5141)

In particular if ϕside is constant in time W can be obtained by analytic continuationfrom the euclidean functional More generally we can write the formal definition

W [ϕΣ] =

int

φ|Σ=ϕ

Dφ eiSR[φ] (5142)

Notice that W [ϕΣ] is a function on the space G defined in Section 333This space represents all possible ensembles of classical field measurementson a closed surface namely the minimal data for a local experimentFormally functions on G define the quantum state space K and W [ϕΣ]defines the preferred covariant vacuum state |0〉 in K

Local Schrodinger equation W [ϕΣ] satisfies a local functional equationthat governs its dependence on Σ Let τ be arbitrary coordinates on ΣRepresent the surface and the boundary fields as Σ τ rarr xμ(τ) andϕ τ rarr ϕ(τ) Let nμ(τ) be the unit length normal to Σ Then

nμ(τ)δ

δxμ(τ)W [ϕΣ] = H(τ) W [ϕΣ] (5143)


where H(x) is an operator obtained by replacing π(x) by minusiδδϕ(x) inthe hamiltonian density

H(x) = gminus12π2(x) + g

12 (|nablaϕ|2 + m2ϕ2) (5144)

Here g is the determinant of the induced metric on Σ and the norm istaken in this metric (see [145 146]) The local HamiltonndashJacobi equation(3186) can be viewed as the eikonal approximation of this equation SinceW is independent from the parametrization we have

partxμ(τ)partτ

δ

δxμ(τ)W [ϕΣ] = P (τ) W [ϕΣ] (5145)

where the linear momentum is P (τ) = nablaφ(τ) δδϕ(τ) If Σ is spacelike(5143) is the (euclidean) TomonagandashSchwinger equation

We expect a local equation like (5143) to hold in any field theory If thetheory is generally covariant the functional W will be independent fromΣ and therefore the left-hand side of the equation will vanish leavingonly the hamiltonian operator acting on the field variables namely aWheelerndashDeWitt equation


Choose Σ to be a cylinder ΣRT with radius R and height T with thetwo bases on the surfaces t = 0 and t = T Given two compact supportfunctions ϕ1 and ϕ2 defined on t = 0 and t = T respectively we canalways choose R large enough for the two compact supports to be includedin the bases of the cylinder Then

limRrarrinfin

W [ϕ1 ϕ2ΣRT ] = W [ϕ1 ϕ2 T ] (5146)

because the euclidean Green function decays rapidly and the effect ofhaving the side of the cylinder at finite distance goes rapidly to zero as Rincreases Equation (5135) illustrates how scattering amplitudes can becomputed from W [ϕ1 ϕ2 T ] In turn (5146) indicates how W [ϕ1 ϕ2 T ]can be obtained from W [ϕΣ] where Σ is the boundary of a finite regionTherefore knowledge of W [ϕΣ] allows us to compute particle scatteringamplitudes We expect this to remain true in the perturbative expansionof an interacting field theory as well where R includes the interactionregion

The limits TR rarr infin seem to indicate that arbitrarily large surfaces Σare needed to compute vacuum and particle scattering amplitudes Butnotice that the convergence of W [ϕ1 ϕ2 T ] to the vacuum projector isdictated by (5130) and is exponential in the mass gap or the Compton


frequency of the particle Thus T at laboratory scales is largely sufficientto guarantee arbitrarily accurate convergence In the euclidean regimerotational symmetry suggests the same to hold for the R rarr infin limitThus the limits can be replaced by choosing R and T at laboratory scales(At least for the vacuum which does not require analytic continuation)

The conventional notions of vacuum and particle states are global innature How is it possible that we can recover them from the local func-tional W [ϕΣ] This is an important question that plays a role in QFTon curved spacetime and in quantum gravity To answer this questionnotice that realistic particle detectors are finitely extended How can afinitely extended detector detect particles if particles are globally definedobjects

The answer is that there exist two distinct notions of particle Fock par-ticle states are ldquoglobalrdquo while the physical states detected by a localizeddetector (eigenstates of local operators describing detection) are ldquolocalrdquoparticle states Local particle states are close to (in a suitable topology)but distinct from the global particle states In conventional QFT we usea global particle state in order to conveniently approximate the local par-ticle state detected by a detector Global particle states indeed are fareasier to deal with

Therefore the global nature of the conventional definition of vacuumand particles is not dictated by the physical properties of particles it isan approximation adopted for convenience Replacing the limits Rrarrinfinand T rarr infin with finite macroscopic R and T we miss the exact globalvacuum or n-particle state but we can nevertheless describe local ex-periments The restriction of QFT to a finite region of spacetime mustdescribe completely experiments confined to this region

Global and local particles in a simple finite system The distinction between globalparticles and local particles can be illustrated in a very simple system Consider twoweakly coupled harmonic oscillators Let the total hamiltonian of the system be

H =1

2(p2

1 + q21) +

1

2(p2

2 + q22) minus 2λq1q2 = H1 + H2 minus λV (5147)

Consider a measuring apparatus that interacts only with the first oscillator and mea-sures the quantity H1 The Hilbert space of the system is H = L2[R

2 dq1dq2] On thisspace the quantity H1 is represented by the operator minush2part2partq2

1 + q21 The operator

has a discrete spectrum E = (n+ 12)h If the result of the measurement is the eigen-value (1 + 12)h let us say that ldquothere is one local particle in the first oscillatorrdquo Inparticular a one-local-particle state is the common eigenstate of H1 and H2

ψlocal(q1 q2) = q1eminus(q21+q22)2h (5148)

in which there is one local particle in the first oscillator and no local particles in thesecond

Next let us diagonalize the full hamiltonian H This can easily be done by findingthe normal modes of the system which are qplusmn = (q1 plusmn q2)

radic2 and have frequencies


ω2plusmn = 1 plusmn λ The eigenvalues of H are therefore E = h(n+ω+ + nminusωminus + 1) We call

|n+ nminus〉 the corresponding eigenstates and N = n+ + nminus the global-particle numberIn particular we call ldquoone-global-particle staterdquo all states with N = 1 namely anystate of the form |ψ〉 = α|1 0〉 + β|0 1〉 Notice that this is precisely the definition ofone-particle states in QFT a one-particle state is an arbitrary linear combination ofstates |k〉 where there is a single quantum in one of the modes In particular considerthe one-global-particle state |ψ〉 = (|1 0〉+ |0 1〉)

radic2 This is a global particle which is

maximally localized on the first oscillator A straightforward calculation gives to firstorder in λ

ψglobal(q1 q2) = (q1 +λ

4q2)e

minus(q21+q22minus2λq1q2)2h (5149)

The two states ψlocal and ψglobal are different and have different physical meaningThe state ψglobal is the kind of state that is called a one-particle state in QFT It is theone-particle state which is most localized on the first oscillator On the other hand ifour measuring apparatus interacts only with the first oscillator then what we measureis not ψglobal it is ψlocal which is an eigenstate of an operator that acts only on thevariable q1

In QFT we confuse the two kinds of states In the formalism we use global-particlestates such as ψglobal However particle detectors are localized in space (A local mea-suring apparatus can only interact with the components of the field in a finite regionlike the apparatus that interacts only with the variable q1 in the example) Thereforethey measure particle states such as ψlocal Strictly speaking therefore the interpreta-tion of the particle states measured by particle detectors as global-particle states is amistake because a global-particle state can never be an eigenstate of a local measuringapparatus and therefore cannot be detected by a local apparatus

The reason we can nevertheless use this interpretation successfully is that the statesψlocal and ψglobal are very similar In the example their distance in the Hilbert normvanishes to first order in lambda

(ψglobal ψlocal) = 0(λ) (5150)

The error we make in using ψglobal to describe the physical state ψlocal is small if λVis small In the field theoretical case λV represents the interaction energy betweenthe region inside the detector and the region outside the detector this energy is verysmall compared to the energy of the state itself for all the states of interest We caneffectively approximate the local-particle states that are detected by our measuringapparatus by means of the global-particle states which are easier to deal with

On the other hand the argument shows that global-particle states are not required fordealing with the realistic observed particles they are just a convenient approximationIf we can define local-particle states by means of a local formalism we are not makinga mistake rather we are simply not using an approximation that was convenient onflat space but may not be viable in a generally covariant context

535 Boundary state space K and covariant vacuum |0〉Finally consider the space G of the variable α = (Σ ϕ) where Σ is a closed3d surface in spacetime Call K the space of functions ψ[α] = ψ[Σ ϕ] Thisspace represents all possible outcomes of ensembles of measurements onthe boundary of a finite region of spacetime The measurements includespacetime localization measurements that determine the surface Σ as well


as field (or particle) measurements that determine ϕ (or a function of ϕ)K is the boundary quantum space

There is a preferred state |0〉 in K given by

〈Σ ϕ|0〉 = W [Σ ϕ] (5151)

If the functional integral can be defined this is given by (5142) The state|0〉 expresses the dynamics entirely As we shall see this formulation ofQFT makes sense in quantum gravity

In general K is a space of functions over G Recall that G is the spaceof data needed to determine a classical solution two events in the finite-dimensional case a set of events forming a 3d closed surface in the fieldcase

In the case of a finite-dimensional theory a classical solution in some in-terval is determined by two events in C In the quantum theory a completeexperiment consists of two events a preparation and a quantum measure-ment In this case K = L2[G] = L2[C times C] sim L2[C] otimes L2[C] = K otimes K isthe space representing two quantum events while K = L2[C] is the spacerepresenting a single quantum event

In the field theoretical case a classical solution in a region R is deter-mined by infinite events in C forming a closed 3d surface namely by a3d surface Σ = partR in spacetime and the field ϕ on it In the quantumtheory a complete experiment requires measurements (or assumptions)on the entire Σ In this case K sim L2[G] is the space describing the obser-vation of the entire boundary surface Σ and the measurements on it

The boundary of R can be formed by two (or even more) connectedcomponents Σ In this case we can decompose K into the tensor productof one factor K associated with each component The space K is then aspace of functionals of the connected surface Σ and the field on it Sincethe WheelerndashDeWitt equation is local it looks the same on K and on KTherefore the distinction between K and K is of much less importance inthe field theoretical context than in the finite-dimensional case The spaceK is associated with the idea of the full data characterizing an experimenton a closed surface Σ while the space K is associated with the idea of anldquoinitial datardquo surface Σ


An interacting quantum field theory can be constructed as a perturbation expansion

around a free theory An alternative is to define a cut-off theory with a large but finite

number of degrees of freedom using a lattice One expects then to recover physical

predictions as suitable limits as the lattice spacing is taken to zero I illustrate here the

definition of the scalar product in a lattice gauge theory since the same technique is

used in quantum gravity


Consider a three-dimensional lattice Γ with L links l and N nodes nTo define a YangndashMills theory for a compact YangndashMills group G on thislattice we associate a group element Ul to each link l and we considerthe Hilbert space KΓ = L2[GLdUl] where GL is the product of L copiesof G and dUl equiv dUl dUl is the Haar measure on the group Quantumstates in KΓ are functions Ψ(Ul) of L group elements The scalar productof two states is given by

〈Ψ|Φ〉 equivint

dU1 dUL Ψ(U1 UL) Φ(U1 UL) (5152)

An orthonormal basis of states in KΓ can be obtained as follows Let jlabel unitary irreducible representations of G and let (Rj(U))αβ be thematrix elements of the representation The PeterndashWeyl theorem tells usthat the states |j β α〉 defined by 〈U |j β α〉 = (Rj(U))αβ form an or-thonormal basis in L2[GdU ] A basis in KΓ is therefore given by thestates

|jl βl αl〉 equiv |j1 jL β1 βL α1 αL〉 (5153)

defined by 〈Ul|jl βl αl〉 =prod

l(Rjl(Ul))

αlβl

The theory is invariant under local YangndashMills transformations on the

lattice These depend on a group element λn for each node n The variablesUl transform under a gauge transformation as Ul rarr λli

minus1Ulλlf where thelink l goes from the initial node li to the final node lf Hence the gauge-invariant states are the ones satisfying

Ψ(Ul) = Ψ(λliminus1Ulλlf ) (5154)

These states form a linear subspace K0Γ of KΓ the space K0

Γ is the(fixed-time) Hilbert space of the gauge-invariant states of the theory Anorthonormal basis of states in K0

Γ can be obtained using the notion of anintertwiner

Intertwiners Consider N irreducible representations j1 jN Considerthe tensor product of their Hilbert spaces

Hj1jN = Hj1 otimes otimesHjN (5155)

This space can be decomposed into a sum of irreducible components Inparticular let H0

j1jNbe the subspace formed by the invariant vectors

namely the subspace that transforms in the trivial representation Thisspace is k-dimensional where k is the multiplicity with which the trivialrepresentation appears in the decomposition It is of course a Hilbertspace and therefore we can choose an orthonormal basis in it We call


the elements i of this basis ldquointertwinersrdquo between the representationsj1 jN

More explicitly elements of Hj1jN are tensors vα1 αN with one index in eachrepresentation Elements of H0

j1jN are tensors vα1αN that are invariant under theaction of G on all their indices That is they satisfy

R(j1)α1β1(U) R(jN )αN

βN (U) vβ1 βN = vα1 αN (5156)

The intertwiners vα1 αNi are a set of k such invariant tensors which are orthonormal

in the scalar product of H0j1jN That is they satisfy

vα1αNi viprimeα1αN = δiiprime (5157)

If the space Hj carries the representation j its dual space Hlowastj carries the

dual representation jlowast An intertwiner i between n dual representationsjlowast1 j

lowastn and m representations j1 jm is an invariant tensor in the

space (otimesi=1nHlowastji) otimes (otimesk=1mHjk) that is a covariant map

i (otimesi=1nHji) minusrarr (otimesk=1mHjk) (5158)

or an invariant tensor with n lower indices and m upper indicesNow associate a representation jl to each link l and an intertwiner in

in each node n (in the tensor product of the representations associatedwith the links adjacent to the node) of the lattice The set s = (Γ jl in)is called a ldquospin networkrdquo Each spin network s defines a state |s〉 by

〈Ul|s〉 = ψs(Ul) =prod

l

Rjl(Ul) middotprod

n

in (5159)

where the raised dot indicates index contraction Notice that the indices(not indicated in the equation) match as on each side of the dot there isone index for each couple node-link The states |s〉 form a complete andorthonormal basis in K0

Γ〈s|sprime〉 = δssprime (5160)

This basis will play a major role in quantum gravity

54 Quantum gravity

Finally I sketch here the formal structure of quantum gravity The actual mathematicaldefinition of the quantities mentioned here is the task I undertake in the second partof the book


In the presence of a background QFT yields scattering amplitudes andcross sections for asymptotic particle states and these are compared with


data obtained in a lab The conventional theoretical definition of theseamplitudes involves infinitely extended spacetime regions and relies onsymmetry properties of the background In a background-independentcontext this procedure becomes problematic For instance backgroundindependence implies immediately that any 2-point function W (x y) isconstant for x = y as mentioned in Section 114 How can the formalismcontrol the localization of the measuring apparatus

We have seen above that in the context of a simple scalar field theory lo-cal physics can be expressed in terms of a functional W [ϕΣ] that dependson field boundary eigenstates ϕ and the geometry of the 3d surface Σthat bounds R Physical predictions concerning measurements performedin the finite region R including scattering amplitudes between particlesdetected in the lab can be expressed in terms of W [ϕΣ] The functionalsatisfies a local version of the Schrodinger equation The geometry of Σcodes the relative spacetime localization of the particle detectors W [ϕΣ]can be expressed as a functional integral over a finite spacetime regionR of spacetime In the euclidean regime the functional integral is welldefined and can be used to determine the Minkowski vacuum state

This technique can be extended to quantum gravity namely to adiffeomorphism-invariant context The effect of diffeomorphism invarianceis that the functional W turns out to be independent of the location of ΣAt first sight this seems to leave us in the characteristic interpretative ob-scurity of background-independent QFT the independence of W from Σ isequivalent to the independence of W (x y) from x and y mentioned above

But a closer look reveals it is not so The boundary field includes thegravitational field which is the metric and therefore the argument of Wdoes describe the metric of the boundary surface that is the relativespacetime location of the detectors as explained in Section 412 There-fore the relative location of the detectors lost with Σ because of generalcovariance comes back with ϕ as this now includes the boundary valueof the gravitational field The boundary value of the gravitational fieldplays the double role previously played by ϕ and Σ In fact this is pre-cisely the core of the conceptual novelty of general relativity there is no apriori distinction between localization measurements and measurementsof dynamical variables

More formally in a background-dependent theory the space G is a spaceof couples (Σ ϕ) but in a general-relativistic theory the space G is just aspace of fields on a closed differential surface In pure GR we can take Gas the space of the gravitational connections A on a closed surface Ac-cordingly the space K is a space of functionals of the field A on a closedsurface These functionals are invariant under 3d diffeomorphisms of thesurface In the second part of the book the space K will be built explic-itly As explained in the previous section the functional W determines a


preferred state |0〉 in K This is the covariant vacuum state which containsthe dynamical information of the theory

A key result of the theory developed in the second part of the book isthat the eigenstates of the gravitational field on a 3d surface are notsmooth fields They present a characteristic Planck-scale discretenessThese eigenstates determine a preferred basis |s〉 in K labeled by theldquospin networksrdquo s that will be described in detail in the second part ofthe book Each state |s〉 describes a ldquoquantum geometry of spacerdquo namelythe possible result of a complete measurement of the gravitational fieldon the 3d surface We shall express W in this preferred basis

W (s) = 〈0|s〉 (5161)

Therefore because of the Planck-scale discreteness of space in the gravi-tational context the analog of W [ϕΣ] is the functional W (s) A definitionof W (s) in the canonical quantum theory will be given below in (737)As we shall see the covariant vacuum state |0〉 will simply be related tothe spin network state with no nodes and no links A sum-over-historiesdefinition of W (s) will be given below in (921)

A case of particular interest is the one in which we can separate theboundary surface Σ into two components For instance these can be dis-connected Accordingly we can write s as (sout sin) and the associatedamplitude as

W (sout sin) = 〈0|sout sin〉 = 〈sout|P |sin〉 (5162)

where P is the projector on the solutions of the WheelerndashDeWitt equa-tion A sum-over-histories expression of W (sout sin) is given in terms ofhistories that go from sin to sout


The notion of ldquovacuum staterdquo plays a central role in QFT on a backgroundspacetime The vacuum is the basis over which Fock space is built In grav-ity on the other hand the notion of vacuum is very ambiguous This factcontributes to make quantum gravity sharply different from conventionalQFTs However this is not a difficulty a preferred notion of vacuum isnot needed for a quantum theory to be well defined The quantum theoryof a harmonic oscillator has a vacuum state but the quantum theory ofa free particle does not In this respect general relativity resembles morea free particle than a harmonic oscillator

Notice that even the terminology of classical GR is confusing with re-spect to the notion of vacuum in relativistic parlance all solutions of theEinstein equations without a source term are called ldquovacuum solutionsrdquo


We use three distinct notions of vacuum in quantum gravity

Covariant vacuum The first is the nonperturbative or covariant vac-uum state |0〉 defined in Sections 514 and 532 This is the statein the boundary state space that defines the dynamics Intuitivelyit is defined by the sum-over-histories on a region bounded by thegiven boundary data If the metric boundary data are chosen to bespacelike this is the HartlendashHawking state In the context we areconsidering instead the boundary surface bounds a finite 4d regionof spacetime and the state |0〉 is a background-independent way ofcoding quantum dynamics

Empty state The state |empty〉 is the kinematical quantum state of thegravitational field in which the volume of space is zero namely inwhich there is no physical space As we shall see it is related to thecovariant vacuum state |0〉

Minkowski vacuum A different notion of vacuum is the Minkowskivacuum state |0M〉 The quantum state |0M〉 that describes theMinkowski vacuum is not singled out by the dynamics alone In-stead it is singled out as the lowest eigenstate of an energy HT

which is the variable canonically conjugate to a nonlocal functionof the gravitational field defined as the proper time T along a givenworldline This is analogous to the identification of the energy witha momentum p0 under the choice of a specific Lorentz time x0 Tofind this state in quantum gravity we can use the procedure em-ployed in (549) and (5139) This will be briefly discussed at theend of Chapter 9 Alternatively in an asymptotically flat contextwe expect |0M 〉 to be the lowest eigenstate of the ADM energy

The notion of vacuum is strictly connected to the notion of energy Thevacuum can be defined as the state with lowest energy In GR the notionof energy is ambiguous and the ambiguity in the definition of energy isreflected in the ambiguity in defining the vacuum Indeed we can identifyseveral notions of energy in GR

Canonical energy The canonical energy namely the generator H oftranslations in coordinate time vanishes identically in any general-relativistic theory In this sense all physical states of quantum grav-ity are vacuum states

Matter energy The energy-momentum tensor T Iμ of the nongravita-

tional fields is well defined and therefore the energy Ematter = T 00

of the nongravitational fields is well defined In classical GR a vac-uum solution is a solution with Ematter = 0 In this sense vacuumstates are all the pure gravity physical states without matter


Gravitational energy The energy of the gravitational field Egravity isstrictly speaking (minus) the left-hand side of the timendashtime com-ponent of the Einstein equations so the timendashtime component ofthe Einstein equations reads Egravity+Ematter = 0 That is the totalenergy vanishes see for instance [147]

ADM energy We can associate an energy EADM to an isolated sys-tem surrounded by a region where the gravitational field is approxi-mately minkowskian Such a system can be described by asymptot-ically flat solutions of the Einstein equations For such a system wecan identify the energy with the generator EADM of time transla-tions in the asymptotic Minkowski space Given asymptotic flatnessEADM is minimized by the Minkowski solution In this sense theMinkowski solution is ldquothe vacuumrdquo of the asymptotic minkowskiantheory

The fact that the notions of energy and vacuum are so ambiguous in GRshould not be disconcerting There is nothing essential in these notions aquantum theory and its predictions are meaningful also in the absence ofthem The notions of energy and vacuum play an important role in non-general-relativistic physics just because of the accidental fact that we livein a region of the Universe which happens to have a peculiar symmetrytranslation invariance in newtonian or special-relativistic time

55 Complements


The thermal time hypothesis discussed in Section 34 extends nicely toQM and very nicely to QFT

QM In QM the time flow is given by

At = αt(A) = eitH0AeminusitH0 (5163)

A statistical state is described by a density matrix ρ It determines theexpectation values of any observable A via

ρ[A] = tr[Aρ] (5164)

This equation defines a positive functional ρ on the observablesrsquo algebraThe relation between a quantum Gibbs state ρ0 and H0 is the same as in(3202) That is

ρ0 = NeminusβH0 (5165)

55 Complements 205

Correlation probabilities can be written as

WAB(t) = ρ0[αt(A)B] = tr[eitH0A eminusitH0B eminusβH0 ] (5166)

Notice that it follows immediately from the definition that

ρ0[αt(A)B] = ρ0[α(minustminusiβ)(B)A] (5167)

namely

WAB(t) = WBA(minustminus iβ) (5168)

A state ρ0 over an algebra satisfying the relation (5167) is said to beKMS (KubondashMartinndashSchwinger) with respect to the flow αt

We can generalize easily the thermal time hypothesis Given a genericstate ρ the thermal hamiltonian is defined by

Hρ = ln ρ (5169)

and the thermal time flow is defined by

Atρ = αtρ(A) = eitρHρAeminusitρHρ (5170)

ρ is a KMS state with respect to the thermal time flow

QFT Tomita flow In QFT finite-temperature states do not live in thesame Hilbert space as the zero-temperature states H0 is a divergent oper-ator on these finite-temperature states This is to be expected since in athermal state there is a constant energy density and therefore a divergingtotal energy H0 Therefore (5165) makes no sense in QFT How thendo we characterize the Gibbs states The solution to this problem is wellknown equation (5167) can still be used to characterize a Gibbs stateρ0 in the algebraic framework and can be taken as the basic postulate ofstatistical QFT a Gibbs state ρ0 over an algebra of observables is a KMSstate with respect to the time flow α(t)

It follows that if we want to extend the thermal time hypothesis tofield theory we cannot use (5169) Can we get around this problem Isthere a flow αtρ which is KMS with respect to a generic thermal stateρ Remarkably the answer is yes A celebrated theorem by Tomita statesprecisely that given any2 state ρ over a von Neumann algebra3 there isalways a flow αt called the Tomita flow of ρ such that (5167) holds

2Any separating state ρ A separating density matrix has no zero eigenvalues This isthe QFT equivalent of the condition stated in footnote 11 of Chapter 3

3The observablesrsquo algebra is in general a Clowast algebra We obtain a von Neumann algebraby closing in the Hilbert norm of the quantum state space


This theorem allows us to extend (3205) to QFT the thermal timeflow αtρ is defined in general as the Tomita flow of the statistical state ρ

Thus the thermal time hypothesis can be readily extended to QFTwhat we call the ldquoflow of timerdquo is simply the Tomita flow of the statisticalstate ρ in which the world happens to be when it is described in termsof macroscopic parameters

The flow αtρ depends on the state ρ However a von Neumann algebra possessesalso a more abstract notion of time flow independent of ρ This is given by the one-parameter group of outer automorphisms formed by the equivalence classes of auto-morphisms under inner (unitary) automorphisms Alain Connes has shown that thisgroup is independent of ρ It only depends on the algebra itself Connes has stressedthe fact that this group provides an abstract notion of time flow that depends only onthe algebraic structure of the observables and nothing else

The thermal time hypothesis and the notion of thermal time have notyet been extensively investigated They might provide the key by whichto relate timeless fundamental mechanics with our experience of a worldevolving in time


The solutions of the WheelerndashDeWitt equation (564) form the linear space H Thisspace is naturally equipped with a scalar product that makes it a Hilbert space Thisscalar product is often denoted the ldquophysicalrdquo scalar product in order to distinguishit from the scalar product in K denoted the ldquokinematicalrdquo scalar product

The relation between kinematical and physical scalar product depends on the hamil-tonian H The space H is the eigenspace of H corresponding to the eigenvalue zeroIn order for solutions to exist the spectrum of H must therefore include zero If zerois part of the discrete spectrum of H then H is a proper subspace of K that is thesolutions of the WheelerndashDeWitt equation (564) are normalizable states in K In thiscase the physical scalar product is the same as the kinematical scalar product andthere is no complication But if zero is part of the continuum spectrum of H then H isformed by generalized eigenvectors which are in S prime and not in K That is the solutionsof the WheelerndashDeWitt equation (564) are nonnormalizable states in K In this casethe physical scalar product is different from the kinematical scalar product What isit

In the quantum gravity (and quantum cosmology) literature there is a certain con-fusion regarding the issue of the definition of the physical scalar product For instanceone often reads that this issue has to do with the notion of time This is a conceptualmistake that derives from the observation that in a nonrelativistic theory there is a pre-ferred time variable and the problem of defining H starting from K does not appearBut the fact that the issue of defining the product in H appears in timeless systemsdoesnrsquot imply that it cannot be resolved unless there is a time variable

In fact there are a large number of solutions to this issue all essentially equivalentPreferences vary here are some of the solutions proposed

(i) The scalar product can be defined on H using the matrix elements of the projectoras illustrated above

55 Complements 207

(ii) Here is a general theorem on the issue If H is a self-adjoint operator on a Hilbertspace K then we can write

K =

int

S

ds Hs (5171)

Here S is the spectrum of H ds is a measure on this spectrum and Hs is a family ofHilbert spaces labeled by the eigenvalues s The meaning of this integral over Hilbertspaces is the following any vector ψ isin K can be written as a family ψs where forevery s ψs isin Hs and

(ψ φ)K =

int

S

ds (ψs φs)Hs (5172)

where ( )H is the scalar product in the Hilbert space H and in this instance theintegral is a standard numerical integral The relevance of this theorem is that it statesthat there is a Hilbert space H0 That is a scalar product on the space of the solutionsof Hψ = 0

Here is a simple example of how the theorem works Consider the space K =L2[R

2 dxdy] and the self-adjoint operator H = minusiddy The solutions of Hψ = 0or

minusid

dyψ(x y) = 0 (5173)

are functions ψ(x y) constant in y and are nonnormalizable in K However the decom-position (5171) (5172) is immediate

K =

int

R

dy Hy (5174)

where H(y) = L2[Rdx] In fact

(ψ φ)K =

int

R2dx dy ψ(x y) φ(x y) =

int

R

dy (ψy φy)Hy (5175)

where ψy(x) = ψ(x y) and

(ψy φy)Hy =

int

R

dx ψy(x) φy(x) (5176)

The space of the solutions of (5173) is H(0) and has the natural Hilbert structureH(0) = L2[R dx]

(iii) Here is another solution Pick a set of self-adjoint operators Ai in K thatcommute with H These are well defined on the space H because if Hψ = 0 thenH(Aiψ) = AiHψ = 0 Now require that the operators Ai be self-adjoint in the phys-ical scalar product For a sufficient number of operators this requirement fixes thescalar product of H

In the example given in (ii) above the obvious self-adjoint operators that commutewith H = minusiddy are x and minusiddx These are well defined on the space of the functionsof x alone There is only one scalar product on this space of functions that makes xand minusiddx self-adjoint the one of L2[R dx]

(iv) A convenient way of addressing the problem especially in the case in which His not a single operator but has many components is given by the ldquogroup averagingrdquotechnique Assume the WheelerndashDeWitt equation has the form Hiψ = 0 where theself-adjoint operators Hiψ = 0 are the generators of a unitary action of a group U on


K Assume also that S is invariant under this action and that we can find an invariantmeasure on the group or at least on the orbit of the group in K Then we can generalizethe operator P S rarr H of (558)

P =

int

U

dτ U(τ) (5177)

and write the physical scalar product as

(Pψ Pφ)H equiv (Pψ)(φ) =

int

U

dτ (ψ|U(τ)|φ)K (5178)

There certainly are other techniques as well This is a field in which the same ideashave independently reappeared many times under different names (and with differentlevels of mathematical precision) All these techniques are generally equivalent If thereis a case in which they differ wersquoll have to resort to physical arguments to find thephysically correct choice


Section 327 illustrated the possibility of using mixed complex and real dynamicalvariables a strategy that will turn out to be useful in GR Here I illustrate whathappens with the same choice in quantum theory In particular I illustrate the key rolethat the reality conditions play in quantum theory Recall the simple example discussedin Section 327 a free particle described in the coordinates x and z = xminus ip We canwrite the quantum theory in terms of wave functionals ψ(z) of the complex variable zThe Schrodinger equation gives immediately (see (3134))

ihpartψ(z t)

partt= H0

(hpart

partz z

)ψ(z t) = minus 1

2m

(hpart

partzminus z

)2

ψ(z t) (5179)

A complete family of solutions is given by

ψk(z t) = eminusihS(ztk) (5180)

where S(z t k) is given in (3135) Observe now that in the quantum theory the realitycondition (3138) becomes a relation between operators

z + zdagger = 2hpart

partz (5181)

Notice that classical complex conjugation is translated into the adjoint operation thisis necessary in order for real quantities to be represented by self-adjoint operators Now(5181) makes sense only after we have specified the scalar product because the daggeroperation is defined in terms of and therefore depends on the scalar product Indeedrequiring the reality condition (5181) to hold amounts to posing a condition on thescalar product of the theory Let us search for a scalar product of the form

(ψ φ) =

intdzdz f(z z) ψ(z) φ(z) (5182)

where f is a function to specify Imposing (5181) gives the condition on f

(z + z)f(z z) = minus2hpart

partzf(z z) (5183)

This gives

f(z z) = eminus(z+z)24h (5184)


Let us check whether the states (5180) are well defined with respect to this productInserting (5180) (at t = 0 for simplicity) and (5184) in (5182) gives

(ψk ψkprime) =

intdzdz eminus(z+z)24h e

ih

(kzminus i2 z2) eminus

ih

(kprimez+ i2 z2) (5185)

A simple change of variables shows that the integral in the imaginary part of z is finiteand the integral in the real part of z is proportional to δ(k kprime) Therefore the statesψk form a standard continuous orthogonal basis of generalized states They are clearlyeigenstates of the momentum since

pψk = i(xminus z)ψk = i

(part

partzminus z

)ψk = kψk (5186)

In fact what we have developed is a simple rewriting of the standard Hilbert space ofa quantum particle

Notice that appearances can be misleading For instance for k = 0 the state ψk

readsψ0(z t) = e+z22h (5187)

This looks like a badly nonnormalizable state but it is not It is a well-defined general-ized state since the negative exponential in the measure compensates for the positiveexponential in the state


Quantum mechanics is one of the most successful scientific theories everHowever its interpretation is controversial What does the theory actuallytell us about the physical world This question sparked off a lively debatewhich was intense during the 1930s the early days of the theory and isgenerating new interest today

The possibility that the interpretation of an empirically successful the-ory could be debated should not surprise examples abound in the historyof science For instance the great scientific revolution was fueled by thedebate on whether the efficacy of the copernican system should be takenas an indication that the Earth really moves In more recent times Ein-steinrsquos celebrated contribution to special relativity consisted to a largeextent just in understanding the physical interpretation (simultaneityis relative) of an already existing effective mathematical formalism (theLorentz transformations) In these cases as in the case of quantum me-chanics an overly strictly empiricist position could have circumventedthe problem altogether by reducing the content of the theory to a list ofpredicted numbers But science would not then have progressed

Quantum theory was first constructed for describing microscopic ob-jects (atoms electrons photons) and the way these interact with macro-scopic apparatuses built to measure their properties Such interactionswere called ldquomeasurementsrdquo The theory is formed by a mathematicalformalism which allows probabilities of alternative outcomes of such mea-surements to be calculated If used just for this purpose the theory raises


no difficulty But we expect the macroscopic apparatuses themselves ndash infact any physical system in the world ndash to obey quantum theory andthis seems to raise contradictions within the theory Here I discuss theseapparent contradictions and a possible resolution This resolution offersa precise answer to the question of what the quantum theory actually tellsus about the physical world


Measurements A ldquomeasurementrdquo of the variable A of a system S is aninteraction between the system S and another system O whose effecton O depends on the value that the variable A has at the time of theinteraction We say that the variable A is ldquomeasuredrdquo and that its valuea is the ldquooutcome of the measurementrdquo For instance let S be a particlethat impacts on O let the effect of this impact depend on the positionof the particle and let q be the value of the position at the moment ofthe impact Then we say that the position Q is measured and that theoutcome of the measurement is q

The term ldquomeasurementrdquo and the common terminology used to de-scribe measurement situations (S for ldquoSystemrdquo and O for ldquoObserverrdquo)are very misleading because they evoke a human intentionally ldquoobservingrdquoS and using an apparatus to gather data about it There is nothingldquohumanrdquo or ldquointentionalrdquo in the definition of measurement given aboveThe system O does not need to be human nor to be a special ldquoapparatusrdquonor to be macroscopic The measured value need not be stored Any in-teraction between two physical systems is a measurement The measuredvariable of the system S is the variable that determines the effect that theinteraction has on O This is true in classical as well as in quantum theory

Classical states and quantum states In classical mechanics a system Sis described by a certain number of physical variables ABC For in-stance a particle is described by its position Q and velocity V Thesevariables change with time They represent the contingent properties ofthe system We say that the values of these variables determine at everymoment the ldquostaterdquo of the system If the value of the position Q of theparticle is q and the value of its velocity V is v we say that the state is(q v) In classical mechanics a state is therefore a list of values of physicalvariables

Quantum mechanics differs from classical mechanics because it assumesthat the variables of the system do not have a determined value at alltimes Werner Heisenberg introduced this key idea According to quantumtheory an electron does not have a well-determined position at every


time When it is not interacting with an external system sensitive to itsposition the electron can be ldquospread outrdquo over different positions It is ina ldquoquantum superpositionrdquo of different positions

It follows that in quantum mechanics the state of the system cannotbe captured by giving the value of its variables Instead quantum theoryintroduces a novel notion of the ldquostaterdquo of a system different from theclassical list of variable values The new notion of ldquoquantum staterdquo wasintroduced in the work of Erwin Schrodinger in the form of the ldquowavefunctionrdquo of the system Paul Adrien Maurice Dirac gave it its generalabstract formulation in terms of a vector Ψ moving in an abstract vectorspace From the knowledge of the state Ψ we can compute the probabilityof the different measurement outcomes a1 a2 of any variable A Thatis the probability of the different ways in which the system S can affecta system O in the course of an interaction

The theory prescribes that at every such measurement we must updatethe value of Ψ to take into account which of the different outcomes hasbeen realized This sudden change of the state Ψ depends on the outcomeof the measurement and is therefore probabilistic This is the ldquocollapse ofthe wave functionrdquo

The notion of ldquostate of the systemrdquo of classical mechanics is there-fore split into two distinct notions in quantum theory (i) the state Ψthat expresses the probability for the different ways the system S caninteract with its surroundings and (ii) the actual sequence of valuesq1 q2 q3 that the variables of S take in the course of the interac-tions These are the called ldquomeasurement outcomesrdquo I prefer calling themldquoquantum eventsrdquo

We can either think that Ψ is a ldquorealrdquo entity or that it is nothing morethan a theoretical bookkeeping for the quantum events which are theldquorealrdquo events The choice of the relative ontological weight we attribute tothe state Ψ or the quantum events q1 q2 q3 is a matter of convenienceempirical evidence alone does not uniquely determine what is ldquorealrdquoI think the second choice is cleaner but in the following I refer to both

The observer observed The key problem of the interpretation of quantummechanics is illustrated by the following situation Consider a real physicalsituation illustrated in Figure 51 in which at some time t a system Ointeracts with a system S and then at a later time tprime a third system Oprime

interacts with the coupled system [S + O] formed by S and O togetherLet the effect on O of the first interaction depend on the variable A ofthe system S and the effect on Oprime of the second interaction depend onthe variable B of the coupled system [S+O] (That is we can say that O


Fig 51 The observer observed

measures the variable A of S at time t and then Oprime measures the variableB of [S +O] at time tprime) Before the first interaction say S was in a statewhere A is a quantum superposition of two values a1 and a2 Say thatat the first interaction O measures the value a1 of the variable O Thepuzzling question can be formulated in various equivalent manners

bull What is the state of S and O between the two interactions

bull Has the quantum event a1 happened or not

bull Does the quantity A have a determined value after the first interac-tion or not

Say that before the first interaction the state of S was Ψ = c1Ψ1 +c2Ψ2

where Ψ1Ψ2 are states where A has values a1 a2 respectively Then attime t we have

c1Ψ1 + c2Ψ2 rarr Ψ1

A takes the value a1(5188)

However the system O obeys the laws of quantum theory as well There-fore we can also give a quantum description of the evolution of the coupledquantum system (S + O) formed by S and O together If we do so no


collapse happens Instead the effect of the interaction is the Schrodingerevolution

(c1Ψ1 + c2Ψ2

)otimes Φ rarr

(c1Ψ1 otimes Φ1 + c2Ψ2 otimes Φ2

)

A is still in the superposition of the two values a1 a2

(5189)

for suitable states ΦΦ1Φ2 of OWhat is real seems to depend on how we choose to describe the world

What is the real state of affairs of the world after the interaction betweenS and O (5188) or (5189) In either case we get a difficulty If we saythat after t the state has collapsed as in (5188) and A has the value a1we get the wrong predictions about the second measurement at time tprimeIn fact quantum theory allows us to predict the probability distributionof the possible outcomes of the second measurement but to computethis we have to use the state (5189) and not the state (5188) Indeedif B and A do not commute this probability distribution can be affectedby the interference between the two different ldquobranchesrdquo in (5189) Inother words we have to assume that the variable A was in a quantumsuperposition of the values a1 a2 and not determined

But if we do so and say that after the first measurement the state is(5189) then we must say that A has no determined value at time t Butthe situation is general any measurement can be thought of as the firstmeasurement of the example and therefore we must conclude that novariable can take a definite value ever

Thus we seem to get a contradiction in both cases whether we thinkthat the wave function has collapsed and a1 was realized or whether wethink it hasnrsquot This is the core of the difficulty of the interpretation ofquantum theory

Real wave functions or real quantum events Let us examine the abovedifficulty in a bit more detail from the two points of view of the twopossible ontologies of quantum theory

If we think that Ψ is real but it never truly collapses there is no sim-ple and compelling reason why the world should appear as described byvalues of physical quantities that take determined values at each interac-tion as it does We experience particles in given positions not particlewavefunctions The relation between a noncollapsing wavefunction ontol-ogy and our experience of the world is very indirect and involuted Weneed some complicated story to understand how specific observed valuesq1 q2 q3 can emerge from the sole Ψ If this story is given (which ispossible) we are then in a situation similar to the one of a quantum eventontology to which I now turn


I think it is preferable to take the quantum events as the actual elementsof reality and view Ψ just as a bookkeeping device coding the events thathappened in the past and their consequences For instance I prefer to saythat the ldquorealityrdquo of a subatomic particle is expressed by the sequence ofthe positions of the particle revealed by the bubbles in a bubble chambernot by the spherically symmetric wave function emerging from the inter-action area The reality of the electron is in the events where it revealsitself interacting with its surrounding not in the abstract probabilityamplitude for such events From this perspective the real events of theworld are the ldquorealizationsrdquo (the ldquocoming to realityrdquo the ldquoactualizationrdquo)of the values q1 q2 q3 in the course of the interactions between physicalsystems These quantum events have an intrinsically discrete ldquoquantizedrdquogranular structure

This perspective however does not solve the above puzzle either Thekey puzzle of quantum mechanics becomes the fact that the statementthat the quantum event a1 ldquohas happenedrdquo can be at the same time trueand not-true has the quantum event a1 happened or not If we answer nothen we are forced to say that no quantum event ever happens becausethe situation described above is completely general any quantum eventhappening in the interaction of two systems S and O is ldquonon-happeningrdquoas far as the effect of (S + O) on a further system Oprime is concerned If wesay yes then we contradict the predictions of quantum mechanics (aboutthe second interaction)

The ldquosecond observerrdquo puzzle captures the core conceptual difficultyof the interpretation of quantum mechanics reconciling the possibilityof quantum superposition with the fact that the world we observe anddescribe is characterized by determined values of physical quantities Moreprecisely the puzzle shows that we cannot disentangle the two accordingto the theory a quantum event (a1) can be at the same time realized andnot realized

A possible escape from the puzzle is to assume that there are ldquospecialrdquosystems that produce the collapse and cause quantum events to happenFor instance these could be ldquomacroscopicrdquo systems or ldquosufficiently com-plexrdquo systems or ldquosystems with memoryrdquo or the ldquogravitational fieldrdquo orhuman ldquoconsciousnessrdquo All these systems and others have been sug-gested as causing quantum collapse and generating quantum events Ifthis were correct at some point we shall be able to measure violations ofthe predictions of QM That is QM as we know it would break down forthose systems

So far this breaking down of QM has never been observed We canfancy a phenomenology that we have not yet observed that could bringback reality to the way we used to think it is It is certainly worthwhile to


investigate this possibility theoretically and experimentally But we shouldnot forget that reality might be truly different from what we thought andmight be simply demanding us to renounce some old prejudice I thinkthat the history of physics indicates that the productive attitude is not toresist the conceptual novelty of empirically successful theories but ratherto make an effort to understand it We should not force reality into ourprejudices but rather try to adapt our conceptual schemes to what welearn about the world


I think that the key to the solution of the difficulty can be found inthe observation that the two descriptions (5188) and (5189) refer todifferent systems the first to O the second to Oprime More precisely the firstis relevant for describing the effects of interactions on O the second fordescribing the effects of interactions on Oprime

The solution of the puzzle can be found in the idea that quantum eventsare the elements of reality but they are always relative to a physicalsystem the quantum event a1 happens with respect to O but it does nothappen with respect to Oprime

In other words the way out from the puzzle is that the values of thevariables of any physical system are relational They do not express aproperty of the system S alone but rather refer to the relation betweenthis system and another system The variable A has value a1 with respectto O but it has no determined values with respect to Oprime This pointof view is called the relational interpretation of quantum mechanics orsimply relational quantum mechanics

The central idea of relational quantum mechanics is that there is nomeaning in saying that a certain variable of the system S takes the valueq There is only meaning in saying that a variable has value q with respectto a system O In the example discussed above for instance the fact thatA takes the value a1 with respect to O does not imply that A has thevalue a1 also with respect to Oprime

If we avoid all statements that are not referred to a physical systemwe can get rid of all apparent contradictions of quantum theory Theapparent contradiction between the two statements that a variable hasor hasnrsquot a value is resolved by referring the statements to the differentsystems with which the system in question interacts If I observe anelectron at a certain position I cannot conclude that the electron isthere I can only conclude that the electron as seen by me is there

Indeed quantum theory must be understood as an account of the waydistinct physical systems affect one another when they interact and not


the way physical systems ldquoarerdquo This account exhausts all that can besaid about the physical world The physical world can be described asa network of interacting components where there is no meaning to ldquothestate of an isolated systemrdquo The state of a physical system is the networkof its relationships with the surrounding systems The physical structureof the world is identified with this network of relationships

The unique account of the state of the world of the classical theoryis thus shattered into a multiplicity of accounts one for each possibleldquoobservingrdquo physical system Quantum mechanics is a theory about thephysical description of physical systems relative to other systems and thisis a complete description of the world

Of course we can pick a system O once and for all as ldquothe observersystemrdquo and be concerned only with the effects of the rest of the worldon this system Each interaction between the rest of the world and O iscorrectly described by standard quantum mechanics In this descriptionthe quantum state Ψ collapses at each interaction with O This descriptionis completely self-consistent but it treats O as if it were a special systema classical nonquantum system If we want to describe O itself quantummechanically we can but we have to pick a different system Oprime as theobserver and describe the way O interacts with Oprime In this descriptionthe quantum properties of O are taken into account but not the ones ofOprime because this description describes the effects of the rest of the worldon Oprime

Consistency This relativisation of actuality is viable thanks to a remark-able property of the formalism of quantum mechanics

John von Neumann was the first to notice that the formalism of thetheory treats the measured system (S) and the measuring system (O) dif-ferently but the theory is surprisingly flexible on the choice of where to putthe boundary between the two Different choices give different accounts ofthe state of the world (for instance the collapse of the wave function hap-pens at different times) but this does not affect the predictions about thefinal observations This flexibility reflects a general structural propertyof quantum theory which guarantees consistency among all the distinctldquoaccounts of the worldrdquo of the different observing systems The mannerin which this consistency is realized however is subtle

As a simple illustration of this phenomenon consider the case in whicha system O with two states Φ1 and Φ2 (say a light-bulb which can be onor off ) interacts with a system S with two states Ψ1 and Ψ2 (say thespin of the electron which can be up or down) Assume the interactionis such that if the spin is up (down) the light goes on (off ) To start


with the electron can be in a superposition of its two states In theaccount of the state of the electron that we can associate with the lightthe wave function of the electron collapses to one of two states duringthe interaction as in (5188) and the light is then either on or off Butwe can also consider the lightelectron composite system as a quantumsystem and study the interactions of this composite system with anothersystem Oprime In the account associated to Oprime there is no collapse at the timeof the interaction and the composite system is still in the superpositionof the two states [spin uplight on] and [spin downlight off ] after theinteraction as in (5189) As remarked above it is necessary to assumethis superposition because it accounts for measurable interference effectsbetween the two states if quantum mechanics is correct these interferenceeffects are truly observable by Oprime

So we have two discordant accounts of the same events the one associ-ated to O where the spin has a determined value and the one associatedto Oprime where the spin is in a superposition Now can the two discordantaccounts be compared and does the comparison lead to a contradiction

They can be compared because the information on the first accountis stored in the state of the light and Oprime has access to this informationTherefore O and Oprime can compare their accounts of the state of the worldHowever the comparison does not lead to a contradiction because the com-parison is itself a physical process that must be understood in the contextof quantum mechanics

Indeed Oprime can physically interact with the electron and then with thelight (or equivalently with the light and then with the electron) If forinstance Oprime finds the spin of the electron up quantum mechanics predictsthat the observer will then consistently find the light on because in thefirst measurement the state of the composite system collapses on its [spinuplight on] component namely on the first term of the right-hand sideof (5189)

That is the multiplicity of accounts does not lead to a contradictionprecisely because the comparison between different accounts can only be aphysical quantum interaction Many common paradoxes of quantum me-chanics follow from assuming that the communication between differentobservers violates quantum mechanics4 This internal self-consistency ofthe quantum formalism is general and is perhaps its most remarkable

4The EPR (EinsteinndashPodolskindashRosen) apparent paradox might be among these Thetwo observers far from each other are physical systems The standard account neglectsthe fact that each of the two is in a quantum superposition with respect to the otheruntil the moment they physically communicate But this communication is a physicalinteraction and must be strictly consistent with causality


aspect5 This self-consistency is a strong indication of the relational na-ture of the world

563 Information

What appears with respect to O as a measurement of the variable A (witha specific outcome) appears with respect to Oprime simply as a dynamicalprocess that establishes a correlation between S and O As far as theobserver O is concerned the variable A of a system S has taken a certainvalue As far as the second observer Oprime is concerned the only relevantelement of reality is that a correlation is established between S and O

Concretely this correlation appears in all further observations that Oprime

would perform on the [S + O] system That is the way the two systemsS and O will interact with Oprime is characterized by the fact that thereis a correlation Oprime will find some properties of O correlated with someproperties of S

On the other hand until it physically interacts with [S+O] the systemOprime has no access to the actual outcomes of the measurements performedby O on S These actual outcomes are real only with respect to O

The existence of a correlation between the possible outcomes of a mea-surement performed by Oprime on S and the outcomes of a measurementperformed by Oprime on O can be interpreted in terms of information In factthis correlation corresponds precisely to Shannonrsquos definition of informa-tion According to this definition ldquoO has information about Srdquo meansthat we shall observe O and S in a subset of the set formed by the carte-sian product of the possible states of O and the possible states of S Thusa measurement of S by O has the effect that ldquoO has information aboutSrdquo This statement has a precise technical meaning which refers to thepossible outcomes of the observations by a third system Oprime

On the other hand if we interact a sufficient number of times with aphysical system S we can then predict (the distribution probability ofthe) future outcomes of our interactions with this system In this senseby interacting with S we can say we ldquohave informationrdquo about S (This

5In fact one may conjecture that this peculiar consistency between the observationsof different observers is the missing ingredient for a reconstruction theorem of theHilbert space formalism of quantum theory Such a reconstruction theorem is stillunavailable On the basis of reasonable physical assumptions one is able to derivethe structure of an orthomodular lattice containing subsets that form Boolean alge-bras which ldquoalmostrdquo but not quite implies the existence of a Hilbert space and itsprojectorsrsquo algebra Perhaps an appropriate algebraic formulation of the condition ofconsistency between subsystems could provide the missing hypothesis to complete thereconstruction theorem


information need not be stored or utilized but its existence is the neces-sary physical condition for being able to store it or utilize it for predic-tions)

Therefore we have two distinct senses in which the physical theory isabout information But a moment of reflection shows that the two simplyreflect the same physical reality as it affects two different systems On theone hand O has information about S because it has interacted with S andthe past interactions are sufficient to ldquogive informationrdquo namely to deter-mine (the probability distribution of) the result of future interactions Onthe other hand O has information about S in the sense that there are cor-relations in the outcomes of measurements that Oprime can make on the two

There is a crucial subtle difference that can be figuratively expressedas follows O ldquoknowsrdquo about S while Oprime only knows that O knows aboutS but does not know what O knows As far as Oprime is concerned a physicalinteraction between S and O establishes a correlation it does not selectan outcome

These observations are sufficient to conclude that what precisely quan-tum mechanics is about is the information that physical systems haveabout one another

The common unease with taking quantum mechanics as a fundamentaldescription of Nature referred to as the measurement problem can betraced to the use of an incorrect notion in the same way that uneasewith Lorentz transformations derived from the notion shown by Einsteinto be mistaken of an observer-independent time The incorrect notionthat generates the unease with quantum mechanics is the notion of anobserver-independent state of a system or observer-independent valuesof physical quantities or an observer-independent quantum event

We can assume that all systems are equivalent there is no a prioriobserverndashobserved distinction the theory describes the information thatsystems have about one another The theory is complete because thisdescription exhausts the physical world

In physics the move of deepening our insight into the physical worldby relativizing notions previously treated as absolute has been appliedrepeatedly and very successfully Here are a few examples

The notion of the velocity of an object has been understood as mean-ingless unless it is referred to a reference body with respect to whichthe object is moving With special relativity simultaneity of two distantevents has been understood as meaningless unless referred to a specificstate of motion of something (This something is usually denoted as ldquotheobserverrdquo without of course any implication that the observer is humanor has any other peculiar property besides having a state of motion Simi-larly the ldquoobserver systemrdquo O in quantum mechanics need not be humanor have any other property beside the possibility of interacting with the


ldquoobservedrdquo system S) With general relativity the position in space andtime of an object has been understood as meaningless unless it is referredto the gravitational field or to another dynamical physical entity

The step proposed by the relational interpretation of quantum mechan-ics has strong analogies with these In a sense it is a longer jump sinceall the contingent (variable) properties of all physical systems are takento be meaningful only as relative to a second physical system This is notan arbitrary step It is a conclusion which is difficult to escape followingfrom the observation ndash explained above in the example of the ldquosecond ob-serverrdquo ndash that a variable (of a system S) can have a well-determined valuea1 for one observer (O) and at the same time fail to have a determinedvalue for another observer (Oprime)

This way of thinking of the world has perhaps heavy philosophical im-plications But it is Nature that is forcing us to this way of thinking If wewant to understand Nature our task is not to frame Nature into our philo-sophical prejudices but rather to learn how to adjust our philosophicalprejudices to what we learn from Nature


I close with a very speculative suggestion As discussed in Section 23 themain idea underlying GR is the relational interpretation of localizationobjects are not located in spacetime They are located with respect toone another In the present section I have observed that the lesson ofQM is that quantum events and states of systems are relational theymake sense only with respect to another system Thus both GR and QMare characterized by a form of relationalism Is there a connection betweenthese two forms of relationalism

Let us look closer at the two relations In GR the localization of anobject S in spacetime is relative to another object (or field) O to whichS is contiguous Contiguities or equivalently Einsteinrsquos ldquospacetime coin-cidencesrdquo are the basic relations that construct spacetime In QM thereare no absolute properties or facts properties of a system S are relativeto another system O with which S is interacting Facts are interactionsThus interactions form the basic relations between systems

But there is a strict connection between contiguity and interaction Onthe one hand S and O can interact only if they are contiguous if they arenearby in spacetime this is locality Interaction requires contiguity Onthe other hand what does it mean that S and O are contiguous Whatelse does it mean besides the fact that they can interact6 Therefore

6The very word ldquocontiguousrdquo derives from the Latin cum-tangere to touch each otherthat is to inter-act


contiguity is manifested by interacting In a sense the fact that inter-actions are local means that there is a sort of identity between beingcontiguous and interacting

Thus locality ties together very strictly the spacetime relationalism ofGR with the relationalism underlying QM It is tempting to try to developa general conceptual scheme based on this observation This could be aconceptual scheme in which contiguity is nothing else than a manifesta-tion or can be identified with the existence of a quantum interactionThe spatiotemporal structure of the world would then be directly deter-mined by who is interacting with whom This is of course very vagueand might lead nowhere but I find the idea intriguing

mdashmdash


Textbooks on quantum theory are numerous I think the best of all is thefirst of them Dirac [148] because of Diracrsquos crystal-clear thinking In theearlier editions Dirac uses a relativistic notion of state (that does notevolve in time) as is done here He calls these states ldquorelativisticrdquo as isdone here In later editions he switches to Schrodinger states that evolvein time explaining in a preface that it is easier to calculate with thesebut it is a pity to give up relativistic states which are more fundamental

I have discussed the idea that QM remains consistent also in the absenceof unitary time evolution in [98] and [149] The same idea is developed bymany authors see [26] [150] and references therein

In the past I have discussed relativistic systems only in terms of ldquoevolv-ing constantsrdquo The two-oscillators example used in the text was consid-ered in these terms in [151 152] The probabilistic interpretation of thecovariant formulation presented in this chapter is an evolution of thispoint of view and derives from [144]

I have taken the discussion on the boundary formulation of QFT from[145] The idea that quantum field theory must be formulated in termsof boundary data on a finite surface has been advocated by Robert Oeckl[153] The derivation of the local Schrodinger equation is in [146] and[154] The TomonagandashSchwinger equation was introduced in [155] On thedifficulty of a direct interpretation of the n-point functions in quantumgravity see for instance [156] The HartlendashHawking state was introducedin [157]

The possibility of defining the physical scalar product on the space ofthe solutions of the WheelerndashDeWitt equation even when these solutionsare nonnormalizable in the kinematical Hilbert space has been discussed


by many authors using a variety of techniques A nice mathematical con-struction has been given by Don Marolf see [158] and references therein

The thermal time hypothesis was extended to QM and QFT in [125]The relational interpretation presented here is discussed in [159 160]

see also [161 162] An overview of similar points of view is in the onlineStanford Encyclopedia of Philosophy [163] on a possibly related point ofview see also [164] The role of information in the foundations of quantumtheory has been stressed by John Wheeler in [165 166] For a recent dis-cussion on the role of information in the foundation of quantum theorysee for instance [167] and references therein An original and fascinat-ing point of view on the relational aspects of quantum and relativity isexplored by David Finkelstein in [168]

Part II

Loop quantum gravity

ndash Now itrsquos time to leave the capsuleif you dare

ndash This is Major Tom to Ground ControlIrsquom stepping through the doorAnd Irsquom floating in a most peculiar wayAnd the stars look very different today

David Bowie Space Oddity

6Quantum space

It is time to begin to put together the tools developed in the first part of the bookand build the quantum theory of spacetime The strategy is simple We ldquoquantizerdquothe canonical formulation of GR described at the beginning of Chapter 4 accordingto the relativistic QM formalism detailed in Chapter 5 This chapter deals with thekinematical part of the theory states partial observables and their eigenvalues Thenext chapter deals with dynamics namely with transition amplitudes


In Chapter 4 we have seen that GR can be formulated as the dynamicalsystem defined by the HamiltonndashJacobi equation (49)

F ijab(τ)

δS[A]δAi

a(τ)δS[A]

δAjb(τ)

= 0 (61)

where the functional S[A] is defined on the space G of the 3d SU(2)connections Ai

a(τ) and is invariant under internal gauge transformationsand 3d diffeomorphisms That is

δfAia(τ)

δS[A]δAi

a(τ)= 0 δλA

ia(τ)

δS[A]δAi

a(τ)= 0 (62)

where the variations δfAia and δλA are given in (412) Equivalently the

theory is defined by the hamiltonian H[AE] = F ijab E

ai E

bj on T lowastG

Following the prescription of Chapter 5 a quantization of the theorycan be obtained in terms of complex-valued Schrodinger wave functionalsΨ[A] on G The quantum dynamics is inferred from the classical dynamicsby interpreting S[A] as times the phase of Ψ[A] Namely interpretingthe classical HamiltonndashJacobi theory as the eikonal approximation of aquantum wave equation as in [142] semiclassical ldquowave packetsrdquo will thenbehave according to the classical theory This can be obtained defining

225

226 Quantum space

the quantum dynamics by replacing derivatives of the HamiltonndashJacobifunctional S[A] with derivative operators The two equations (62) remainunchanged they simply force Ψ[A] to be invariant under SU(2) gaugetransformations and 3d diffeomorphisms Equation (61) gives

F ijab (τ)

δ

δAia(τ)

δ

δAjb(τ)

Ψ[A] = 0 (63)

This is the WheelerndashDeWitt equation or EinsteinndashSchrodinger equationIt governs the quantum dynamics of spacetime In other words the dy-namics is defined by the hamiltonian operator H = H[Aminusi δδA]

More precisely we want a rigged Hilbert space S sub K sub S prime where S is asuitable space of functionals Ψ[A] Partial observables are represented byself-adjoint operators on K Their eigenvalues describe the quantization ofphysical quantities The operator P formally given by the field theoreticalgeneralization of (558)

P simint

[DN ] eminusiint

d3τ N(τ)H(τ) (64)

sends S to the space of the solutions of (63) Its matrix elements be-tween eigenstates of partial observables define the transition amplitudesof quantum gravity These determine all (probabilistic) dynamical rela-tions between any measurement that we can perform

A preferred state in K is |empty〉 the eigenstate of the geometry with zerovolume and zero area The covariant vacuum is given by |0〉 = P |empty〉 If weassume that the surface Σ coordinatized by τ is the entire boundary of afinite spacetime region then we can identify K with the boundary spaceK The correlation probability amplitude associated with a measurementof partial observables on the boundary surface is A = W (s) = 〈0|s〉where |s〉 is the eigenstate of the partial observables corresponding to themeasured eigenvalues

We must now build this structure concretely As we shall see in a precisetechnical sense this structure is unique

62 The kinematical state space KI construct here the quantum state space defined by the real connection (see Section42) There are three reasons for this First the physical lorentzian theory can beformulated in terms of this connection the only difference is that the hamiltonianoperator is slightly more complicated than (63) as explained in Section 422 Secondthings are far easier with the real connection and it is better to do easy things first inthe construction of the quantum state space defined by the complex connection thereare still some open technical complications [169] Third the real connection with thehamiltonian operator (63) defines the quantum euclidean theory namely the quantumtheory which has the theory defined in Section 421 as its classical limit this is aninteresting model by itself and is likely to be related to the physical theory as wasdiscussed in Section 42


Cylindrical functions Let G be the space of the smooth 3d real connec-tions A defined everywhere on a 3d surface Σ except possibly at isolatedpoints1 Fix the topology of Σ say to a 3-sphere I now define a space Sof functionals on G

We are now going to make use of the geometric interpretation of thefield A as a connection The so(3) Lie algebra is the same as the su(2)Lie algebra and it is convenient to view A as an su(2) connection Let τibe a fixed basis in the su(2) Lie algebra I choose τi = minus i

2 σi where σiare the Pauli matrices (A14) Write

A(τ) = Aia(τ) τi dxa (65)

Recall from Section 215 that an oriented path γ in Σ and a connectionA determine a group element U(A γ) = P exp

intγ A called the holonomy

of the connection along the path For a given γ the holonomy U(A γ)is a functional on G Consider an ordered collection Γ of smooth orientedpaths γl with l = 1 L and a smooth function f(U1 UL) of L groupelements A couple (Γ f) defines a functional of A

ΨΓf [A] = f(U(A γ1) U(A γL)) (66)

S is defined as the linear space of all functionals ΨΓf [A] for all Γ and f We call these functionals ldquocylindrical functionsrdquo In a suitable topologywhich is not important to detail here S is dense in the space of all con-tinuous functionals of A

I call Γ an ldquoordered oriented graphrdquo embedded in Σ I call simplyldquographrdquo an ordered oriented graph up to ordering and orientation anddenote it by the same letter Γ Clearly as far as cylindrical functionsare concerned changing the ordering or the orientation of a graph is justthe same as changing the order of the arguments of the function f orreplacing arguments with their inverse

Scalar product I now define a scalar product on the space S If twofunctionals ΨΓf [A] and ΨΓg[A] are defined with the same ordered ori-ented graph Γ define

〈ΨΓf |ΨΓg〉 equivint

dU1 dUL f(U1 UL) g(U1 UL) (67)

where dU is the Haar measure on SU(2) Notice the similarity with thelattice scalar product (5152) equation (67) is the scalar product of aYangndashMills theory on the lattice Γ

1The reason for this technical choice will become clear below

228 Quantum space

The extension of this scalar product to functionals defined on the samegraph but with different ordering or orientation is obvious But also theextension to functionals defined on different graphs Γ is simple In factobserve that different couples (Γ f) and (Γprime f prime) may define the same func-tional For instance say Γ is the union of the Lprime curves in Γprime and Lprimeprime othercurves and that f(U1 ULprime ULprime+1 ULprime+Lprimeprime) = f prime(U1 ULprime)then clearly ΨΓf = ΨΓprimef prime Using this fact it is clear that we can rewriteany two given functionals ΨΓprimef prime and ΨΓprimeprimegprimeprime as functionals ΨΓf and ΨΓg

having the same graph Γ where Γ is the union of Γprime and Γprimeprime Using thisfact (67) becomes a definition valid for any two functionals in S

〈ΨΓprimef prime |ΨΓprimeprimegprimeprime〉 equiv 〈ΨΓf |ΨΓg〉 (68)

Notice that even if (67) is similar to the scalar product of a lattice YangndashMills theory the difference is profound Here we are dealing with a gen-uinely continuous theory in which the states do not live on a single latticeΓ but on all possible lattices in Σ There is no cut-off on the degrees offreedom as in lattice YangndashMills theory

Loop states and the loop transform An important example of a finite norm state isprovided by the case (Γ f) = (α tr) That is Γ is formed by a single closed curve αor a ldquolooprdquo and f is the trace function on the group We can write this state as Ψαor simply in Dirac notation as |α〉 That is

Ψα[A] = Ψαtr[A] = 〈A|α〉 = tr U(Aα) = tr Pe∮α A (69)

The very peculiar properties that these states have in quantum gravity which will beillustrated later on have motivated the entire LQG approach (and its name) The normof Ψα is easily computed from (67)

|Ψα|2 =

intdU |tr U |2 = 1 (610)

A ldquomultilooprdquo is a collection [α] = (α1 αn) of a finite number n of (possiblyoverlapping) loops A ldquomultiloop staterdquo is defined as

Ψ[α][A] = Ψα1 [A] Ψαn [A] = tr U(Aα1) tr U(Aαn) (611)

The functional on loop space

Ψ[α] = 〈Ψα|Ψ〉 (612)

is called the ldquoloop transformrdquo of the state Ψ[A] The functional Ψ[α] represents thequantum state as a functional on a space of loops This formula called the ldquolooptransformrdquo is the formula through which LQG was originally constructed (see forinstance [170]) Using the measure dμ0[A] mentioned below this can be written as

Ψ[α] =

intdμ0[A] tr Pe

∮α AΨ[A] (613)

Intuitively this is a sort of infinite-dimensional Fourier transform from the A space tothe α space


Kinematical Hilbert space Define the kinematical Hilbert space K ofquantum gravity as the completion of S in the norm defined by the scalarproduct (67)2 and S prime as the completion of S in the weak topology definedby (67)3 This completes the definition of the kinematical rigged Hilbertspace S sub K sub S prime

Why this definition The main reason is that the scalar product (67) isinvariant under diffeomorphisms and local gauge transformations (Section622 below) and it is such that real classical observables become self-adjoint operators (Section 65 below) These very strict conditions arethe ones that the scalar product must satisfy in order to give a consistenttheory with the correct classical limit Furthermore the main feature ofthis definition is that the loop states Ψα are normalizable As we shallsee later on loop states are natural objects in quantum gravity Theydiagonalize geometric observables and they are solutions of the WheelerndashDeWitt equation Hence the kinematics as well as the dynamics selectthis space of states as the natural ones in gravity

There are two objections that can be raised against the definition ofK we have given First K is nonseparable This objection would be fatalin the context of flat-space quantum field theory but it turns out tobe harmless in a general-relativistic context because of diffeomorphisminvariance Indeed the ldquoexcessive sizerdquo of the nonseparable Hilbert spacewill turn out to be just gauge the physical Hilbert space H is separable Aswe shall see in Section 64 it is sufficient to factor away the diffeomorphismgauge to obtain a separable Hilbert space Kdiff

Second loop states are normalizable in lattice YangndashMills theory butthey are nonnormalizable in continuous YangndashMills theory By analogyone might object that they should not be normalizable states in continu-ous quantum gravity either As we shall see below however this analogyis misleading again precisely because of the great structural differencebetween a diffeomorphism-invariant QFT and a QFT on a backgroundAs we shall see in continuous YangndashMills theory a loop state describesan unphysical excitation that has infinitesimal transversal physical sizeIn gravity on the other hand a loop state describes a physical excitationthat has finite (planckian) transversal physical size This will become clearbelow in Section 662

Boundary Hilbert space There are two natural ways of defining theboundary space K We can either define K = Klowast otimes K and describe thequantum geometry of a spacetime region bounded by an initial and a

2The space of the Cauchy sequences Ψn where ||Ψm minus Ψn|| converges to zero3The space of the sequences Ψn such that 〈Ψn|Ψ〉 converges for all Ψ in S

230 Quantum space

final surface or simply define K = K interpreting the closed connectedsurface Σ as the boundary of a finite 4d spacetime region

621 Structures in KThe space K has a rich and beautiful structure I mention here only a fewaspects of this structure which are important for what follows referringto the more mathematically oriented literature on this subject (see [20]and references therein) for more details

Graph subspaces The cylindrical functions with support on a given graphΓ form a subspace KΓ of K By definition KΓ = L2[SU(2)L] where L isthe number of paths in Γ The space KΓ is the (unconstrained) Hilbertspace of a lattice gauge theory with spatial lattice Γ as described inSection 536 If the graph Γ is contained in the graph Γprime the Hilbert spaceKΓ is a proper subspace of the Hilbert space KΓprime This nested structureof Hilbert spaces is called a projective family of Hilbert spaces K is ndash andcan be defined as ndash the projective limit of this family

An orthonormal basis The tool for finding a basis in K is the PeterndashWeyltheorem which states that a basis on the Hilbert space of L2 functions onSU(2) is given by the matrix elements of the irreducible representations ofthe group Irreducible representations of SU(2) are labeled by half-integerspin j Call Hj the Hilbert space on which the representation j is definedand vα its vectors Write the matrix elements of the representation j as

R(j)αβ(U) = 〈U |j α β〉 (614)

For each graph Γ choose an ordering and an orientation Then a basis

|Γ jl αl βl〉 equiv |Γ j1 jL α1 αL β1 βL〉 (615)

in KΓ is simply obtained by tensoring the basis (614) That is

〈A|Γ jl αl βl〉 = R(j1)α1β1(U(A γ1)) R(jL)αL

βL(U(A γL)) (616)

This set of vectors in K is not a basis because the same vector appearsin KΓ and KΓprime if Γ is contained in the graph Γprime However it is very easyto get rid of the redundancy because all KΓ vectors belong to the trivialrepresentation of the paths that are in Γprime but not in Γ Therefore anorthonormal basis of K is simply given by the states |Γ jl αl βl〉 definedin (616) where the spins jl = 1

2 132 2 never take the value zero This

fact justifies the following definition


Proper graph subspaces For each graph Γ the proper graph subspace KΓ

is the subset of KΓ spanned by the basis states with jj gt 0 It is easy tosee that all proper subspaces KΓ are orthogonal to each other and theyspan K we can write this as

K simoplus

Γ

KΓ (617)

The ldquonullrdquo graph Γ = empty is included in the sum the corresponding Hilbertspace is the one-dimensional space spanned by the state Ψ[A] = 1 Thisstate is denoted |empty〉 thus 〈A|empty〉 = 1

K as an L2 space I have defined K as the completion of S in the scalarproduct defined by the bilinear form (67) Can this space be viewed asa space of square integrable functionals in some measure The answer isyes and it involves a beautiful mathematical construction that I will notdescribe here since it is not needed for what follows and for which I referthe reader to [20] Very briefly K sim L2[A dμ0] where A is an extensionof the space of the smooth connection The extension includes distribu-tional connections The measure dμ0 is defined on this space and is calledthe AshtekarndashLewandowski measure The construction is analogous to thedefinition of the gaussian measure dμG[φ] sim ldquoeminus

intdxdy φ(x)G(xy)φ(y)[dφ]rdquo

which as is well known needs to be defined on a space of distributionsφ(x) The space A has the beautiful property of being the Gelfand spec-trum of the abelian Clowast algebra formed by the smooth holonomies of theconnection A


The kinematical state space S sub K sub S prime carries a natural representationof local SU(2) and Diff(Σ) simply realized by the transformations ofthe argument A The scalar product defined above is invariant underthese transformations Therefore K carries a unitary representation oflocal SU(2) and Diff(Σ) Let us look at this in some detail

Local gauge transformations Under (smooth) local SU(2) gauge trans-formations λ Σ rarr SU(2) the connection A transforms inhomogeneouslylike a gauge potential ie

A rarr Aλ = λAλminus1 + λdλminus1 (618)

This transformation of A induces a natural representation of local gaugetransformations Ψ(A) rarr Ψ(Aλminus1) on K Despite the inhomogeneous

232 Quantum space

transformation rule (618) of the connection the holonomy transformshomogeneously (see Sec 215)

U [A γ] rarr U [Aλ γ] = λ(xγf )U [A γ]λminus1(xγi ) (619)

where xγi xγf isin Σ are the initial and final points of the path γ For a given

(Γ f) define

fλ(U1 UL) = f(λ(xγ1

f )U1λminus1(xγ1

i ) λ(xγLf )ULλminus1(xγLi )) (620)

It is then easy to see that the transformation of the quantum states is

ΨΓf (A) rarr [UλΨΓf ](A) = ΨΓf (Aλminus1) = ΨΓfλminus1 (A) (621)

Since the Haar measure is invariant under right and left group transforma-tions it follows immediately that (67) is invariant From (621) and fromtheir definition it is easy to see that basis states |Γ jl αl βl〉 transform as

Uλ|Γ jl αl βl〉 = R(j1)α1αprime

1(λminus1(xf1)) R(j1)βprime

1β1(λ(xi1)) middot middot middotR(jL)αL

αprimeL(λminus1(xfL)) R(jL)βprime

LβL(λ(xiL))

|Γ jl αprimel β

primel〉 (622)

where il and fl are the points where the link l begins and ends

(Extended) diffeomorphisms Consider maps φ Σ rarr Σ that are con-tinuous invertible and such that the map and its inverse are smootheverywhere except possibly at a finite number of isolated points Callthese maps ldquoextended diffeomorphismsrdquo (or sometimes loosely justldquodiffeomorphismsrdquo) Call the group formed by these maps Diff lowast

An example of an extended diffeomorphism which is not a proper diffeomorphismin two dimensions is the following In polar coordinates the map

rprime = r φprime = φ +1

2sinφ (623)

is continuous everywhere while it is differentiable everywhere except at r = 0 whereits jacobian is ill defined

Under an extended diffeomorphism the transformation of the connec-tion is well defined (recall A isin G is defined everywhere on Σ except on afinite number of isolated points) A transforms as a one-form

A rarr φlowastA (624)

Hence S carries the representation Uφ of Diff lowast defined by UφΨ(A) =Ψ((φlowast)minus1 A) The holonomy transforms as

U [A γ] rarr U [φlowastA γ] = U [A φminus1γ] (625)


where (φγ)(s) equiv (φ(γ(s)) That is dragging A by a diffeomorphism φis equivalent to dragging the curve γ (Notice that if φ is not a properdiffeomorphism the curve φγ may fail to be smooth at a finite num-ber of points at most) In turn a cylindrical function ΨΓf [A] is sentto a cylindrical function ΨφΓf [A] namely one which is based on theshifted graph Since the right-hand side of (67) does not depend explic-itly on the graph the diffeomorphism invariance of the inner product isimmediate


The kinematical state space K is a space of arbitrary wave functionals ofthe connection Ψ[A] but recall that we need a space of states that arewave functionals invariant under local gauge transformations and diffeo-morphisms More formally the two classical equations (480) and (481)must be implemented in the quantum theory giving

Daδ

δAia(τ)

Ψ[A] = 0 (626)

F iab(τ)

δ

δAia(τ)

Ψ[A] = 0 (627)

The same argument as used in Section 434 shows that these equationsdemand the invariance of Ψ under local SU(2) transformations and diffeo-morphisms More precisely the smearing functions fa(τ) of Section 434must be chosen in an appropriate class here the relevant group is Diff lowast

and therefore fa(τ) can be any infinitesimal generator of Diff lowast Underthis choice (627) is equivalent to the requirement of invariance of thestate under Diff lowast

Call K0 the space of the states invariant under local SU(2) and Kdiff

the space of the states invariant under local SU(2) and Diff lowast Recallingthat we call H the space of solutions of the WheelerndashDeWitt equationswe have therefore the sequence of Hilbert spaces

K SU(2)minusrarr K0Diff lowastminusrarr Kdiff

Hminusrarr H (628)

where the three steps correspond to the implementation of the threeequations that the wave functional must satisfy namely (626) (627) and(63) respectively I construct explicitly K0 in the next section and Kdiff

in the following one (Except for the first one the domain of the mapsin (628) will turn out to be given by the first term of the correspondingrigged Hilbert space)

234 Quantum space

63 Internal gauge invariance The space K0

The space K0 is the space of the states in K invariant under local SU(2)gauge transformations I call S0 the gauge-invariant subspace of S and S prime

0

its dual It is not difficult to see that K0 is a proper subspace of K Exam-ples of finite norm SU(2) invariant states are provided by the loop statesdefined in (69) In fact a moment of reflection shows that the multiloopstates are sufficient to span K0 In the first years of the development ofLQG multiloop states were used as a basis for K0 however this basis isovercomplete and this fact complicates the formalism Nowadays we havea much better control of K0 thanks to the introduction of the spin net-work states which can be seen as finite linear combinations of multiloopstates forming a genuine orthonormal basis

The spin network basis of quantum gravity is a simple extension ofthe spin network basis defined in Section 536 in the context of latticegauge theory As we shall see in the next section however diffeomorphisminvariance will soon make the two cases very different and connect thequantum gravity spin network basis to Penrosersquos old ldquospin networkrdquo ideathat quantum states of the geometry can be described as abstract graphscarrying spins

Spin networks Denote ldquonodesrdquo the end points of the oriented curves in ΓWithout loss of generality assume that each set of curves Γ is formed bycurves γ that if they overlap at all overlap only at nodes Viewed in thisway Γ is in fact a graph immersed in the manifold that is a collectionof nodes n which are points of Σ joined by links l which are curves inΣ The ldquooutgoing multiplicityrdquo mout of a node is the number of links thatbegin at the node The ldquoingoing multiplicityrdquo min of a node is the numberof links that end at the node The multiplicity or valence m = min +mout

of a node is the sum of the twoGiven the graph Γ for which an ordering and an orientation have been

chosen let jl be an assignment of an irreducible representation differ-ent from the trivial one to each link l Let in be an assignment of anintertwiner in to each node n The notion of intertwiner was defined inSection 536 The intertwiner in associated with a node is between therepresentations associated with the links adjacent to the node The tripletS = (Γ jl in) is called a ldquospin network embedded in Σrdquo A choice of jland in is called a ldquocoloringrdquo of the links and the nodes respectively


Consider a spin network S = (Γ jl in) with L links and N nodes Thestate |Γ jl αl βl〉 defined above in (616) has L indices αl and L indices


j

j

2= 12

S =n1

n2

1 = 1

j3

= 12

Fig 61 A simple spin network with two trivalent nodes

βl The N intertwiners in have altogether precisely a set of indices dualto these The contraction of the two

|S〉 equivsum

αlβl

vβ1βn1i1 α1αn1

vβn1+1βn2i2 αn1+1αn2

vβ(nNminus1+1)βL

iN α(nNminus1+1)αL

|Γ jl αl βl〉 (629)

defines the spin network state |S〉 The pattern of the contraction of theindices is dictated by the topology of the graph itself the index αl (respβl) of the link l is contracted with the corresponding index of the inter-twiner vin of the node n where the link l starts (resp ends) The gaugeinvariance of this state follows immediately from the transformation prop-erties (622) of the basis states and the invariance of the intertwiners Asa functional of the connection this state is

ΨS[A] = 〈A|S〉 equiv(otimes

l

R(jl)(H[A γl])

)

middot(otimes

n

in

)

(630)

The raised dot notation indicates the contraction between dual spaces onthe left the tensor product of the matrices lives in the space otimesl (Hlowast

jlotimesHjl)

On the right the tensor product of all intertwiners lives precisely in thedual of this space

Example Letrsquos say Γ has two nodes n1 and n2 and three links l1 l2 l3 each link begin-ning at n1 and ending at n2 Let the coloring of the links be j1 = 1 j2 = 12 j3 = 12see Figure 61 At each of the two nodes we must therefore consider the tensor productof two fundamental and one adjoint representation of SU(2) As is well known the ten-sor product of these representations contains a single copy of the trivial representationtherefore there is only one possible intertwiner A moment of reflection shows that thisis given by the triple of Pauli matrices viAB = 1radic

3σiAB since these have precisely the

invariance property

(R(U))ij UAC UB

D σjCD = σiAB (631)

236 Quantum space

Here (R(U))ij is the adjoint representation with i j = 1 2 3 vector indices and UAC

is the fundamental representation with AB = 0 1 spinor indices The 1radic3

is a nor-malization factor to satisfy (5157) Therefore there is only one possible coloring of thenodes in this case The spin network state is then

ΨS[A] =1

3σiAB (R(H[A γ1]))

ij (H[A γ2])

AC (H[A γ3])

BD σjCD (632)

Let me now enunciate the main fact concerning the spin network statesthe ensemble of the spin network states |S〉 forms an orthonormal basisin K0 Orthonormality can be checked by a direct calculation The basisis labeled by spin networks namely graphs Γ and colorings (jl in)

Some comments First I have assumed the spins jl to be all differentfrom zero (a spin network containing a link l with jl = 0 is identified withthe spin network obtained by removing the link l) Second this result isa simple consequence of the PeterndashWeyl theorem namely of the fact thatthe states |Γ jl αl βl〉 form a basis in K and the very definition of theintertwiners Third the spin network basis is not unique as it dependson the (arbitrary) choice of a basis in each space of intertwiners at eachnode Notice also that in the basis |S〉 = |Γ jl in〉 the label Γ runs overall nonoriented and nonordered graphs However for the definition of thecoloring an orientation and an ordering has to be chosen for each Γ

The space S0 is the space of the finite linear combinations of spinnetwork states which is dense in K0 and S prime

0 is its dual


Orientation There is an isomorphism ε between a representation j and its dual jlowast Ifj is the fundamental

ε C2 rarr (C2)lowast

ψA rarr ψA = εABψB (633)

where εAB is the antisymmetric tensor This extends to all other representations sincethey can be obtained from tensor products of the fundamental Using this we can raiseand lower the indices of the intertwiners and identify intertwiners with the same totalnumber of indices We can then ask what happens if we change the orientation of one ofthe links Using (A6) a straightforward calculation that I leave to the reader showsthat the only change is to the overall sign if the representation has half-integer spinThus the only relevant orientation of the spin network is an overall global orientation

Spin networks versus loop states A spin network state can be decomposed into a finitelinear combination of (multi-)loop states The representation j can be written as thesymmetrized tensor product of 2j fundamental representations Therefore we can writethe elements of Hj as completely symmetric complex tensors ψA1A2j with 2j spinorindices Ai = 0 1 In this basis the representation matrices have the simple form

R(j)A1A2jB1B2j (U) = U (A1

(B1 UA2j)

B2j) (634)


=

g2

g

g

1

12

1

12

+3

Fig 62 Decomposition of a spin network state into loop states

where the parentheses indicate complete symmetrization (see Appendix A1) In thisbasis the intertwiners are simply combinations of the only two SU(2) invariant ten-sors namely εAB and δBA For instance a (nonnormalized) trivalent intertwiner betweenincoming representations j jprime and an outgoing representation jprimeprime is

vA1A2j B1B2jprimeC1C2jprimeprime = εA1B1 εAaBa δC1

Ba+1 δ

CbB2jprime

δCb+1Aa+1

δC2jprimeprimeA2j

(635)

where j = a + c jprime = a + b and jprimeprime = b + c Now when the two holonomy matrices oftwo contiguous links γ1 and γ2 are joined by δBA they give the holonomy of the curveobtained joining γ1 and γ2 which I denote γ1γ2

H[A γ1]ABδ

BCH[A γ2]

CD = H[A γ1γ2]

AD (636)

On the other hand recall that εABUA

CεCD = (Uminus1)DA Therefore

εDBH[A γ1]ABεACH[A γ2]

CE = H[A γminus1

1 γ2]DE (637)

Therefore the tensors εAB and δBA in the intertwiners simply join the segments in thearguments of the holonomies Since the graph of the spin network is finite a line ofjoining must close to a loop A moment of reflection will convince the reader thata spin network state (629) is therefore equal to a linear combination of products ofholonomies of closed lines that wrap along the graph That is it is a linear combinationof multiloop states

The decomposition of a spin network state in loop states can be obtained graphi-cally as follows Replace each link of the graph colored with spin j with 2j parallelstrands Symmetrize these strands along each link The intertwiners at the nodes canbe represented as collections of segments joining the strands of different links By join-ing these segments with the strands one obtains a linear combination of multiloopsThe spin network states can then be expanded in the corresponding loop states Noticethat this is analogous to the construction at the basis of the KaufmanndashLins recouplingtheory illustrated in Appendix A1 (care should be taken with signs) For details on thisconstruction see [171]

Applying this rule to the state (632) illustrated in Figure 61 it is easy to see that

ΨS(A) =1

2

[Ψ

((γ1γminus12 )(γ1γ

minus13 ))

+ Ψγ1γ

minus12 γ1γ

minus13

] (638)

See Figure 62 for a graphical illustration of this decomposition

Details on intertwiners Given a graph and a coloring of its links it may happen thatthere is no nonvanishing intertwiner at all associated with a node For instance a nodewith valence unity cannot exist in a spin network because a single nontrivial irreducible

238 Quantum space

representation does not contain any invariant subspace If the node is bivalent there isan intertwiner only if the incoming and outgoing representations are the same and theintertwiner is the identity The spin network with two links joined by a bivalent nodeis identified with one obtained replacing the two links with a single link A trivalentnode may have adjacent links with colorings j1 j2 j3 only if these satisfy the ClebschndashGordan conditions (A10)ndash(A11) The intertwiners are directly given by the Wigner3j-coefficients (A16) up to normalization Nontrivial intertwiner spaces begin onlywith nodes of valence four or higher Intertwiners of an n-valent node can be labeledby nminus 2 spins as detailed in Appendix A1

64 Diffeomorphism invariance The space Kdiff

Let me now come to the second and far more crucial invariance 3d dif-feomorphism invariance We have to find the diffeomorphism-invariantstates

Transformation properties of spin network states under diffeomorphisms The spin net-work states |S〉 are not invariant under diffeomorphisms A diffeomorphism moves thegraph around on the manifold and therefore changes the state Notice however that adiffeomorphism may change more than the graph of a spin network that is the equa-tion Uφ|Γ jl in〉 = |(φΓ jl in)〉 is not always correct In particular a diffeomorphismthat leaves the graph Γ invariant may still affect a spin network state |Γ jl in〉 This isbecause for each graph the definition of the spin network state requires the choice ofan orientation and ordering of the links and these can be changed by a diffeomorphism

Here is an example Let Γ be an ldquoeyeglassesrdquo graph formed by two loops α and βin the j = 12 representation connected by a path γ in the j = 1 representation Thespace of the intertwiners at each node is one-dimensional but this does not imply thatthere is no choice to be made for the basis since if i is a normalized intertwiner so isminusi With one choice the state is

ΨS[A] = (U(Aα))AB σiB

A (R(1)(U(A γ)))ij σjDC (U(A β))CD (639)

Using elementary SU(2) representation theory this can be rewritten (up to a normal-ization factor) as

ΨS[A] sim trH[Aαγβγminus1] minus trH[Aαγβminus1γminus1] (640)

Now consider a diffeomorphism φ that turns the loop β around namely it reverses itsorientation while leaving α and γ as they are Clearly this diffeomorphism will sendthe two terms of the last equation into each other giving

UφΨS[A] = minusΨS[A] (641)

while φΓ = ΓGiven an oriented and ordered graph Γ there is a finite discrete group GΓ of maps

gk such as the one of the example that change its order or orientation and that canbe obtained as a diffeomorphism The elements gk of this group act on KΓ

A moment of reflection will convince the reader that the diff-invariantstates are not in K0 they are in S prime

0 We are therefore in the commonsituation in which the solutions of a quantum equation must be searched


in the extension of the Hilbert space and the scalar product must be ap-propriately extended to the space of the solutions as explained in Section552

The elements of S prime0 are linear functionals Φ on the functionals Ψ isin S0

The requirement of diff invariance makes sense in S prime0 because the action

of the diffeomorphism group is well defined in S prime0 by duality

(UφΦ)(Ψ) equiv Φ(Uφminus1Ψ) (642)

Therefore a diff-invariant element Φ of S prime0 is a linear functional such that

Φ(UφΨ) equiv Φ(Ψ) (643)

The space Kdiff is the space of these diff-invariant elements of S prime0 Remark-

ably we have a quite good understanding of this space whose elementscan be viewed as the quantum states of physical space

The space Kdiff I now define a map Pdiff S0 rarr S prime0 and show that the

(closure in norm of the) image of this map is precisely Kdiff Let the statePdiffΨ be the element of S prime

0 defined by

(Pdiff Ψ)(Ψprime) =sum

Ψprimeprime=UφΨ


The sum is over all states Ψprimeprime in S0 for which there exist a φ isin Diff lowast

such that Ψprimeprime = UφΨ The key point is that this sum is always finite andtherefore well defined To see this notice that since Ψ and Ψprime are in S0they can be expanded in a finite linear combination of spin network statesIf a diffeomorphism changes the graph of a spin network state ΨS thenit takes it to a state orthogonal to itself If it doesnrsquot change the graphthen either it leaves the state invariant so that no multiplicity appears in(644) or it changes the ordering or the orientation of the links but theseare discrete operations giving at most a discrete multiplicity in the sum in(644) Therefore the sum in (644) is always well defined Clearly Pdiff Ψis diff invariant namely it satisfies (643) Furthermore it is not difficultto convince oneself that the functionals of the form (644) span the spaceof the diff-invariant states Therefore the (closure in norm of the) imageof Pdiff is Kdiff States related by a diffeomorphism are projected by Pdiff

to the same element of Kdiff

Pdiff ΨS = Pdiff (UφΨS) (645)

Finally the scalar product on Kdiff is naturally defined by

〈Pdiff ΨS Pdiff ΨSprime〉Kdiffequiv (Pdiff ΨS)(ΨSprime) (646)

240 Quantum space

(see section 552) This completely defines Kdiff Equivalently Kdiff isdefined by the bilinear form

〈ΨΨprime〉Kdiffequiv 〈Ψ|Pdiff |Ψprime〉 equiv

sum

Ψprimeprime=φΨ


in S0

To understand intuitively why the above definition works consider the followingformal argument Imagine that we were able to define a measure dφ on Diff lowast Wecould then write diffeomorphism-invariant states by simply integrating an arbitrarystate on the orbit of the diffeomorphism group

PdiffΨ equivint

Diff lowast[dφ] UφΨ (648)

Therefore


int

Diff lowast[dφ] (ΨprimeUφΨ) (649)

Let us assume for simplicity that Ψ isin KΓ and Ψprime isin KΓprime (the general case following bylinearity) Then the right-hand side of (649) vanishes unless there is a φ that sends Γin Γprime If this is the case the integral has support just on the subgroup of Diff lowast thatleaves Γ invariant Elements of this subgroup can either change the state Ψ or leave itinvariant Thus we can rewrite (649) as


sum

Ψprimeprime=UφprimeΨ

int

DΨprimeprime[dφ] (ΨprimeUφΨprimeprime) (650)

where the integral is over the subgroup DΨprimeprime of Diff lowast that leaves Ψprimeprime invariant Butthen we can take the scalar product out of the integral and write


sum

Ψprimeprime=UφΨ

(ΨprimeUφΨprimeprime)

(int

DΨprime[dφ]

)

(651)

If we now assume that the measure dφ is such that the volume of DΨprime is unity werecover the definition given above Therefore the definition (644) can be seen as arigorous implementation of the intuitive ldquointegration of the diffeomorphism grouprdquo of(648)


To understand the structure of Kdiff consider the action of Pdiff on thestates of the spin network basis To this aim observe that a diffeomor-phism sends a spin network state |S〉 to an orthogonal state or to a stateobtained by a change in the order of the orientation of the links Denotegk|S〉 the states that are obtained from |S〉 by changes of orientation orordering and that can be obtained via a diffeomorphism as in the exam-ple above The maps gk form the finite discrete group GΓ therefore therange of the discrete index k is finite Then it is easy to see that

〈S|Pdiff |Sprime〉 =

0 if Γ = φΓprimesum

k 〈S|gk|Sprime〉 if Γ = φΓprime (652)


An equivalence class K of unoriented graphs Γ under diffeomorphisms iscalled a ldquoknotrdquo Knots without nodes have been widely studied by thebranch of mathematics called knot theory with the aim of classifyingthem Knots with nodes have also been studied in knot theory but to alesser extent From the first line of (652) we see that two spin networksS and Sprime define orthogonal states in Kdiff unless they are knotted in thesame way That is unless they are defined on graphs Γ and Γprime belongingto the same knot class K Therefore the basis states in Kdiff are first ofall labeled by knots K We call KK the subspace of Kdiff spanned by thebasis states labeled by the knot K That is

KK = Pdiff KΓ (653)

for any Γ isin KThe states in KK are then distinguished only by the coloring of links and

nodes As observed before the colorings are not necessarily orthonormaldue to the nontrivial action of the discrete symmetry group GΓ To findan orthonormal basis in KK we have therefore to further diagonalize thequadratic form defined by the second line of (652) Denote |s〉 = |K c〉 theresulting states The discrete label c is called the coloring of the knot KUp to the complications due to the discrete symmetry GΓ it correspondsto the coloring of the links and the nodes of Γ The states |s〉 = |K c〉 arecalled spin-knot states or s-knot states


The key property of knots is that they form a discrete set Therefore thelabel K is discrete It follows that Kdiff admits a discrete orthonormalbasis |s〉 = |K c〉 Thus Kdiff is a separable Hilbert space The ldquoexcessivesizerdquo of the kinematical Hilbert space K reflected in its nonseparabilityturns out to be just a gauge artifact

The fact that knots without nodes form a discrete set is a classic result of knottheory It is easy to understand it intuitively first if two loops without nodes can becontinuously deformed into each other without crossing then there is a diffeomorphismthat sends one into the other second to change the node class we have to deform a linkacross another link and this is a discrete operation On the other hand the fact thatknots with nodes form a discrete set is nontrivial Indeed it depends on the fact thatwe have chosen the extension Diff lowast of the diffeomorphism group Diff Had we chosenDiff as the invariance group the space of the knot classes would have been continuous

To understand this recall that the action of Diff on the tangent space is linearConsider a graph Γ that can be deformed continuously into a graph Γprime Let p be thelocation of an n-valent node of Γ and pprime the corresponding node on Γprime Is there adiffeomorphism φ in Diff sending Γ to Γprime The answer in general is negative for thefollowing reason The diffeomorphism must send p to pprime Hence φ(p) = pprime The tangentspace to p is sent into the tangent space to pprime by the jacobian Jp of φ at p which is a

242 Quantum space

linear transformation in 3d Let vi i = 1 n be the tangents of the n links at p andvprimei i = 1 n be the tangents of the n links at pprime For φ to send the two nodes intoeach other we must have

Jpvi = vprimei (654)

But in general there is no linear transformation sending n given directions into n othergiven directions In other words (654) gives n linear conditions on the nine degreesof freedom of the jacobian matrix (Jp)

ab = partφa(x)partxb|p Therefore in general two

graphs that can be transformed into each other continuously cannot be transformed intoeach other by a diffeomorphism The equivalence classes are characterized by continuousparameters at the nodes

On the other hand maps in Diff lowast can freely transform these parameters The reasonis that thanks to the relaxation of the differentiability condition an extended diffeo-morphism φ isin Diff lowast can act nonlinearly on the tangents In Section 67 after thediscussion of the physical interpretation of the knot states I will discuss the physicalreasons why Diff lowast is more appropriate than Diff as gauge group

This concludes the construction of the kinematical quantum state spaceof LQG The physical meaning of the s-knot states in Kdiff will becomeclear later on It is now time to define the operators

65 Operators

There are two basic field variables in the canonical theory from which allmeasurable quantities can be constructed the connection Ai

a(τ) and itsmomentum Ea

i (τ) I now define quantum operators corresponding to sim-ple functions of these Quantum states are functionals Ψ[A] of the connec-tion A The momentum conjugate to the real connection A is (18πG)E(see (440)) We can therefore define the two field operators

Aia(τ)Ψ[A] = Ai

a(τ) Ψ[A] (655)1

8πGEa


δAia(τ)

Ψ[A] (656)

on functionals of Ψ[A] In the following I choose units in which 8πG = 1I will then restore physical units when needed The first is a multiplica-tive operator the second a functional derivative However both theseoperators send Ψ[A] out of the state spaces that we have constructed Inparticular they are not well defined in K This can be easily cured bytaking instead of A and E some simple function of these


The holonomy U(A γ) is well defined on S More precisely let UAB(A γ)

be the matrix elements of the group element U(A γ) Then

(UAB(A γ)Ψ)[A] = UA

B(A γ)[A] Ψ[A] (657)

65 Operators 243

The right-hand side is clearly in S if Ψ[A] is In fact any cylindricalfunction of the connection is immediately well defined as a multiplicativeoperator in K

For instance consider a closed loop α and let Tα[A] = trU(Aα) Consider the actionof this operator on a spin network state |S〉 with a graph that does not intersect withα Then clearly

Tα|S〉 = |S cup α〉 (658)

where S cup α is the spin network formed by S plus the loop α in the j = 12 represen-tation

Notice that all this is quite different from quantum field theory on abackground spacetime where field operators are operator-valued distri-butions and therefore are well defined only when smeared in three dimen-sions In (657) a well-defined operator is obtained by simply smearing(the path-ordered exponential of) the field in just one dimension alongthe loop γ This is a characteristic feature of diff-invariant quantum fieldtheories


To understand the action of E we have to compute the functional deriva-tive of the holonomy the building block of the cylindrical functions It isnot hard to show that

δ

δAia(x)

U(A γ) =int

ds γa(s) δ3(γ(s) x) [U(A γ1) τi U(A γ2)] (659)

Here s is an arbitrary parametrization of the curve γ γa(s) are the coor-dinates of the curve γa(s) equiv dγa(s)ds is the tangent to the curve in thepoint s γ1 and γ2 are the two segments in which γ is separated by thepoint s This is a crucial formula that plays a major role in what followsThe diligent reader is therefore invited to derive it and understand it indetail There are several possible derivations The naive one is just to usea formal functional derivation of the expression (280) The rigorous one isto consider variations of the defining equation (278) See [172] for details

Notice that the right-hand side of (659) is a distribution but only atwo-dimensional one since one of the three deltas in δ3 is in fact integratedover ds It is therefore natural to search for an operator well defined onK by smearing E in two-dimensions To this purpose consider a two-dimensional surface S embedded in the 3d manifold

244 Quantum space

g

S

P

g

2

1

Fig 63 A curve that intersects the surface at an individual point P

Let σ = (σ1 σ2) be coordinates on the surface S The surface is definedby S (σ1 σ2) rarr xa(σ1 σ2) Consider the operator

Ei(S) equiv minusiint

Sdσ1dσ2 na(σ)

δ

δAia

(x(σ)) (660)

where

na(σ) = εabcpartxb(σ)partσ1

partxc(σ)partσ2

(661)

is the normal one-form on S and εabc is the completely antisymmetricobject (for the relativists the LevindashCivita tensor of density weight (minus1))

The ldquograsprdquo Let us now compute the action of the operator Ei(S) onthe holonomy U(A γ) Assume for the moment that the end points of γdo not lie on the surface S For simplicity let us also begin by assumingthat the curve γ crosses the surface S at most once and denote P theintersection point (if any) see Figure 63

The curve is separated into two parts γ = γ1 cup γ2 by P By using(659) and (660) we obtain

Ei(S)U(A γ)

= minusiint

Sdσ1dσ2 εabc

partxa(σ)partσ1

partxb(σ)partσ2

δ

δAic

(x(σ)) U(A γ)

= minusiint

S

int

γdσ1dσ2ds εabc

partxa

partσ1

partxb

partσ2

partxc

partsδ3(x(σ) x(s)

)

times U(A γ1) τi U(A γ2) (662)

A closer look at this result reveals a great simplification of the last integralThe integral vanishes unless the surface and the curve intersect Assume

65 Operators 245

that there is a single intersection point and further assume that it hascoordinates xa = 0 In the neighborhood of this point consider the map(σ1 σ2 s) rarr (x1 x2 x3) from the integration domain to coordinate spacedefined by

xa(σ1 σ2 s) = xa(σ1 σ2) + xa(s) (663)

The jacobian of this map

J equiv part (x1 x2 x3)part (σ1 σ2 s3)

= εabcpartxa

partσ1

partxb

partσ2

partxc

parts(664)

appears in the integral We can therefore make the change of variables(σ1 σ2 s) rarr (x1 x2 x3) in the integral

The jacobian is nonvanishing since I have required that there is only a single non-degenerate point of intersection The jacobian (664) and the integral (662) wouldvanish if the tangent vectors given by the partial derivatives in (664) were coplanarie if a tangent partxab(σ)partσ12 to the surface were parallel to the tangent partxc(s)partsof the curve This happens for instance if the curve lies entirely in S Then therewouldnrsquot be just a single intersection point I consider these limiting cases later on

With a change of variables in the integral we can easily perform theintegration and get rid of the delta function We obtain remarkablyint

S

int

γdσ1dσ2ds εabc

partxa(σ)partσ1

partxb(σ)partσ2

partxc(s)parts

δ3(x(σ) x(s)

)= plusmn1 (665)

In fact this integral is a well-known analytic coordinate-independent ex-pression for the intersection number between the surface S and the curveγ It vanishes if there is no intersection The sign is dictated by the relativeorientation of the surface and the curve Hence we obtain the simple result

Ei(S)U(A γ) = plusmni U(A γ1) τi U(A γ2) (666)

The action of the operator Ei(S) on holonomies consists of just insertingthe matrix (plusmni τi) at the point of intersection We say that the operatorEi(S) ldquograspsrdquo γ

The generalization to multiple intersections is immediate Using P tolabel different intersection points we have

Ei(S)U(A γ) =sum

Pisin(S cap γ)

plusmni U(A γP1 ) τi U(A γP2 ) (667)

For later use I give here also the action of the operator Ei(S) on the holonomy inan arbitrary representation j

Ei(S) Rj(U(A γ))

= plusmni Rj(U(A γ1)) (j)τi R

j(U(A γ2)) (668)

where (j)τi is the SU(2) generator in the spin-j representation

246 Quantum space

Thus Ei(S) is a well-defined operator on K The fact that it is a surfaceintegral of Ea

i (τ) which is well defined can be understood as follows Geo-metrically Ea

i (τ) is not a vector field but rather a vector density The nat-ural associated geometric quantity is the two-form Ei = εabcE

ai dxb and dxc

But a two-form can be naturally integrated over a surface giving an ob-ject which is well behaved under diffeomorphisms In fact the operatorwe have defined corresponds precisely to the classical quantity

Ei(S) =int

SEi (669)

The same is true in the quantum theory The functional derivative is avector density therefore a two-form and this is naturally integrated overa surface Geometry and operator properties begin here to go nicely handin hand

The operators Tα and Ei(S) defined on the Hilbert space K form a rep-resentation of the corresponding classical Poisson algebra A major resultof the mathematically rigorous approach to loop quantum gravity is theproof of a unicity theorem for this representation The theorem is usu-ally called the ldquoLOSTrdquo theorem from the initials of the people that havediscovered it (more precisely one of its versions see the Bibliographicalnotes below) The theorem is analogous to the Stone-vonNeuman theo-rem in nonrelativistic quantum mechanics which shows the unicity of theSchrodinger representation It relies heavily on the hypothesis of diffeo-morphism invariance The theorem shows under certain general hypothe-ses that the loop representation which was built largely ldquoby handrdquo isthe only possible way of quantizing a diffeomorphism invariant theoryNo such theorem hold in conventional quantum field theory In particu-lar this shows that in the diffeomorphism invariant context the theory israther tightly determined

66 Operators on K0

To be well defined on K0 an operator must be invariant under internalgauge transformations As far as the connection is concerned this is veryeasy to obtain We noticed above that any cylindrical function gives awell-defined operator The cylindrical function needs simply to be gaugeinvariant to be well defined in K0 The operator Tα for instance definedin (658) is well defined on K0


The situation with E is slightly more complicated The operator Ei(S)clearly cannot be gauge invariant as the index i transforms under internal


gauges On the other hand we cannot obtain a gauge-invariant quantityby simply contracting this index as

E2(S) equivsum

i

Ei(S)Ei(S) (670)

because the transformation property of Ei(S) is complicated by the in-tegral over S Let us nevertheless compute its action on a spin networkstate S since this is a crucial step for what follows Let us assume thatthere is a single intersection P between the surface S and (the graph Γ)of the spin network S Let j be the spin of the link at the intersectionUsing (668) we see that the first operator Ei(S) inserts a matrix (j)τi atthe intersection So does the second but minus(j)τi

(j)τi = j(j + 1) times 1 is theCasimir operator of SU(2) Therefore

E2(S)|S〉 = 2 j(j + 1) |S〉 (671)

This beautiful result however is completely spoiled if Γ intersects S morethan once because in this case the τi matrices at different points getcontracted and we do not get a gauge-invariant state

To circumvent this difficulty let us define a gauge-invariant operatorA(S) associated to the surface S as follows For any N partition thesurface S into N small surfaces Sn that become smaller and smaller asN rarr infin and such that for each N

⋃n Sn = S Then define

A(S) equiv limNrarrinfin

sum

n

radicE2(Sn) (672)

Do not confuse the A chosen to denote this operator with the A thatdenotes the connection there is no relation between these two quantitiesThe reason for choosing the letter A to denote the operator (672) willbecome clear shortly

In the classical case by Riemannrsquos very definition of the integral wehave

A(S) =int

S

radicnaEa

i nbEbi d2σ (673)

which is a well-defined gauge-invariant quantity In the quantum casethe action of the operator (672) is easy to compute Let us evaluate iton a spin network state under the simplifying assumption that no spinnetwork node lies on S For sufficiently high N no Sn will contain morethan one intersection with Γ see Figure 64 Therefore the sum over nreduces to a sum over the intersection points P between S and Γ and isindependent from N for N sufficiently high Using (671) we have then

248 Quantum space

Γ

Sn

S

Fig 64 A partition of S

P2

SP3P1

Fig 65 A simple spin network S intersecting the surface S

immediately (see Figure 65)

A(S)|S〉 =

sum

Pisin(ScupΓ)

radicjP (jP + 1) |S〉 (674)

where jP is the color of the link that crosses S at P This is a key resultFirst of all the operator A(S) is well defined in K This is the operatorcorresponding to the classical quantity (673) Second spin network statesare eigenfunctions of this operator

To summarize we have obtained for each surface S isin M a well-definedSU(2) gauge-invariant and self-adjoint operator A(S) which is diagonalon the spin networks that do not have a node on S The correspondingspectrum (with the restrictions mentioned) is labeled by multiplets j =(j1 jn) i = 1 n and n arbitrary of positive half-integers ji This


is called the main sequence of the spectrum and is given by

Aj =

sum

i


We will compute the rest of the spectrum which is also real and discretein Section 664

Since the operator is diagonal on spin network states (with appro-priate choices of intertwiners) and all its eigenvalues are real it is alsoself-adjoint In fact this operator is well defined at the level of rigor-ous mathematical physics For a completely rigorous detailed construc-tion see for instance [20] and [173] The fact that this operator can berigorously constructed at all it is finite and its spectrum can even becomputed is a rather striking result considering that its definition in-volves an operator product and a square root This is the first remarkablepay-off of a well-defined diffeomorphism-invariant formalism for quantumfield theory As we will see in a moment this result has major physicalsignificance

In principle the operators Tα and A(S) are sufficient to define thequantum theory In practice it will be convenient later on to define otheroperators as well Before doing so however let us discuss the physicalmeaning of the mathematical result achieved so far

662 Quanta of area

In the previous section I have constructed and diagonalized the SU(2)gauge-invariant and self-adjoint operator A(S) What is the physical in-terpretation of this operator A direct comparison of (673) with (428)shows that A(S) is precisely the physical area of the surface S

Therefore we obtain immediately an important physical result Thepartial observable given by the area of a fixed two-dimensional surfaceis represented in the quantum theory by a self-adjoint operator with adiscrete spectrum But this yields immediately a physical prediction anymeasurement of the area of any physical surface can only give an outcomewhich is in the spectrum of this operator Since the spectrum is discretethis means that the physical area is a quantized partial observable Ameasurement of the area can only give a result contained in the spectrum(675)ndash(6125) of A(S)

Restoring physical units for 8πG and c the area operator is 8πGcminus3

times (672) and its eigenvalues are 8πGcminus3 times the ones in (675)ndash(6125) The main sequence for instance gives

Aj = 8πGcminus3sum

i


250 Quantum space

Had we used the more general Barbero connection described in Section423 instead of the real connection then from (446) we would have theoperator E given by

c3

8πγGEa


δAia(τ)

Ψ[A] (677)

instead of (656) In this case the spectrum is modified by an overallconstant factor

Aj = 8πγGcminus3sum

i


Up to the single Immirzi parameter γ this is a precise and quantitativeprediction of LQG It can in principle be verified or falsified Alter-natively indirect consequences of this prediction could have observableeffects In fact this quantization of the area is the basis of many resultsof the theory such as the derivation of black-hole entropy

The smallest (nonvanishing) eigenvalue in (676) taking the Immirziparameter equal to 1 is

A0 = 4radic

3πGcminus3 sim 10minus66 cm2 (679)

This is a sort of elementary quantum of area of the order of the Planckarea It is the quantum of area carried by a link in the fundamental j =12 representation The fact that there is no area that can be measuredbelow a minimum amount indicates that there is a sort of minimal sizeof physical space at the Planck scale

An intrinsic discreteness of physical space at the Planck length haslong been expected in quantum gravity Notice that in the context ofLQG this discreteness is not imposed or postulated Rather it is a directconsequence of a straightforward quantization of GR Space geometryis quantized in the same manner in which the energy of an harmonicoscillator is quantized


The two-hand loop operator The area operator can be defined also in a different man-ner which is of interest because it employs a technique that we will use below For eachsmall surface S define E2(S) in an SU(2) gauge-invariant manner as follows Given apath γ with end points r and s define the ldquotwo-handed loop operatorrdquo

T abγ = Ea

i (r)R(1)(U(A γ))ijEbj (s) (680)

where R(1)(U)ij is the adjoint j = 1 representation Given two points r and s in asmall surface S let γrs be a straight path (in the coordinate chosen) from r to s and

T ab(r s) = T abγrs

(681)


Then define

E2(S) =

int

Sd2σ

int

Sd2σprimena(σ)nb(σ

prime)T ab(σ σprime) (682)

In the limit in which the surface is small only the first term of the holonomy whichis the identity survives and therefore this definition converges to the one in (670) forsmall surfaces

The advantage of using this kind of regularization in the quantum theory is that itsimplifies the SU(2) representation calculations This is because the regularized oper-ator is itself SU(2) invariant

The action of the quantum operator (681) on a spin network state is easy to computeThere is a contribution for each intersection of a spin network link with the surface S foreach E Each of the two intersections is called a grasp For each of these contributionsthe spin network is modified by the creation of two nodes at the points r and s onefor each grasp and the addition of the loop γrs to the spin network We say that theldquohandsrdquo of the operator ldquograsprdquo the spin network Each node is trivalent with two linksbeing the ones of the grasped spin network say in a representation j and the otherbeing γrs in the representation j = 1 The intertwiner between these representationsis the SU(2) generator (j)(τi)

αβ in the representation j This is not normalized If we

call the normalized intertwiner iiαβ we have

(j)(τi)αβ = nj ii

αβ (683)

where nj can be computed easily by taking the norm of this equation This gives

n2j = tr((j)τ i (j)τi) = j(j + 1) tr(1) = j(j + 1) (2j + 1) (684)

Recoupling theory In the limit in which the surface is small the two grasps are onthe same link and at the same point and the line between them is infinitesimal It isnevertheless useful to write the two grasps as separated and the lines between them asfinite lines just remembering that the connection on these lines is trivial namely thatthey are associated to identities In this representation the result of the grasp on a linkof spin j of the spin network can therefore be represented as follows

E2(S) j sim 2j(j + 1) (2j + 1)

j

j

j 1

(685)

This picture can be directly interpreted in terms of recoupling theory which is a sim-ple graphical way of making calculations with SU(2) representation theory In thisrepresentation lines represent contraction of representation indices and nodes repre-sent normalized intertwiners

j

α

β

equiv δβα

γ δ ε

j jprime jprimeprime equiv vγδε (686)

Here α β and γ are indices in an orthonormal basis in the representation j and δand ε in the representations jprime and jprimeprime respectively Index contraction is represented byjoining open ends Since the picture in (685) represents an overall intertwiner between

252 Quantum space

the representation j and the representation j it must be proportional to the identityin the representation j Namely

j

j

j 1 = c j

(687)

The coefficient c can be computed by closing both sides namely tracing the matricesThis gives

c =

j

j

1

j13

(688)

The theta-shaped diagram in the numerator has value unity because it is the norm ofan intertwiner while the denominator is the trace of the identity namely the dimensionof the representation Therefore

c =1

2j + 1 (689)

Putting everything together the action of E2 gives

E2(S) j sim 2 j(j + 1) j (690)

In the present case this result was obtained earlier in a simpler way But the idea canbe used to define a general method of computing in LQG which is very convenientThe general idea is that an operator such as (680) can be represented by the picture

αr s

(691)

where the dots represent the operator E that can grasp a link The result of a graspis the formation of a node and multiplication by the factor

radicj(j + 1) (2j + 1) More

precisely if we include also the numerical part the action of the grasp of a hand locatedat a point x over a link γ with spin j is

x

j

γ

= njΔa[γ x] j

γ

(692)

where

Δa[γ x] equivint

ds γa(s) δ3(γ(s) x) (693)

Calculations have mostly been done with a slightly different notation which derivesfrom [174] It is the KauffmanndashLins (KL) notation and is explained in Appendix A2


Tables of formulas exist in this notation Let us therefore now change to the KLnotation in this section In it one uses the ldquocolorrdquo p = 2j of a link which is twice itsspin4 and is integer and the trivalent nodes are not normalized to unity The relationis given in (A68) In the case of a vertex with spins j j 1 namely colors p p 2 as theone above the normalization factor of the node easily derived from the formulas ofAppendix A2 is

⎛

⎝

j

j

1

⎞

⎠

spin network

=

⎛

⎜⎜⎜⎜⎝

radicj

(j+1)(2j+1)

p = 2j

p = 2j

2

⎞

⎟⎟⎟⎟⎠

KL

(694)

and using (692) the action of the grasp operator in this notation is therefore

x

p

γ

= p Δa[γ x] γ

p (695)

In the next section I give an example of a full calculation using this grasp operator forcomputing the complete spectrum of the area operator

Paths with many hands The definition (680) can be generalized to paths with anarbitrary number of ldquohandsrdquo For instance let

T abc(x r s t) =1

3εijkR

(1)(U(A γxr))il Ea

l (r)

timesR(1)(U(A γxs))jm Eb

m(s)R(1)(U(A γxt))kn Ec

n(t) (696)

Given a closed surface S define the three-hand generalization of the operator (690) as

E3(S) =

int

Sd2σ

int

Sd2σprime

int

Sd2σprimeprime |na(σ) nb(σ

prime) nc(σprimeprime) T abc(x σ σprime σprimeprime)| (697)

where x is a point in the interior of S (whose exact position is irrelevant as we willalways consider the limit of small S) The absolute value in the definition is for laterconvenience This can be represented by the picture

(698)

As we will see in a moment this operator also plays an important physical role

4The expression ldquocolorrdquo is routinely used with two distinct meanings It indicates twicethe spin as here Or it may designate any label of links or nodes (or later edges orfaces of a spinfoam) as in ldquothe links of the spin network are colored with represen-tations and the nodes with intertwinersrdquo

254 Quantum space


The simplification that we took above in order to compute the spectrum of the areawas to assume that no node is on the surface Here we drop this assumption in orderto find the full spectrum If we drop this assumption the regularization of the areaoperator considered above is not sufficient because we obtain ill-defined expressions ofthe kind int 1

0

dx δ(x) = (699)

We need a better regularization of the operator To this end it is sufficient to smearthe operator transverse to the surface Introduce a smooth coordinate τ over a finiteneighborhood of S in such a way that S is given by τ = 0 Consider then the three-dimensional region around S defined by minusδ2 le τ le δ2 Partition this region into anumber of blocks D of coordinate height δ and square horizontal section of coordinateside ε For each fixed choice of ε and δ we label the blocks by an index I Later we willsend both δ and ε to zero In order to have a one-parameter sequence we now choose δas a fixed function of ε For technical reasons the height of the block D must decreasemore rapidly than ε in the limit thus we put δ = εk with any k greater than 1 andsmaller than 2

Consider one of the blocks The intersection of the block and a τ = constant surfaceis a square surface let AI(τ) be the area of such a surface Let AIε be the average overτ of the areas of the surfaces in the block namely

AIε equiv 1

δ

int δ2

minusδ2

AI(τ)dτ =1

δ

int

DI

d3xradic

EaiEbinanb (6100)

Summing over the blocks yields the average of the areas of the τ = constant surfacesand as ε (and therefore δ) approaches zero the sum converges to the area of the surfaceS Therefore we have

A(S) = limεrarr0

sum

I

AIε equiv limεrarr0

Aε(S) (6101)

The quantity AIε associated with each block can be expressed as follows Write

AIε =radic

A2Iε (6102)

and notice that

A2Iε =

1

δ2

int


ab(x y) + O(ε5) (6103)

Equation (6103) holds because of the following We have

T ab(x y) = Eai(xI)Ebi (xI) + O(ε) (6104)

for any three points x y and xI in D It follows that

ε4 na(xI)nb(xI)Eai(xI)E

bi (xI) =

1

2δ2

int


ab(x y) + O(ε5)

(6105)

Equation (6103) follows from

A2I =

(1

δ

int δ2

minusδ2

AI(τ)dτ

)2

=

(1

δ

int

Dd3xradic

EaiEbinanb

)2

= ε4 na(xI)nb(xI)Eai(xI)E

bi (xI) + O(ε5) (6106)


jd

ju

j tP

Fig 66 The three classes of links that meet at a node on the surface

Equations (6101) (6102) and (6103) define the regularization of the area The quan-tum operator A(S) is defined by (6101) where

A2Iε equiv 1

2δ2

int


ab(x y) (6107)

The action of A(S) on the quantum states is found from the action of the T ab

operators The operator T ab(x y) annihilates the state |S〉 unless its hands x and y fallon some links of the graph of S If this happens the action of the operator on the stategives the union of S and α with two additional nodes at the points x and y Moreprecisely if x and y fall over two edges of β with color p and q respectively using thegrasp operator (695) we have

T ab(x y)p

q = 2 p qΔa[β x] Δb[β y] 2

p

q x

y

(6108)

Since the loop α runs back and forth between the intersection points x and y (the twograsps) it has spin one or color 2

Consider now the action of the operator A(S) on a generic spin network state |S〉Due to the limiting procedure involved in its definition the operator A(S) does notaffect the graph of |S〉 Furthermore since the action of T ab inside a specific coordinateblock D vanishes unless the graph of the state intersects D the action of A(S) ulti-mately consists of a countable sum of terms one for each intersection P of the graphwith the surface

Consider an intersection P between the spin network and the surface For the purposeof this discussion we can consider a generic point on a link as a ldquobivalent noderdquo andthus say without loss of generality that P is a node In general there will be n linksemerging from P Some of these will emerge upward(u) some downward(d) and sometangential(t) to the surface S see Figure 66 Since we are taking the limit in which theblocks shrink to zero we may assume without loss of generality that the surface andthe links are linear around P (see below for subtleties concerning higher derivatives)Due to the two integrals in (6107) the positions of the two hands of the area operatorare integrated over each block As the action of T ab is nonvanishing only when bothhands fall on the spin network we obtain n2 terms one for every couple of graspedlinks Consider one of these terms in which the grasped links have color p and q Let uswrite the result of the action of T ab with a finite ε on the links p and q of an n-valent

256 Quantum space

intersection P (up to the prefactor) as

p

q

2P

ε (6109)

The irrelevant links are not shown The links labeled p and q are generic in the sensethat their angles with the surface do not need to be specified at this point (the two linksmay also be identical) From the definition (6101) and (6107) of the area operator andthe definition of the T ab operator each term in which the grasps run over two links ofcolor p and q is of the form

T =1

2δ2

int

DotimesDd3xd3y na(x)Δa[β x]nb(y)Δ

b[β y] p q p

q

2P

ε (6110)

giving

T =1

2δ2

int

DotimesD

(

na(x)

int

β

ds βa(s)δ3[β(s) x] (6111)

times nb(y)

int

β

dt βb(t)δ3[β(t) y] p q p

q

2P

ε

)

d3xd3y

=1

2δ2

int

β

ds na(s) βa(s)

int

β

dt nb(t) βb(t)p q p

q

2P

ε

=p q

2δ2

(int

β

ds na(s) βa(s)

int

β

dt nb(t) βb(t)

)

p

q

2P

+ O(ε)

In the last step I have pulled the state out of the integral This is possible because the

ε-dependent states p

q

2P

ε all have the same limit state as ε rarr 0 I write this limit

simply as p

q

2P

without ε that is

p

q

2P

ε = p

q

2P

+ O(ε) (6112)

Hence the substitution of the ε-dependent states with their limit in the integral ispossible up to terms of order O(ε) Note that

int

β

dt nb(t) βb(t) =

0 if β is tangent to Sδ2 otherwise

(6113)

This result is independent of the angle the link makes with the surface because δcan always be chosen sufficiently small so that β crosses the top and bottom of thecoordinate block D (This is the reason for requiring that δ goes to zero faster than ε)Also since we have chosen k smaller than 2 it follows that any link tangential to the


PeP

eP

eP

d

u

t

Fig 67 Trivalent expansion of an n-valent node The dashed lines indicate thelines tangent to the surface

surface exits the box from the side irrespective of its second (and higher) derivativesfor sufficiently small ε and gives a vanishing contribution as ε goes to zero Thereforein the limit the links tangent to the surface do not contribute to the action of the areawhereas every nonvanishing term takes the form

2 p q

8 p

q

2P

(6114)

Generically there will be several links above below and tangential to the surface SExpand the node P into a virtual trivalent spin network We choose to perform theexpansion in such a way that all links above the surface converge to a single ldquoprincipalrdquovirtual link eu all links below the surface converge to a single principal virtual linked and all links tangential to the surface converge to a single principal virtual link etThe three principal links join in the principal trivalent node This trivalent expansionis shown in Figure 67

This choice simplifies the calculation of the action of the area since the sum of thegrasps of one hand on all real links above the surface is equivalent to a single grasp oneu (and similarly for the links below the surface and ed) This follows from the identity

p p q

r

2 +q

p q

r

2 = r

p q

r

2 (6115)

which can be proven as follows Using the recoupling theorem (A65) the left-hand sideof (6115) can be written as

sum

j

(

p

2 p jq r p

minus q λ2r

j

r p jq 2 q

)

2

p

r

q

j (6116)

where j can take the values r minus 2 r and r + 2 A straightforward calculation using

258 Quantum space

(A61) gives

p

2 p jq r p

minus q λ2r

j

r p jq 2 q

= r δjr (6117)

and (6115) follows A repeated application of the identity (6115) allows us toslide all grasps from the real links down to the two virtual links eu and ed Thuseach intersection contributes as a single principal trivalent node regardless of itsvalence

We are now in a position to calculate the action of the area on a generic intersectionFrom the discussion above the only relevant terms are as follows

A2P

q

p

r =

2

8

(

p2

q

p

r

p

p 2

+ q2

q

p

r

q

q2

+ 2 p q

q

p

rp

q

2

)

(6118)

where the first term comes from grasps on the links above the surface the second fromgrasps on two links below the surface and the third from the terms in which one handgrasps a link above and the other grasps a link below the surface Each term in thesum is proportional to the original state (see (A63) (A64)) Therefore we have

A2P

q

p

r = minus l408

( p2 λu + q2 λd + 2 p q λt )

q

p

r (6119)

The quantities λu λd and λt are easily obtained from the recoupling theory Using theformulas in Appendix A2 we obtain

λu =θ(p p 2)

Δ(p)= minus (p + 2)

2p (6120)

λd is obtained by replacing p with q in (6120) λt has the value

λt =

Tet

[p p rq q 2

]

θ(p q r)=

minus2p(p + 2) minus 2q(q + 2) + 2r(r + 2))

8pq (6121)

(Tet is defined in (A60))Substituting in (6118) we have

A2P

q

p

r =

2

16( 2 p (p + 2) + 2 q (q + 2) minus r (r + 2) )

q

p

r (6122)


Since A2P is diagonal the square root can be easily taken

AP

q

p

r =radic

A2P

q

p

r

=

radicradicradicradic2

4

(

2p

2

(p2

+ 1)

+ 2q

2

( q2

+ 1)minus r

2

( r2

+ 1))

q

p

r (6123)

Adding over the intersections and getting back to the spin notation p2 = ju q2 = jd

and r2 = jt the final result is

A(S)|S〉 =

⎛

⎝

2

sum

PisinScapS

radic2ju

P (juP +1) + 2jd

P (jdP +1) minus jt

P (jtP +1)

⎞

⎠ |S〉 (6124)

This expression provides the complete spectrum of the area It reduces to the earlierresult (675) for the case jt

P = 0 and jdP = ju

P (for every P )The complete spectrum of A(S) is therefore labeled by n-tuplets of triplets of positive

half-integers ji namely ji = (jui j

di j

ti ) i = 1 n and n arbitrary It is given

restoring natural units and the Immirzi parameter as

Aji(S) =

4πGγ

c3

sum

i

radic2ju

i (jui + 1) + 2jd

i (jdi + 1) minus jt

i (jti + 1) (6125)

It contains the previous case (675) which corresponds to the choice jui = jd

i andjti = 0 The eigenvalues which are contained in (6125) but not in (675) are called the

degenerate sector


A second operator that plays a key role in the physical interpretationof the quantum states of the gravitational field is the operator V(R)corresponding to the volume of a region R As for the area operatorconstructed above this quantity requires a bit of work to be defined inthe quantum theory because of the care to be taken in the definition ofthe operator products involved in detE and the square root Consider athree-dimensional region R The volume of R is

V (R) =int

Rd3x

radic13

∣∣∣∣εabcεijkEaiEbjEck

∣∣∣∣ (6126)

To construct a regularized form of this expression consider the classicalquantity (696) In the limit in which r s and t converge to x we have

T abc(x s t r) rarr 2εijk Eai(x)Ebj(x)Eck(x) = 2 εabc detE(x)

(6127)

We can therefore use the 3-hand loop operator to regularize the volume

260 Quantum space

Fix an arbitrary chart of the 3-manifold and consider a small cubicregion RI of coordinate volume ε3 Let xI be an arbitrary but fixed pointin RI Since classical fields are smooth we have E(s) = E(xI) + O(ε) forevery s isin RI and Hα(s t) B

A = 1 BA + O(ε) for any s t isin RI and straight

segment α joining s and t Consider the quantity

WI =1

16ε6 3E3(partRI) (6128)

where E3 is defined in (697) Because of (6127) we have to lowest orderin ε

WI =1

8ε6 3

∣∣det(E(xI)

∣∣int

partRI

d2σ

int

partRI

d2τ

int

partRI

d2ρ∣∣na(σ)nb(τ)nc(ρ)εabc

∣∣

=∣∣detE(xI)

∣∣ (6129)

Thus WI is a nonlocal quantity that approximates the volume elementfor small ε Using the Riemann theorem as in the case of the area we canthen write the volume V(R) of the region R as follows For every ε wepartition R into cubes RIε of coordinate volume ε3 Then

V(R) = limεrarr0

Vε(R) (6130)

Vε(R) =sum

Iε

ε3W12Iε

(6131)

Volume operator Returning now to the quantum theory we have thenimmediately a definition of the volume operator as

V(R) = limεrarr0

Vε(R) (6132)

Vε(R) =sum

Iε

ε3W12Iε

(6133)

WIε =1

16ε6 3E3(partRI) (6134)

where these quantities now are operators Notice the crucial cancellationof the ε6 factor when inserting (6134) into (6133)

The meaning of the limit in (6132) needs to be specified The specification of thetopology in which the limit is taken is an integral part of the definition of the oper-ator As is usual for limits involved in the regularization of quantum field theoreticaloperators the limit cannot be taken in the Hilbert space topology where in general itdoes not exist The limit must be taken in a topology that ldquoremembersrdquo the topologyin which the corresponding classical limit (6130) is taken This is easy to do in thepresent context We say that a sequence of quantum states Ψn converges to the stateΨ if Ψn[A] converges to Ψ[A] for all smooth connections A We use the correspondingoperator topology On rarr O if OnΨ rarr OΨ for all Ψ in the domain

An important consequence of the use of this topology is that a sequence of cylindricalfunctions converges to a cylindrical function defined on the limit graph The graphs


Γn converge to Γ in the topology of the 3-manifold This fact allows us to separate thestudy of a limit into two steps First we study the graph of the limit state Secondwe can study what happens to the coloring of states in order to express the limitrepresentation in terms of the spin network basis

Let us now begin to compute the action of this operator on a spinnetwork state The three surface integrals on the surface of the cube andthe line integrals along the loops combine as in the case of the areato give three intersection numbers which select three intersection pointsbetween the spin network and the boundary of the cube At these threepoints which we denote as r s and t the small graph γστρ of the operatorgrasps the spin network

Notice that the integration domain of the (three) surface integrals is a six-dimensional space ndash the space of the possible positions of three points on the surfaceof a cube Let us denote this integration domain as D6 The absolute value in (6134)plays a crucial role here contributions from different points of D6 have to be taken astheir absolute value while contributions from the same point of D6 have to be summedalgebraically before taking the absolute value The position of each hand of the operatoris integrated over the surface and therefore each hand grasps each of the three pointsr s and t producing 33 distinct terms However because of the absolute value a termin which two hands grasp the same point say r vanishes This happens because theresult of the grasp is symmetric but the operator is antisymmetric in the two hands ndashas follows from the antisymmetry of the trace of three sigma matrices Thus only termsin which each hand grasps a distinct point give nonvanishing contributions For eachtriplet of points of intersection r s and t between spin network and cube surface thereare 3 ways in which the three hands can grasp the three points These 3 terms havealternating signs because of the antisymmetry of the operator but the absolute valueprevents the sum from vanishing and yields the same contribution for each of the 3terms

If there are only two intersection points between the boundary of thecube and the spin network then there are always two hands graspingat the same point contributions have to be summed before taking theabsolute value and thus they cancel Thus the sum in (6133) reducesto a sum over the cubes Iiε whose boundaries have at least three distinctintersections with the spin network and the surface integration reducesto a sum over the triple-grasps at distinct points For ε small enough theonly cubes whose surfaces have at least three intersections with the spinnetwork are cubes containing a node i of the spin network Therefore thesum over cubes reduces to a sum over the nodes n isin S capR of the spinnetwork contained inside R Let us denote by Inε the cube containing thenode n We then have


sum

nisinScapVε3radic

|WInε | |S〉

WInε |S〉 =1

16 ε6 3E3(partRI)|S〉 (6135)

262 Quantum space

The action of the operator E3(partRI) is the sum over the triplets (r s t) ofdistinct intersections between the spin network and the boundary of thecube For each such triplet let T (r s t)|S〉 be the result of this actionThen


sum

nisinScapVε3radic

|WInε | |S〉

WInε |S〉 =1

16 ε6 3

sum

rst

T (r s t) |S〉 (6136)

Next the key point now is that in the limit ε rarr 0 the operator does notchange the graph of the spin network state nor the coloring of the linksThe only possible action of the operator is therefore on the intertwinersTherefore

V(R) |Γ jl i1 iN 〉 = (16πG)32sum

nisinScapVVin

iprimen |Γ jl i1 iprimen iN 〉

(6137)The computation of the numerical matrices Vin

iprimen is an exercise in recou-pling theory For instance for a trivalent node we have to compute W in

j1 j2j3

1

1

113

13 = W j1

j2

j3

(6138)

and more complicated diagrams for higher-valence nodes The completecalculation is presented in great detail in [175] where a list of eigenvaluesis also given One of the interesting outcomes of the detailed calculationis that the node must be at least quadrivalent in order to have a non-vanishing volume

The operator can be shown to be a well-defined self-adjoint nonnegativeoperator with discrete spectrum For each given graph and labeling weshall choose from now on a basis in of intertwiner that diagonalizesthe matrices Vin and therefore the volume operator We denote Vin thecorresponding eigenvalues

67 Quantum geometry

Physical interpretation of the spin network states The essential propertyof the volume operator is that it has contribution only from the nodes of aspin network state |S〉 That is the volume of a region R is a sum of termsone for each node of S inside R Therefore each node of a spin networkrepresents a quantum of volume That is we can interpret a spin network


with N nodes as an ensemble of N quanta of volume or N ldquochunksrdquo ofspace located in the manifold ldquoaroundrdquo the node each with a quantizedvolume Vin

The elementary chunks of quantized volume are then separated fromeach other by surfaces The area of these surfaces is governed by the areaoperator The area operator A(S) has contribution from each link of Sthat crosses S Therefore the following interpretation follows Two chunksof space are contiguous if the corresponding nodes are connected by a linkl In this case there is an elementary surface separating them and thearea of this surface is determined by the color jl of the link l to be

Al = 8πcminus3Gradic

jl(jl + 1) (6139)

Therefore the intertwiners associated with the nodes are the quantumnumbers of the volume and the spins associated with the links are quan-tum numbers of the area Volume is on the nodes and area is on thelinks separating them The graph Γ of the spin network determines theadjacency relation between the chunks of space

In other words the graph Γ can be interpreted as the graph dual to acellular decomposition of physical space in which each cell is a quantumof volume

Thus a spin network state |S〉 determines a discrete quantized 3d metricThis physical picture is beautiful and compelling However its full beautyreveals itself only in going to the space of the diffeomorphism-invariantstates Kdiff

Physical interpretation of the s-knot states Consider an s-knot state |s〉For simplicity consider the generic case in which its symmetry group istrivial so that we can disregard the technicalities due to the diffeomor-phisms that change orientation and ordering Then we can view |s〉 as theprojection under Pdiff of a spin network state |S〉 In going from the spinnetwork state |S〉 to the s-knot state |s〉 we preserve the entire informationin |S〉 except for its localization on the 3d space manifold This is preciselyas the implementation of diffeomorphism invariance in the classical theorywhere a physical geometry is an equivalence class of metrics under diffeo-morphisms In the quantum case |s〉 retains the information about thevolume and the adjacency of the chunks of volumes and about the areaof the surfaces that separate these volumes But any information of thelocalization of the chunks of volume on the 3d manifold is lost under Pdiff

The physical interpretation of the resulting state |s〉 is therefore ex-tremely compelling it represents a discrete quantized geometry This isformed by abstract chunks of space which do not live on the 3d manifold

264 Quantum space


they are only localized with respect to one another Their spatial relationis only determined by the adjacency defined by the links see Figure 68These are not quantum excitations in space they are quantum excitationsof space itself Volume of the chunks and area of the surfaces are givenby the coloring of the s-knot The spins jl are the quantum numbers ofthe area and the intertwiners in are the quantum numbers of the volume

These are quantum states defined in a completely 3d diffeomorphism-invariant manner and with a simple physical interpretation These are thequantum states of space

Surfaces and regions on s-knots Recall that in classical GR we distin-guish between a metric g and a geometry [g] A geometry is an equiv-alence class of metrics under diffeomorphism For instance in three di-mensions the euclidean metric gab(x) = δab and a flat metric gprimeab(x) = δabare different metrics but define the same geometry [g] = [gprime] The no-tion of geometry is diffeomorphism invariant while the notion of metricis not On a given manifold with coordinates x we can define a surfaceby S = (σ1 σ2) rarr xa(σi) Then it makes sense to ask what is the areaof S in a given metric gab(x) but it makes no sense to ask what is thearea of S in a given geometry because the relative location of S and thegeometry is not defined

However given a geometry it is meaningful to define surfaces on thegeometry itself For instance (in 2d) given the geometry of the surface ofthe Earth (an ellipsoid) the equator is a well-defined (1d) surface and


Fig 69 Regions and surfaces defined on an s-knot the set of the thick blacknodes define a ldquoregionrdquo of space the set of the thick black links define theldquosurfacerdquo surrounding this region

so is the parallel 1 km north of the equator their location is determinedwith respect to the geometry itself Concretely a surface on a geometrycan be defined in various manners For instance it can be defined by thecouple (S g) with g isin [g] The couple (φS φlowastg) defines the same surfaceAlternatively the surface can be defined intrinsically (the equator is thelongest geodesic)

Now in quantum gravity we find precisely the same situation Abovewe have defined coordinate surfaces S and regions R and their areasand volumes Such coordinate surfaces and regions are not defined atthe diffeomorphism-invariant level However we can nevertheless definesurfaces and regions on the abstract quantum state |s〉 itself and associateareas and volumes with them A region R is simply a collection of nodesIts boundary is an ensemble of links and defines a surface we can say thatthis surface ldquocutsrdquo these links A moment of reflection will convince thereader that this is precisely the same situation as in the classical theory

For instance consider an s-knot state with two four-valent nodes andfour links with spins 12 12 1 1 On this quantum geometry we canidentify a closed surface separating the two quanta of volume This surfacecuts the four links and has area A = (8

radic3 + 16

radic2)πGcminus3 A more

complex situation is illustrated in Figure 69

Eigenvalues and measurements Suppose we had the technological capa-bility to measure the area of a surface or the volume of a region withPlanck-scale precision An example of an area measurement for instanceis the measurement of the cross section of an interaction Shall we obtainone of the eigenvalues computed in this section If the theory developed

266 Quantum space

so far is physically correct the answer is yes In fact area and volume arepartial observables Partial observables can be measured and the theorypredicts that the possible outcomes of a measurement are the numbers inthe spectrum of the corresponding operator

Therefore these spectra are precise quantitative physical predictions ofLQG

This prediction has raised a certain discussion The objection has been made thatin the classical theory area and volume of coordinate surfaces and regions are notdiffeomorphism-invariant quantities and therefore we cannot interpret them as trueobservables The objection is not correct it is generated by the obscurity of the con-strained treatment of diffeomorphism-invariant systems To clarify this point considerthe following simple example Consider a particle moving on a circle subject to a forceLet φ be the angular coordinate giving the position of the particle and pφ its conju-gate momentum As we know well pφ turns out to be quantized Now if we write thecovariant formulation of this system we have the WheelerndashDeWitt equation

Hψ(t φ) =

(i

part

parttminus

2 part2

partφ2+ V (φ)

)ψ(t φ) = 0 (6140)

which in the language of constrained systems theory is the hamiltonian constraintequation Notice that the momentum pφ is not a gauge-invariant quantity it doesnot commute with the operator H that is [pφ H] = 0 This happens precisely forthe same reason for which area and volume are not gauge-invariant quantities in GRBut this does not affect the simple fact that we can measure pφ and we do predictthat it is quantized The confusion originates from the distinction between completeobservables and partial observables which was explained in detail in Chapter 3 pφ isnot a complete observable but it is nevertheless a partial observable We cannot predictthe physical value of pφ from a physical state namely from a solution of the WheelerndashDeWitt equation because we do not know at which time this is to be measured Butwe can nevertheless compute and predict its eigenvalues

Alternatively we can define the evolving constant of the motion pφ(T ) as in Section515 This is a gauge-invariant quantity The spectral properties of pφ(T ) howeverare the same as those of pφ More importantly they are not affected by the potentialV (φ) namely they are not affected by the dynamics Similarly we could in principle usegauge-invariant definitions of areas of surfaces that would be genuinely diffeomorphisminvariant but this complicated exercise is useless because the spectral properties canbe directly determined by the partial observable operators

Why Diff It is now time to address the question of the choice of the precisefunctional space of coordinate transformations or active gauge maps φ M rarr M and to justify the fact that I have chosen Diff lowast instead of Diff Notice that in theclassical theory the precise functional space in which we choose the fields is dictated bymathematical convenience not by the physics In fact we always make measurementssmeared in spacetime which cannot be directly sensitive to what happens at pointsIndeed in classical field theory we choose to change freely the class of functions whenconvenient For instance twice differentiable fields allow us to write the equations ofmotion more easily but then we prefer to work with distributional fields in certainapplications Analytic fields are usually considered too rigid because we do not liketoo much the idea that a field in a small finite neighborhood could uniquely determinethe field everywhere The smooth category (Cinfin) is often easy and convenient and it


has been generally taken as the natural point of departure in quantum gravity but itis not God-given If we use smooth fields it is natural to consider smooth coordinatetransformations and Diff as the gauge group In fact this was the traditional choicein quantum gravity However at the end of Section 64 I pointed out that if we chooseDiff as the gauge group then the knot classes are labeled by continuous parameters(moduli) and the space Kdiff turns out to be nonseparable At first we may thinkthat these moduli represent physical degrees of freedom If they did there would beobservable quantities that are affected by them However none of the operators that wehave constructed in this chapter is sensitive to these moduli In particular a momentof reflection shows that all the geometric operators are only sensitive to features ofgraphs (and surfaces and volumes) that are invariant under continuous deformationsof the graphs (surfaces and regions) Therefore it is possible that these moduli are anartifact of the mathematics they have nothing to do with the physics They just reflectthe fact that we have not chosen the functional space of the maps φ appropriately Theφ in Diff are too ldquorigidrdquo in the sense that they leave invariant the linear structure ofthe tangent space at a node while this linear structure has no physical significance Thechoice of Diff lowast as gauge group is a simple extension of the gauge group that gets rid ofthe redundant parameters Accordingly we have to work with a space of fields slightlylarger that Cinfin Nothing changes in the classical theory while the quantum theory iscured of the double problem of having a nonseparable Hilbert space and redundantphysically meaningless moduli Of course choices other than Diff lowast are possible

Noncommutativity of the geometry I close this section with an observa-tion Consider a spin network state containing a four-valent node n Letl1 l2 l3 l4 be the four links adjacent to the node n Let i be the intertwineron this node Consider a surface S(12)(34) such that n is on S(12)(34) thelinks l1 and l2 are on one side of the surface while the links l3 and l4 are onthe other side of it We can choose a basis in the space of the intertwinersby splitting the node n into two trivalent nodes joined by a virtual link l(see Appendix A1) Let us do so by pairing the links as (l1 l2) and (l3 l4)That is the two trivalent nodes are between (l1 l2 l) and (l l3 l4) A basisin the space of intertwiners is then given by

vα1 α2 α3 α4j = vα1 α2 αjvαj α3 α4 (6141)

where the indices αi are in the representations of the links and the index αj

is in the representation j It is not hard to show that in order for the stateto be an eigenstate of the area of Sa the intertwiner must be one of thesebasis elements In other words the basis (6141) diagonalizes the areaof Sa Now consider a surface S(13)(24) such that n is on it and the linksl1 and l3 are on one side of the surface while the links l2 and l4 are onthe other side of it Clearly in this case it will be a different basis in thespace of the intertwiners that diagonalizes the area It will be the basis

wα1 α2 α3 α4k = vα1 α3 αkvαk α2 α4 (6142)

The two bases are related by a 6j symbol In general they are different Itfollows that the operator A(S(12)(34)) and the operator A(S(13)(24)) do

268 Quantum space

not commute (if they did they would be diagonalized by the same basis)Therefore the 3d geometry is in a sense noncommutative area operatorsof intersecting surfaces do not commute with each other


What is the connection between the discrete and quantized geometrydescribed above and the smooth structure of physical geometry that weperceive around us The answer requires a few steps

Weaves Ordinary measurements of geometric quantities ndash that is mea-surements of the gravitational field ndash are macroscopic we observe thegeometry of space at a scale l much larger than the Planck length lP Atthis large scale planckian discreteness is smoothed out

Consider the fabric of a T-shirt as an analogy At a distance it is asmooth curved two-dimensional geometric surface At a closer look it iscomposed of thousands of one-dimensional linked threads The image ofspace given by LQG is similar Consider a very large spin network formedby a very large number of nodes and links each of Planck scale Micro-scopically it is a planckian-size lattice But probed at a macroscopic scaleit appears as a three-dimensional continuous metric geometry Physicalspace around us can therefore be described as a very fine weave Thehidden texture of reality is a weave of spins

This intuitive picture can be made precise Fix a classical macro-scopic 3d gravitational field e which determines a macroscopic 3d metricgab(x) = eia(x) eib(x) It is possible to construct a spin network state |S〉that approximates this metric at a scale l lP The precise relationbetween |S〉 and e is the following Consider a region R (or a surface S)with a size larger than l (in the metric g) and slowly varying at this scaleRequire that |S〉 is an eigenstate of the volume operator V(R) (and ofthe area operator A(S)) with eigenvalues equal to the volume of R (andof the area of S) determined by e up to small corrections in lPl That is

V(R)|S〉 =(V[eR] + O(lPl)

)|S〉

A(S)|S〉 =(A[eS] + O(lPl)

)|S〉 (6143)

where V[eR] (resp A[eS]) given in (273) (and (270)) is the volume(the area) of the region (the surface) determined by the gravitationalfield e

A spin network state |S〉 that satisfies these equations for any largeregion and surface is called a ldquoweaverdquo state of the metric g At largescale the state |S〉 determines precisely the same volumes and areas as g


This definition is given at the nondiff-invariant level but it can be easilycarried over to the diff-invariant level the s-knot state |s〉 = Pdiff |S〉 iscalled the weave state of the 3-geometry [g] the equivalence class of 3-metrics to which the metric g belongs

Several weave states were constructed and studied in the early daysof LQG for various 3d metrics including the ones of flat spaceSchwarzschild and gravitational waves They satisfied (6143) or equa-tions similar to these (at the time the area and volume operators werenot known and other operator functions of the gravitational field wereused to play the same role) Most of these weave states were constructedbefore the discovery of the spin network basis working with the morecumbersome loop basis Equations (6143) do not determine a weave stateuniquely from a given 3-metric There is a large freedom in constructinga weave state for a given metric because only the averaged properties areconstrained by (6143) The weave states constructed should not be takenas realistic proposals for the microstates of a given macroscopic geome-try They are only a proof of existence of microstates that have specifiedmacroscopic properties

On the other hand the weave states have played a very important rolein the historical development of the LQG I recall this role below becauseit contains an important physical lesson on the physics of Planck-scalediscreteness

The failure of the a rarr 0 limit and the emergence of Planck-scale dis-creteness There is a gap of several years between the construction of theloop representation of quantum GR (c 1988) and the calculation of theeigenvalues of area and volume (c 1995) which revealed that the theorypredicts a discrete structure of space During these years the fact that theloops have ldquoPlanck sizerdquo was not known and at first not even suspectedThe intuition was that a macroscopic geometry could be constructed bytaking a limit of an infinitely dense lattice of loops ndash roughly as a con-ventional QFT can be defined by taking the limit of a lattice theory asthe lattice size a goes to zero To construct a weave state approximatinga classical metric therefore the aim was at first to satisfy equations like(6143) by quantum states defined as limits where the spatial density ofthe loops was taken to infinity But something unexpected and very re-markable happened With increasing density of loops the accuracy of theapproximation did not increase Instead the eigenvalue of the operatorincreased

Let me be more precise Suppose we start with a 3d manifold withcoordinates x We want to define a weave on this manifold that approxi-mates the flat 3d metric g(0)

ab(x) = δab that is the field e(0)ia(x) = δia

270 Quantum space

We construct a spatially uniform weave state |Sa0〉 formed by a tangle ofloops of coordinate density ρ = aminus2

0 (The coordinate density ρ can bedefined as the ratio between the total coordinate length L of the loopsand the total coordinate volume V ) The loops are then at an average dis-tance a0 from each other Therefore one expects that the approximation(6143) would break down at the scale l sim a0 The idea was therefore toimprove the approximation by decreasing the ldquolattice spacingrdquo a0 namelyby increasing the coordinate density of the loops But decreasing a0 tobecome a lt a0 the calculations yielded instead

A(S) |Sa〉 sima2

0

a2

(A[e(0)S] + O(lPl)

)|S〉 (6144)

instead of a decrease in the error the area increases In other wordsby adding loops we do not obtain a better approximation Rather weapproximate a different field Since

a20

a2A[e0S] = A[(a0a) e(0)S] = A[e(a)S] (6145)

where e(a)ia(x) = a2

0a2 δai the weave with increased loop density approxi-

mates the metric

g(a)ab (x) =

a20

a2δab (6146)

But notice that the physical density of the loops ρa does not changewith decreasing a The physical density ρa is the ratio between the totallength of the loops and the total volume determined by the metric g(a)namely by the metric that the state |Sa〉 itself determines via equation(6143) This is

ρa =La

Va=

(a0a)L(a0a)3V

=a2

a20

ρ =a2

a20

aminus2 = aminus20 (6147)

The physical density remains aminus20 irrespective of the density of the loops

a chosen But then if a0 is not determined by the density of the loopsit must be given by a dimensional constant of the theory itself and sincethe only scale in the theory is the Planck scale we have necessarily thatup to numerical factors

a0 sim lP (6148)

At first this result was disconcerting The theory refused to approximatea smooth geometry at a physical scale lower than lP Then the reasonbecame clear there is no physical scale lower than lP Each loop carries


a quantum of geometry of Planck size more loops give more size not abetter approximation to a given geometry This was the first unexpectedhint that the loops themselves have an intrinsic geometric size and thatin the theory there is no spatial structure at physical scales smaller thanthe Planck scale

Quantum and classical discreteness superposition of weaves The weavepicture of space resembles the space of a lattice theory But there is astrong difference between the two and the analogy should be taken withgreat care

Planck-scale discreteness is predicted by loop quantum gravity on thebasis of a standard quantization procedure in the same manner in whichthe quantization of the energy levels of an atom is predicted by nonrela-tivistic quantum mechanics while the discretization of space in a latticetheory is assumed

But the difference is far more substantial than this In a lattice theorythe lattice is a fixed structure on which the theory is defined A weaveon the other hand is one of many quantum states that have a certainmacroscopic property and a very peculiar one since it is a single elementof the spin network basis There is no reason for the physical state of spacenot to be in a generic state and the generic quantum state that has thismacroscopic property is not a weave state it is a quantum superpositionof weave states Therefore it is reasonable to expect that at small scalespace is a quantum superposition of weave states

Therefore the picture of physical space suggested by LQG is not trulythat of a small-scale lattice Rather it is a quantum probabilistic cloudof such lattices

The Minkowski vacuum To a certain approximation macroscopic spacearound us is described by the Minkowski metric We should thereforeexpect that in the quantum theory there is a state |0M〉 (M for Minkowski)that reproduces the Minkowski metric at large scales

At fixed time Minkowski 3d space is described by a flat 3-metric gnamely by the gravitational field eia(x) = δia Does this imply that weshould expect |0M〉 to be a weave state of e It does not In quantumMaxwell theory the state that corresponds to the classical solution withvanishing electric and magnetic fields E = B = 0 is the vacuum state|0〉 But |0〉 is not an eigenstate of the electric field E Since E and Bdo not commute E and B have no common eigenstates Eigenstates ofE are maximally spread in B Instead E and B have vanishing meanvalue and minimal spread on |0〉 The situation is precisely the same asfor the vacuum state of an harmonic oscillator The classical solutionx(t) = p(t) = 0 corresponds to the vacuum state a state which is neither

272 Quantum space

an eigenstate of the position x nor an eigenstate of the momentum p Aneigenstate of x has Δp very large and would spread instantaneously

Similarly |0M〉 cannot be an eigenstate of the gravitational field EAn eigenstate of E has maximum spread in the gravitational magneticfield and would spread instantaneously Accordingly |0M〉 is not a weavestate of the flat geometry nor a superposition of weave states of the flatgeometry

|0M〉 must be a state with vanishing mean value and minimal spreadof the gravitational electric and magnetic fields On the other hand itshould be concentrated around weave states in the same sense in whichthe vacuum of the harmonic oscillator is concentrated around x = 0

We do not yet know the form of the state |0M〉 explicitly In fact theexploration of this functional is one of the main open problems in thetheory I will come back to this problem in Chapter 9

We can obtain some hints about the state |0M〉 from the classical fieldtheory obtained linearizing GR around the Minkowski solution In theclassical theory let us write

eIμ(x) = δIμ + hIμ(x) (6149)

and restrict our considerations to solutions of the Einstein equationswhere hIμ(x) 1 To first order in h these solutions satisfy the lin-earized Einstein equations The linearized Einstein equations are free waveequations on Minkowski space for a spin-2 field The solutions are super-positions of plane waves They can be gauge-fixed fixing h0

μ = hI0 = 0and restricting hia(x) to the sole transverse traceless components that ispartah

ia = partih

ia = haa = 0 For each momentum k there are then two inde-

pendent polarizations εplusmn In this gauge the linearized Einstein equationsdescribe simply a collection of uncoupled harmonic oscillators of frequencyω(k) = |k| =

radickaka one for each polarization and for each Fourier mode

hia(k) = (2π)minus32

intdx eikx hia(x) (6150)

The quantum field theory of this free system is a fully conventionalfree QFT In its vacuum state |0lin〉 (lin for linearized) all oscillators arein their ground state The Hilbert state of the theory is the Fock spacespanned by the states

|k1 ε1 kn εn〉 (6151)

containing n quanta with momenta and polarizations (ki εi) Thesequanta are called gravitons hia(x) is a conventional field operator on thisFock space (formed by creation and annihilation parts) For each (gauge-fixed) field configuration h we can write the (generalized) eigenstates


|h〉 These allow us to write a generic state |ψ〉 of the Fock space in theSchrodinger representation

Ψ[h] = 〈h|ψ〉 (6152)

In particular the vacuum state is given in this representation by

Ψ0[h] = 〈h|0lin〉 = Neminus12

intdk |k| ha

i (minusk) hia(k) (6153)

This is a gaussian functional concentrated around the field configurationh = 0 We can rewrite this state as a functional of the gravitational fieldas

Ψ0[e] = 〈eminus δ|0lin〉 = Neminus12


which is a functional concentrated around g = δIt is then reasonable to suspect that |0M〉 should satisfy

ΨM[Se] equiv 〈Se|0M〉 sim Ψ0[e] = Neminus12


for all spin network states Se that are weaves for the field e This is astate concentrated around the flat weave S0

The ldquoemptyrdquo state The Minkowski vacuum state |0M〉 should not be con-fused with the covariant vacuum state |0〉 and with the empty state |empty〉(see Section 542) The state

Ψempty[A] = 〈A|empty〉 = 1 (6156)

is an eigenstate of A(S) and V(R) with vanishing eigenvalue Thereforeit describes a space with no volume and no area Spin network statescan be constructed acting on |empty〉 with the holonomy operator In thisparticular sense |empty〉 is analogous to the Fock vacuum The state |empty〉 isgauge invariant and diff invariant hence this state is also in K0 and inKdiff In fact it represents the quantum state of the gravitational field inwhich there is no physical space at all As we shall see in the next chapter|empty〉 is a solution of the WheelerndashDeWitt equation and therefore it is in Has well

mdashmdash


The ldquoloop representation of quantum general relativityrdquo was introducedin [176 177] These papers present the first surprising results of the ap-proach solutions of the WheelerndashDeWitt equation and general solutions

274 Quantum space

of the diffeomorphism constraint The loop transform mapping betweenfunctionals of the connection and loop functionals was illustrated in [170]The approach was motivated by Ted Jacobsonrsquos and Lee Smolinrsquos discov-ery [178] of loop solutions of the WheelerndashDeWitt equation written inAshtekar variables Rodolfo Gambini and his collaborators have indepen-dently developed a formal loop quantization for YangndashMills theory [179]The importance for the theory of the nodes ndash where loops intersect ndash wasstressed by Jorge Pullin nodes were studied in [180] An account of thisfirst stage of LQG can be found in the review [2]

The importance for the theory of the graphs was understood by JerzyLewandowski The spin network basis was introduced in quantum grav-ity in [171] solving the longstanding difficulty of the overcompleteness ofthe loop basis The inspiration came from Penrosersquos speculations on thecombinatorial structure of space [181] For motivations and the history ofthe idea of spin networks see [182] The mathematical systematization ofthis idea is due to John Baez [183] The fact that the equivalence classesunder Diff of graphs with intersections are not discrete and the associatedproblem of the nonseparability of the state space was pointed out in [171]and studied in [184] where the structure of the corresponding modulispaces is analyzed The solution of the problem in terms of Diff lowast is dis-cussed in [185] A different approach to obtain a separable Hilbert space isin [186] The uniqueness theorem for the loop representation in quantumgravity (the ldquoLOSTrdquo theorem) has been proven by Jerzy LewandowskiAndrzej Okolow Hanno Sahlmann and Thomas Thiemann [187] and ina slightly different version by Christian Fleischhack [188]

The idea that LQG could predict a discrete Planck-scale geometryemerged in studying the weave states [189] The first explicit claim thatthe eigenvalues of the area represent a physical prediction of the theoryobservable in principle is in [190] The definition of the area and vol-ume operators and the first calculations of their discrete eigenvalues arein [191] The main sequence of the spectrum of the area was calculatedin this work The degenerate sector was computed in [173] and in [192]which I have followed here A calculation mistake in the spectrum of thevolume in the first version of [191] was soon found by Renate Loll [193]in developing the lattice version of the volume operator Loll noticed thatthe node must be at least quadrivalent in order to have a nonvanishingvolume There exist a number of equivalent constructions of the volumeoperator in the literature they all define the same operator except forone possible variant A systematic study of the variants in the definition ofthe volume operator has been completed by Jerzy Lewandowski in [194]Systematic techniques to compute geometry eigenvalues were developedin [175] using the mathematics of [174] For more details (and another


application) see also [195] A length operator (which is surprisingly farmore complicated than area and volume to deal with) is studied byThiemann in [196] The noncommutativity of the areas of intersectingsurfaces has been studied in detail in [197] On angle operators see [198]An applealing and well written introduction to spin networks and theirgeometry is Seth Majorrsquos [199]

The mathematical-physics version of LQG started from the seminalwork of Abhay Ashtekar Chris Isham and Jerzy Lewandowski [200 201]See Ashtekarrsquos 1992 Les Houches lectures [202] and John Baez [203] Thisdirection led to the paper [204] where the loop representation of [176 177]was mathematically systematized the relation between the different vari-ants of the formalisms is still very confused in [204] and was elucidatedby Roberto De Pietri [205] and Thomas Thiemann For a detailed intro-duction and full references see [9] The relation between the full theoryand the linearized theory has been explored in [206] The notion of weavestate was introduced in [189] for a recent discussion and references see[207]

7Dynamics and matter

In the previous chapter I constructed the Hilbert space and the basic operators ofLQG In this chapter I discuss the dynamical aspects of the theory For this we mustwrite a well-defined version of the WheelerndashDeWitt equation (63) in order to constructthe hamiltonian operator H

The construction of the hamiltonian operator of LQG has taken a long time pro-ceeding through a number of steps The first was to realize that the operator can bedefined via a simple regularization and that any loop state Ψα solves HΨα = 0 pro-vided that α is a loop without self-intersections This observation opened the door toLQG The result remains true in all subsequent definitions of H With this first simpleregularization however H diverges on intersections

The second step was to realize that a finite operator H could be obtained providedthat (i) its action is defined on diffeomorphism-invariant states and (ii) its densitycharacter is appropriately dealt with By a sort of magic the action of the operatorwith the correct density weight on a diff-invariant state converges trivially in the limitin which the cut-off is removed This result is the major pay-off of the background-independent approach to QFT It is a manifestation of the relation between backgroundindependence and the absence of UV divergences

The third step was the idea of writing H as a commutator a technique that allowsus to write the operator avoiding square roots and inversion of matrices This techniquesolved at once the remaining roadblocks on the one hand it made it possible to usethe real Barbero connection thus avoiding the difficulties of implementing nontrivialreality conditions in the quantum theory on the other hand it made it possible tocircumvent the difficulties due to the nonpolynomiality of H such as the square rootpreviously used to get a density-weight-one hamiltonian

In this chapter I do not follow the contorted historical path but rather define theoperator directly I discuss only the operator corresponding to the first term of thehamiltonian (443) that defines lorentzian GR with a real connection This first termalone defines euclidean quantum GR The technique described here extends directly tothe second term for which I refer the reader to [20]

Following common GR parlance I call ldquomatterrdquo anything which is not the gravita-tional field At best as we know the content of the Universe is the one described byGR and the standard model fermions YangndashMills fields gravitational field and pre-sumably Higgs scalars As explained in Chapter 1 LQG has no ambition of providinga naturally ldquounifiedrdquo theory or explaining the reasons of the content of the Universe

276


In this chapter I assume that the four entities noted above make up the Universe andI describe how the background-independent quantum theory of the gravitational fielddescribed thus far can be very naturally extended to a background-independent theoryfor all these fields

The remarkable aspect of this extension is that the finiteness of the gravitationaldynamics extends to matter In the theory there is ldquono spacerdquo for UV divergencesneither for gravity nor for matter


Regularization The form of the hamiltonian H which is most convenientfor the quantum theory is the one given in (416) namely

H =int

N tr(F and V A) (71)

The reason is that we have already defined the quantum operator V andthe operators F and A can be defined as limits of holonomy operators ofsmall paths while the classical Poisson bracket can readily be realized inthe quantum theory as a quantum commutator

Fix a point x and a tangent vector u at x consider a path γxu ofcoordinate length ε that starts at x tangent to u Then the holonomy canbe expanded as

U(A γxu) = 1 + ε ua Aa(x) + O(ε2) (72)

Similarly fix a point x and two tangent vectors u and v at x and considera small triangular loop αxuv with one vertex at x and two sides tangentto u and v at x each of length ε Then

U(Aαxuv) = 1 +12ε2 uavb Fab(x) + O(ε3) (73)

Using this and writing hγ = U(A γ) we can regularize the expression ofthe hamiltonian by writing it as

H = limεrarr0

1ε3

intNεijktr

(hγminus1

xukhαxuiuj

V hγxuk)

d3x (74)

Here (u1 u2 u3) are any three tangent vectors at x whose triple productis equal to unity Following the same strategy we used for the area andvolume operators let us partition the 3d coordinate space in small regionsRm of coordinate volume ε3 We can then write the integral as a Riemannsum and write

H = limεrarr0

1ε3

sum

m

ε3Nmεijktr(hγminus1

xmukhαxmuiuj

V(Rm) hγxmuk) (75)


where xm is now an arbitrary point in Rm and Nm = N(xm) The factthat the limit is independent of the choice of this point is assured by theRiemann theorem Notice that the ε3 factors cancel and can therefore bedropped

Definition of the operator Since V(Rm) and hγ are well-defined operatorsin K we can then consider the corresponding quantum operator

H = minus i

limεrarr0

sum

m

Nm εijk tr(hγminus1

xmukhαxmuiuj

[V(Rm) hγxmuk]) (76)

To complete the definition of the operator we have yet to choose the pointxm the three vectors (u1 u2 u3) and the paths γxuk

and αxuiuj in eachregion Rm This must be done in such a way that the resulting quantumoperator is well defined covariant under diffeomorphism invariant underinternal gauges and nontrivial These requirements are highly nontrivialRemarkably there is a choice that satisfies all of them

The key observation to find it is the following When acting on a spinnetwork state this operator acts only on the nodes of the spin networkbecause of the presence of the volume (This is not changed by the pres-ence of the term hγxu in the commutator for the following reason Thevolume operator vanishes on trivalent nodes The operator hγxu can atmost increase the valence of a node by one Therefore there must be atleast a trivalent node in the state for H not to vanish)

Therefore in the sum (76) only the regions Rm in which there is anode n give a nonvanishing contribution Call Rn the region in which thenode n of a spin network S is located Then

H|S〉 = limεrarr0

Hε|S〉 (77)

where

Hε|S〉 = minus i

sum

nisinSNnε

ijk tr(hγminus1

xnukhαxnuiuj

[V(Rn) hγxnuk])|S〉 (78)

The sum is now on the nodes Now the only possibility to have a nontrivialcommutator is if the path γxnuk

itself touches the node We thereforedemand this This can be obtained by requiring that xn is precisely thelocation of the node Recall that the precise location of xn is irrelevant inthe classical theory because of the Riemann theorem but not so in thequantum theory This fixes xn

Finally there is a natural choice of the three vectors (u1 u2 u3) and forthe paths γxuk

and αxuiuj take (u1 u2 u3) tangent to three links l lprime lprimeprime

emerging from the node n (The condition that their triple product is


l

lprime

lprimeprime

γαxlprimelprimeprime

xl

x

Fig 71 The path γxl and the loop αxlprimelprimeprime at a trivalent node in the point x

unity can be satisfied by adjusting the length) Take γxukto be a path

γxl of coordinate length ε along the link l Take αxuiuj to be the triangleαxlprimelprimeprime formed by two sides of coordinate length ε along the other two linkslprime and lprimeprime and take the third side as a straight line (in the coordinates x)connecting the two end points This straight line is called an ldquoarcrdquo Thesum over i j k is a sum over all permutations of the three links see Figure71 If the node has valence higher than three that is if there are morethan three links at the node n we preserve covariance summing over allordered triplets of distinct links Thus we pose

Hε|S〉 = minus i

sum

nisinSNn

sum


(hγminus1


])|S〉

(79)

where εllprimelprimeprime is the parity of the permutation (determined by the sign of

the triple product) This completes the definition of the hamiltonian con-straint (up to one additional detail that we add below) Two major ques-tions are left open First whether the limit is finite Second whetherit is well behaved under diffeomorphisms and gauge transformations Inparticular whether it is independent of the coordinates chosen The twoquestions are intimately connected

A side remark There is a simple intuitive way of understanding why the hamiltonianacts only on nodes Consider the naive (divergent) form (63) of the WheelerndashDeWittoperator and act with this on a spin network state at a point where there are no nodesEquation (659) shows that the result of this action (disregarding the divergence) isproportional to

F ijab (τ)

δ

δAia(τ)

δ

δAjb(τ)

ΨS [A] sim F ijab(τ) γaγb (710)

where γa is the tangent to the spin network links at the point τ where the operatoracts But this expression vanishes because Fab is antisymmetric in ab while γaγb is


symmetric On the other hand acting on a node the two functional derivatives givemixed terms of the kind Fabγ

a1 γ

b2 where γa

1 and γa2 are the tangents of two different

links emerging from the node These terms may be nonvanishing Hence the operatorhas a nontrivial action only on nodes

711 Finiteness

In general the limit (77) does not exist This is no surprise as operatorproducts are generally ill defined in quantum field theory they can bedefined in regularized form but then a divergence develops when removingthe regulator ε The big surprise however is that the limit (77) does existon a subclass of states diffeomorphism-invariant states As these are thephysical states this is precisely what we need and is sufficient to definethe theory Here is where the intimate interplay between diffeomorphisminvariance and quantum field theoretical short-scale behavior begins toshine we are here at the core of diffeomorphism-invariant QFT

To compute H on diff-invariant states recall these are in the dual spaceS prime So far we have only considered H on spin network states or bylinearity on S The action of H on S prime is immediately defined by duality

(HΦ)(Ψ) equiv (Φ)(HΨ) (711)

(To be precise I should call Hdagger the operator in the left-hand side but forsimplicity I do not) Equivalently for every spin network state

(HΦ)(|S〉) = Φ(H|S〉) (712)

The key point is that we want to consider the regularized operator on S prime

and take the limit there Thus instead of simply inserting (77) in thelast equation and writing

(HΦ)(|S〉) = Φ(limεrarr0

Hε|S〉) (713)

I define the hamiltonian operator on S prime by


Φ(Hε|S〉) (714)

Notice that the limit is now a limit of a sequence of numbers (not a limitof a sequence of Hilbert space vectors) I now show that the limit exists(namely is finite) if Φ isin Kdiff namely if Φ is a diffeomorphism-invariantstate

The key to see this is the following crucial observation Given a spinnetwork S the operator in the parentheses modifies the state |S〉 in twoways by changing its graph Γ as well as its coloring The volume operatordoes not change the graph The graph is modified by the two operators


hγxnland hαxnllprime The first superimposes a path of length ε to the link l

of Γ The second superimposes a triangle with two sides of length propor-tional to ε along the links lprime and lprimeprime of Γ and a third side that is not onΓ as in Figure 71 The fundamental observation is that for ε sufficientlysmall changing ε in the operator changes the resulting state but not itsdiffeomorphism equivalence class (The maximal εm for this to happenis the value of ε such that the added paths cross or link other nodes orlinks of S) This is rather obvious adding a smaller triangle is the sameas adding a larger triangle and then reducing it with a diffeomorphismTherefore for ε lt εm the term in the parentheses remains in the samediffeomorphism equivalence class as ε is further reduced But Φ is invari-ant under diffeomorphisms and therefore the dependence on ε of theargument of the limit becomes constant for ε lt εm

Therefore the value of the limit (714) is simply given by


Φ(Hε|S〉) = Φ(H|S〉) (715)

where

H|S〉 = minus i

sum

nisinS

Nn

sum


(hγminus1


])|S〉

(716)

and the size ε of the regularizing paths is simply taken to be small enoughso that the added arc does not run over other nodes or link other links ofS The finiteness of the limit is then immediate

Discussion relation between regularization and background independenceThis result is very important and deserves a comment The first key pointis that the coordinate space x has no physical significance at all The phys-ical location of things is only location relative to one another not thelocation with respect to the coordinates x The diffeomorphism-invariantlevel of the theory implements this essential general-relativistic require-ment The second point is that the excitations of the theory are quan-tized This is reflected in the short-scale discreteness or in the discretecombinatorial structure of the states This is the result of the quantummechanical properties of the gravitational field When these two featuresare combined there is literally no longer room for diverging short-distancelimits The limit ε rarr 0 is a limit of small coordinate distance it becomesfinite simply because making the regulator smaller cannot change any-thing below the Planck scale as there is nothing below the Planck scaleOnce the regulating small loop αxlprimelprimeprime is smaller than the size needed tolink or cross other parts of the spin network any further decrease of its


size is gauge not physics This is how diffeomorphism invariance cures indepth the ultraviolet pathologies of quantum field theory

One last technicality In the definition given there is a residual (discrete) dependenceon the coordinates x In two different coordinate systems the arc may link the originallinks of the graph differently For instance a fourth link lprimeprimeprime may pass ldquooverrdquo or ldquounderrdquothe arc If the node is n-valent the possible alternatives are labeled by the homotopyclasses of lines (without intersection) going from the north to the south pole on asphere with n minus 2 punctures To have a fully diffeomorphism-invariant definition wemust therefore sum over these Nn alternatives On the other hand we can consistentlyexclude coordinate systems in which three links are coplanar and the arc intersects alink for all values of ε We can do this because we are using extended diffeomorphismsThus calling αr

xlprimelprimeprime r = 1 Nn a representative of the rth homotopy class wearrive at

(HΦ)(|S〉) = Φ(H|S〉) (717)

where

H|S〉 = minus i

sum

nisinSllprimelprimeprimer

Nnεllprimelprimeprime tr(hγminus1xnl

hαrxnlprimelprimeprime

[V(Rn) hγxnl ])

|S〉 (718)

This is the final form of the operator

712 Matrix elements

The resulting action of H on s-knot states is simple to derive and toillustrate (i) The action gives a sum of terms one for each node n of thestate (ii) For each node H gives a further sum of terms one term foreach triplet of links arriving at the node and for each triplet one termfor every permutation of the three links l lprime lprimeprime Each of these terms acts asfollows on the s-knot state (see Figure 72) (iii) It creates two new nodesnprime and nprimeprime at a finite distance from n along the links lprime and lprimeprime The exactlocation of these nodes is of course irrelevant for the s-knot state (iv) Itcreates a new link of spin-12 connecting nprime and nprimeprime (without linking anyother node) This new link is called ldquoarcrdquo (v) It changes the coloring j prime

of the link connecting n and nprime and the coloring j primeprime of the link connectingn and nprimeprime These turn out to be the colors of the links lprime and lprimeprime increasedor decreased by 12 (vi) It changes the intertwiner at the node n thenew intertwiner is between the representations corresponding to the newcolorings of the adjacent links

Remark Again it is easy to understand the origin of this action of the hamiltonianoperator on the basis of the simple form (63) of the hamiltonian constraint Thetwo functional derivatives ldquograsprdquo a spin network and as explained before the graspvanishes except in the vicinity of a node The curvature term Fab is essentially aninfinitesimal holonomy Therefore it creates a small loop next to the node This loop


D+ minus

j prime

j primeprime+

j primeprime j primeprime

j prime

j primen

nprime

nprimeprime

j j

1minus2

1minus2

1minus2

minus=^

Fig 72 Action of Dnlprimelprimeprimerεεprime

must be in the plane of two grasped links and can be identified with the triangle definedby the added arc

Notice that the ClebschndashGordan conditions always hold at the modified node Thisfollows immediately from the fact that the modified node is obtained from recouplingtheory the matrix element associated with nodes not satisfying the ClebschndashGordanconditions turns out to vanish

Call Dnlprimelprimeprimerplusmnplusmn an operator that acts around the node n by acting asdescribed in (iii) (iv) (v) This is illustrated in Figure 72 Then

H|S〉 =sum

nisinS

Nn

sum

llprimelprimeprimer

sum

εprimeεprimeprime=plusmnHnlprimelprimeprimeεprimeεprimeprime Dnlprimelprimeprimerεprimeεprimeprime |S〉 (719)

The operator Hnlprimelprimeprimeεprimeεprimeprime acts as a finite matrix on the space of the inter-twiners at the node n The explicit computation of its matrix elements isa straightforward problem in SU(2) representation theory It is discussedin detail in [208] where its matrix elements are explicitly given for simplenodes

The operator H is defined on S prime and as we have seen is finite when restricted toKdiff Notice however that the operator does not leave Kdiff invariant In general thestate H|s〉 is not a diffeomorphism-invariant state This is because of its dependenceon N To see this compute its action on a generic spin network state

〈s|H|S〉 =sum

nisinS

Nn

sum

llprimelprimeprimer

sum

εprimeεprimeprime=plusmn〈s|Hnlprimelprimeprimeεprimeεprimeprime Dnlprimelprimeprimerεprimeεprimeprime |S〉 (720)

On the right-hand side the quantities Nn are the values of N(x) at the points xn

where the nodes n of the spin network S are located The rest of the expression isdiff invariant but these values obviously change if we perform a diffeomorphism on|S〉 This has of course to be expected because the classical quantity H is itself notdiffeomorphism invariant Therefore there is no reason for the corresponding quantumoperator to be diff invariant and preserve Kdiff The theory this operator defines isnevertheless diffeomorphism invariant because the operator enters the theory via theWheelerndashDeWitt equation HΨ = 0 This equation is well defined on Kdiff Its solutionsare the (possibly generalized) states in Kdiff that are in the kernel of H Since H isfinite on the entire Kdiff this is well defined This is completely analogous to the classical


theory where H is not diff invariant but the equation H = 0 is perfectly sensible asan equation for diff-invariant equivalence classes of solutions (An alternative strategyyielding a version of the hamiltonian operator sending Kdiff into itself has been recentlyexplored by Thiemann in [209])

Recalling the definition of |s〉 we can write the matrix elements of Hamong spin network states

〈Sprime|H|S〉 =sum

nisinS

Nn

sum

|Ψ〉=Uφ|Sprime〉

sum

llprimelprimeprime

sum

εprimeεprimeprime=plusmn〈Ψ|HnlprimelprimeprimeεprimeεprimeprimeDnlprimelprimeprimeεprimeεprimeprime |S〉

(721)

The operator H is not symmetric This is evident for instance from thefact that it adds arcs but does not remove them Its adjoint Hdagger can bedefined simply by the complex conjugate of the transpose of its matrixelements

〈Sprime|Hdagger|S〉 = 〈S|H|Sprime〉 (722)

and a symmetric operator is defined by

Hs =12(H + Hdagger) (723)

This is an operator that adds as well as removes arcs It is reasonableto expect that this operator be better behaved for the classical limitTherefore we take this operator as the basic operator defining the theory

713 Variants

The striking fact about the hamiltonian operator is that it can be definedat all But how unique is it There are a number of possible variants ofthe operator that one may consider These can be seen as quantizationambiguities that is they define different dynamics in the quantum theoryall of which at least at first sight have the same classical limit So far itis not clear if these are truly all viable or whether there are physical ormathematical constraints that select among them

Higher j If we expand the matrix representing the holonomy of a con-nection in a representation j we obtain an expression analogous tothat in (72) Therefore we can regularize the terms F and A in thehamiltonian by using the holonomy in any arbitrary representationj That is we can write up to an irrelevant numerical factor

H = limεrarr0

1ε3

intN tr

(h

(j)

γminus1xw

h(j)αxuv

V h(j)γxw

)d3x (724)


where h(j) = R(j)(h) This has no effect on the classical limit How-ever the corresponding quantum operator

H(j)=minus i

limεrarr0

sum

m

Nm tr(h

(j)

γminus1xmw

h(j)αxmuv

[V(Rm) h(j)γxmw

])

(725)

is different from the operator (76) This can be easily seen by notic-ing that the added arc does not have spin 12 but rather spin j Ingeneral any arbitrary linear combination

H =sum

j

cj H(j) (726)

defines an hamiltonian operator with a correct classical limit Thereare indications that the coefficients cj should be different from zeroin a consistent theory discussed for instance in [210] In the samepaper matrix elements of the operator H(j) are computed and somearguments on criteria to fix the coefficients cj are discussed As weshall see in Chapter 8 this quantization ambiguity plays a role inloop quantum cosmology

Hs or H Both the symmetric operator Hs and the nonsymmetric oper-ator H define a quantum dynamics While there are arguments fortaking the symmetric one there are also arguments for taking thenonsymmetric one [20]

Other regularizing loops Loops different from αxlprimelprimeprime and γxl could bechosen for the regularization The freedom is strongly limited bydiffeomorphism invariance and by the condition that the result-ing operator is finite and nonvanishing But other choices might bepossible

Ordering A different ordering can be chosen between the volume and theholonomy operators

Others More generally no uniqueness theorem exists so far

This freedom in the definition of the hamiltonian operator is not aproblem it is an asset No complete and completely consistent theory ofquantum gravity with a well-understood low-energy limit exists so farHaving more than one is for the moment the very least of our worriesThe remarkable result is the existence of a finite and interesting hamil-tonian operator The fact that we have a certain residual latitude in itsdefinition might very well turn out to be helpful The ldquocorrectrdquo variantcould be selected by some internal consistency requirement that has notyet been considered or by requiring the correct classical limit If theseconditions turn out to be insufficient we shall simply have nonequivalentquantum theories with the same classical limit The physically correct onewill have to be determined by experiments I wish we were already there



The evolution of the mathematical description of the matter fields dur-ing the twentieth century has slowly converged with the evolution of themathematical description of the gravitational field In both cases differen-tial geometry notions such as fiber bundles sections and automorphismsof the bundle play a role in the description of the classical fields If wetake for instance a coupled EinsteinndashYangndashMills system and describethe gravitational field by means of an SO(3 1) connection the structuresof the two fields the gauge and the gravitational field are barely differentfrom each other

Indeed as I have argued in Section 232 the distinction between matterand spacetime (gravity) is not profound it is largely conventional Thegravitational field is not substantially different from the other matterfields It is the full coupled gravity + matter theory which is profoundlydifferent from a theory on a fixed background When the dynamical grav-itational field is not approximated by a fixed background the full theoryis generally covariant and the physical fields live only ldquoon one anotherrdquoas the animals and the whale of the metaphor

Accordingly the methods developed above for the gravitational fieldextend naturally to other fields The theory of pure quantum gravity andthe theory of quantum gravity and matter do not differ much from eachother The second has just some additional degrees of freedom The sim-ilarity and compatibility of the classical mathematical structures makesthe extension of the quantum theory to these additional degrees of free-dom very natural

721 YangndashMills

The easiest extension of the theory described in the previous chapter isto YangndashMills fields Let GYM be a compact YangndashMills group such asin particular the group SU(3)times SU(2)timesU(1) that defines the standardmodel Let AYM be a 3d YangndashMills connection for this group and A bethe gravitational connection The two 3d connections A (gravitational)and AYM (YangndashMills) can be considered together as a single connectionA=(AAYM) for the group G = SU(2)timesGYM The construction of K K0

and Kdiff of Chapter 6 extends immediately to this connection withoutdifficulties

Holonomies of the YangndashMills field can be defined as operators on Kprecisely as gravitational holonomies Surface integrals of the YangndashMillselectric field can be defined precisely as for the E gravitational field

The diffeomorphism-invariant quantum states of GR + YangndashMills arethen given by s-knot states labeled by abstract knotted graphs carrying

72 Matter kinematics 287

irreducible representations of the group G on the links and the corre-sponding intertwiners on the nodes Since G is a direct product its irre-ducibles are simply given by products of irreducibles of SU(2) and GYMIn other words each link is labeled with a spin jl and an irreduciblerepresentation of GYM

Notice how the quantum theory realizes the relational localization char-acteristic of GR the position of the YangndashMills field is well defined withrespect to the quantum state of spacetime defined by the gravitationalpart of the spin network or equivalently vice versa

Notice also that in the absence of diffeomorphism invariance the aboveconstruction would not yield a sensible quantum state space of the YangndashMills field because it would yield a nonseparable Hilbert space

722 Fermions

Let η(x) be a Grassman-valued fermion field It transforms under a rep-resentation k of the YangndashMills group GYM and under the fundamentalrepresentation of SU(2) It is more convenient for the quantum theoryto take the densitized field ξ equiv

radic|detE| η as the basic field variable

The Grassman-valued field ξ and its complex conjugate take value in a(finite-dimensional) superspace S An integral over S is defined by theBerezin symbolic integral dξ

Define a cylindrical functional Ψ[Aψ] of the connection and the fermionfield as follows Given (i) a collection Γ of a finite number L of paths γl(ii) a finite number N of points xn and (iii) a function f of L groupelements and N Grassman variables a cylindrical functional is defined by

ΨΓf [Aψ] = f(U(A γ1) U(A γL) ξ(x1) ξ(xN )) (727)

Since Grassman variables anticommute cylindrical functionals can be atmost linear in each (component of the) fermion fields in each point n

A scalar product is defined on the space of these functions as followsGiven two cylindrical functionals defined by the same Γ define


GL

dUl

int

SN

dξn g(U1 UN ξ1 ξN )

times f(U1 UN ξ1 ξN ) (728)

The extension of this scalar product to any two cylindrical functions isthen completely analogous to the purely gravitational case This definesthe extension of K to fermions

Basis states are easily constructed by fixing the degree Fn of the mono-mials in ξ at each node n which determine the fermion number in theregion of the node


In the absence of fermions we constructed gauge-invariant function-als by contracting the indices of the holonomies among themselves withintertwiners In the presence of fermions we can also contract SU(2)and YangndashMills indices with the indices of the fermions In other wordsfermions live on the nodes of the graph Γ and the gravitational andYangndashMills lines of flux can end at a fermion For this to happen ofcourse generalized ClebschndashGordan conditions must be satisfied For in-stance a single fermion cannot sit at the open end of a link in the trivialrepresentations of SU(2) because its SU(2) index cannot be saturatedThis physical picture is well known for instance in canonical lattice gaugetheory In fact each Hilbert subspace KΓ can be identified as the Hilbertspace of a lattice YangndashMills theory with fermions defined on a lattice ΓWe are back to the original intuition of Faraday the lines of force canemerge from the charged particles

723 Scalars

The present formulation of the standard model requires also a certainnumber of scalar fields so far unobserved Whether these fields ndash theHiggs fields ndash are in fact present in Nature and observable or whetherthey represent a phenomenological description of some aspect of Naturewe havenrsquot yet fully understood is still unclear Scalar fields can be incor-porated in LQG but in a less natural manner than YangndashMills fields andfermions

Let φ(x) be a suitable multiplet of scalar fields (that we can alwaystake as real) transforming in a representation k of the gauge group GYMand therefore taking value in the corresponding vector space Hk

The complication is that Hk is noncompact ndash it has infinite volumeunder natural invariant measures This makes the definition of the scalarproduct of the theory more difficult (because the Hilbert space associatedto subgraphs is not a subspace of the Hilbert space associated to a graph)One way out of this difficulty is the following Assume k is the adjointrepresentation of GYM We can then exponentiate the field φ(x) definingU(x) = expφ(x) The field U(x) then takes values in GYM which iscompact and carries the Haar invariant measure

Define a cylindrical functional Ψ[Aψ φ] of the connection fermionand scalar fields as follows Given (i) a collection Γ of a finite number Lof paths γl (ii) a finite number N of points xn and (iii) a function f ofL group elements N fermion variables and N other group elements acylindrical functional is defined by

ΨΓf [Aψ φ] = f(U(A γ1) U(A γL) ψ(x1) ψ(xN ) eφ(x1)

eφ(xN )) (729)

73 Matter dynamics and finiteness 289

A scalar product is defined on the space of these functions as followsGiven two cylindrical functionals defined by the same Γ we define


GL

dUl

int

SN

dξnint

GN

dU primen

times f(U1 UL ξ1 ξN U prime1 U

primeN )

times g(U1 UL ξ1 ξN U prime1 U

primeN ) (730)

and extend this to any graph as usual


A state |s〉 in Kdiff can then be labeled by the following quantum numbers

bull An abstract knotted graph Γ with links l and nodes n

bull A spin jl associated with each link l

bull An irreducible representation kl of the YangndashMills group GYM asso-ciated with each link l

bull An integer Fn associated with each node

bull An irreducible representation Sn of the YangndashMills group GYM as-sociated with each node n

bull An SU(2) intertwiner in associated with each node n

bull A GYM intertwiner wn associated with each node n

Thus we can write

|s〉 = |Γ jl kl Fn Sn in wn〉 (731)

This state describes a quantum excitation of the system that has a simpleinterpretation as follows There are N regions n that have volume andwhere fermions and Higgs scalars can be located These are separated byL surfaces l that have area and are crossed by flux of the (electric) gaugefield The quantum numbers are related to observable quantities as inTable 71 This completes the definition of the kinematics of the coupledgravity+matter system


The dynamics of the coupled gravity+matter system is simply definedby adding the terms defining the matter dynamics to the gravitational


Table 71 Quantum numbers of the spin networkstates for gravity and matter

Quantum number Physical quantity

Γ adjacency between the regionsin volume of the node njl area of the surface lFn number of fermions at node nSn number of scalars at node nwn field strength at node nkl electric flux across the surface l

relativistic hamiltonian The hamiltonian for the fields described is givenby

H = HEinstein + HYangndashMills + HDirac + HHiggs (732)

HEinstein is the gravity hamiltonian described in the previous chaptersThe other terms are

HYangndashMills =1

2g2YMe

3tr[EaEb] Tr[EaEb + BaBb]

HDirac =12e

Eai (iπτ iDaξ + Da(πτ iξ) +

i2Ki

aπξ + cc)

HHiggs =12e

(p2 + tr[EaEb] Tr[(Daφ)(Dbφ)] + e2V (φ2)

) (733)

Here π is the momentum conjugate to the fermion field p the momentumconjugate to the scalar field E is the momentum conjugate to the YangndashMills potential (the electric field) B the curvature of the YangndashMillspotential (the magnetic field) D is the SU(2)timesGYM covariant derivativetr is the trace in the SU(2) Lie algebra and Tr is the trace in the Liealgebra of GYM e equiv

radic|detE| V is the Higgs potential p2 = Tr[pp]

and φ2 = Tr[φφ] and gYM is the YangndashMills coupling constant For aderivation and a discussion of these expressions see [211]

To define the quantum hamiltonian the expressions in (733) must beregulated and expressed in terms of the operators well defined on K Thiscan be done following the same strategy we used for the gravitationalquantum hamiltonian in Section 71 I do not present the detailed con-struction here for which I refer the reader to Thiemannrsquos work [211]

The essential result of this construction is that the total hamiltonian(732) can be constructed as a well-defined operator on the Hilbertspace of gravity and matter K The operator acts on nodes as does thegravitational part but its action is more complex than the pure gravity


operator and codes the entire dynamics of the standard model andgeneral relativity

The fact that the total hamiltonian turns out to be finite is extremely re-markable It is perhaps the major pay-off of the background-independentquantization strategy on which LQG is based

I advise the reader to read the beautiful account by Thiemann in[211] and especially in [11] for an explanation of the internal reasonsof this finiteness Thiemann illustrates how the ultraviolet divergencesof ordinary quantum field theory can be directly interpreted as a con-sequence of the approximation that disregards the quantized discretenature of quantum geometry For instance Thiemann shows how the op-erator 1

2β2e3tr[EaEb] Tr[EaEb] the kinetic term of the YangndashMills hamil-

tonian is well defined so long as we treat E as an operator but becomesinfinite as soon as we replace E with a smooth background field


With the definition of the operator H for gravity and matter finite onKdiff the formal definition of the quantum theory of gravity is completedThe theory is finite provides a compelling intuitive description of thePlanck-scale structure of space has definite predictions such as the eigen-values of area and volume and reduces to classical GR in the naive rarr 0limit The transition amplitudes of the theory are defined by


where s and sprime are two s-knot states and P is the projector on the space ofthe solutions of the equation HΨ = 0 The quantity W (s sprime) is interpretedas the probability amplitude of observing the discretized geometry withmatter determined by the s-knot s if the geometry with matter determinedby sprime was observed

If we consider a region of spacetime bounded by two disconnected sur-faces the diff-invariant boundary space is Kdiff = Klowast

diff otimesKdiff and we canrewrite (734) in terms of the covariant vacuum state

〈0|sout sin〉 = 〈sout|P |sin〉 (735)

where |sout sin〉 = 〈sout| otimes |sin〉 isin KdiffMore generally we can consider a finite region of spacetime bounded

by a 3d surface Σ If s represents the outcome of the measurement of thegravitational field and matter fields on Σ then

W (s) = 〈0|s〉 (736)


gives the correlation probability amplitude of the measurement of thestate s This can also be viewed as the transition amplitude from theempty set to the full s hence

W (s) = 〈empty|P |s〉 (737)

or

|0〉 = P |empty〉 (738)

in K This is loop quantum gravityMuch remains to be done Here are some issues that I have not ad-

dressed

(i) Lorentzian theory So far I have dealt only with the euclidean the-ory As already mentioned in Section 422 the lorentzian theorycan be expressed in terms of the same kinematics as the euclideantheory only adding a second term to the hamiltonian I shall notdiscuss the quantization of this second term here This is done indetail in [20] Alternatively the quantization has to be defined us-ing the complex connection but the full quantum state space witha complete operator algebra has not yet been constructed for thecomplex connection as far as I know Another alternative is to de-rive the amplitudes of the lorentzian theory from the amplitudesof the euclidean theory as one can do in flat-space QFT As men-tioned a naive reproduction of the flat-space technique is not viablein quantum gravity but a suitable extension of this might work

(ii) Transition amplitudes The matrix elements of the projector P arenot easy to compute

(iii) Scattering A general technique to connect the transition amplitudesW (s sprime) to particle observables such as gravitonndashgraviton scatteringmust be developed

(iv) Classical limit Can we prove explicitly that classical GR can berecovered from LQG

(v) Form of the dynamics Is the proposed form of the hamiltonian con-straint correct or does it have to be corrected

(vi) Physical consequences What does the theory say about the stan-dard physical problems where quantum gravity is expected to berelevant such as black-hole thermodynamics and early cosmology

(vii) Observable predictions Are there any

Much is known on several of these issues Some of them are discussedbelow In particular Chapter 8 deals with (vi) and (vii) Chapter 9 with(i) (ii) (iii) and (v)


Nevertheless I emphasize the fact that whatever its consequences andits physical correctness the theory developed thus far provides a finiteand consistent general covariant and background-independent quantumfield theory for the gravitational field and the matter fields In it thecore physical insights of GR and QFT merge beautifully Finding such atheory was our major aim

741 Variants

The LQG theory that I have described above is a standard version of thetheory There are a number of possible variants that have been consideredin the literature

Different regularization of H I have described this possibility abovein Section 71

q-deformed spin networks An intriguing possibility is to replace thegroup SU(2) with the quantum group SU(2)q in the quantum the-ory and choose q to be given by qN = 1 where N is a large numberIt is possible to define q-deformed spin networks labeled by repre-sentations of SU(2)q and build the rest of the theory as above Thisis an interesting possibility for several reasons Several of the spin-foam models studied in Chapter 9 are defined using quantum groupsand their states are q-deformed spin networks The spinfoam modelsshow that the use of q-deformed spin networks is naturally connectedwith a cosmological constant λ The quantum group SU(2)q has afinite number of irreducible representations which grows with N This implies that the quantum of area has a maximum value de-termined by N and related to the (large) length determined by thecosmological constant Finally N works like a natural infrared cut-off which is likely to cure any eventuality of infrared divergencesq-deformed LQG has been studied in the literature but a system-atic construction of LQG in terms of q-deformed spin networks isstill missing

Different ordering of the area operator We can quantize the har-monic oscillator choosing an ordering of the hamiltonian such thatthe vacuum energy is zero instead of 1

2ω Similarly we can choosedifferent orderings for the area operator and obtain a different spec-trum The ordering used in this book and in most of the literatureis the natural one for the Casimir operator but alternatives havebeen considered and produce some intriguing effects In particu-lar it is possible to order the operator to obtain an equally spacedspectrum This would reintroduce the BekensteinndashMukhanov effectstudied below in Section 824 and apparently would automati-cally give a dominance of spin-1 quanta for a black hole bringing


the value of the Immirzi parameter to match the frequency of theblack-hole ringing modes (see Section 823)

Different regularization of the area operator The full spectrum ofthe area operator given in (6125) contains the main sequence(675) If we think that a diffeomorphism-invariant notion of a sur-face is truly the boundary of a region and the region is an ensembleof quanta of volume then we are led to the idea that physical sur-faces are described by the mathematical surfaces that cut the linkswithout touching the nodes These surfaces have area given by themain sequence (675) Thus the degenerate sector might be phys-ically spurious To have the eigenvalues in the degenerate sectorwe need a surface that cuts precisely through the node and this isagainst the intuition that the location of the surfaces is only definedup to Planck scale A different regularization of the area operatormight get rid of the degenerate sector

Unknotted spin networks A very interesting possibility is to modifythe definition of the spin network states of the theory droppingthe information on the knotting and linking of the graphs Thatis to define the graphs Γ that form the spin networks solely interms of the adjacency relations between nodes as is usually donein graph theory and not as is done above as equivalence classesof embedded graphs under extended diffeomorphisms The physicaldifferences implied by the two definitions are not clear at present

Different regularization of the volume Two definitions of the vol-ume have been given in the literature The two turn out to be slightlydifferent Originally the two were given in different mathematicallanguages and it was thought that the difference had to do with thedifferent formulations of the theory Later it became clear that bothoperators can be defined in either formulation The volume opera-tor defined here does not distinguish a node in which some tangentsof the adjacent links are coplanar from a node in which they arenot coplanar The other version of the volume operator used forinstance in [20] makes this distinction its action on a node withcoplanar links differs from the one given here This second operatoris covariant under Diff but not under Diff lowast

Extended loop representation Gambini and Pullin have developeda version of LQG in which loop states are not normalizableNormalizable states are obtained smearing loop states The mainmotivation is the fact that in flat-space QFT this is the case I referthe reader to their book [7] for a discussion and details

Lorentz spin networks In the hamiltonian theory on which LQG isbased there is a partial gauge-fixing One of the consequences of


this gauge-fixing is that the connection with the covariant formalismused in the spinfoam models becomes technically more cumbersomeTo avoid this difficulty Sergei Alexandrov has studied the possibilityof defining LQG without making this gauge-fixing and keeping thefull Lorentz group in the hamiltonian formalism [127 212] Thismight give a different regularization of the area operator as well[213]

mdashmdash


The WheelerndashDeWitt equation appeared in [214] The first version of thehamiltonian operator of LQG and its first solutions were constructed in[177] Various other solutions were found see for instance [215] A re-view of early solutions of the hamiltonian operator is in [216] The resultthat diffeomorphism-invariant states make the operator finite appearedin [217] The general structure of the hamiltonian constraint and the op-erator D are illustrated in [218]

The idea of expressing the hamiltonian as a commutator and there-fore the first fully well-defined version of the hamiltonian operator wasobtained by Thomas Thiemann in [133] and systematically developed byThiemann in the remarkable ldquoQSDrdquo series of papers [201 209] Matrix el-ements of this operator were systematically studied in [208] On the hamil-tonian operator with positive cosmological constant and the possibilityof defining the theory in this case see [220] Thiemann and collaboratorsare developing an original and promessing approach to the definition ofthe quantum dynamics called the ldquoMaster Programrdquo The idea is to con-densate a full set of constraints into a single one For an introduction andreferences see [221]

Fermions were introduced in LQG in [222] and in [223] The key stepfor the present formulation of the fermionndashLQG coupling was taken byThomas Thiemann using half-density spinor fields [224] A complete studyof the matter hamiltonian in LQG is due to Thiemann See his [20] andcomplete references therein The fermionic contribution to the spectrumof the area operator was considered in [225] The intriguing possibilitythat fermions are described by the linking of the spin networks has beenrecently explored in [226]

On the area operator with equispaced eigenvalues and its effect on theblack-hole entropy see [227] A q-deformed version LQG was consideredin [228] see [220] and references therein For q-deformed spin networkssee also [229] [230] and [231]

8Applications

In this chapter I briefly mention some of the most successful applications of LQG toconcrete physical problems I have no ambition of completeness and I will not presentany detailed derivation For these I refer to original papers and review articles I onlyillustrate the main ideas and the main results

The two traditional applications of quantum gravity are early cosmology and black-hole physics In both these fields LQG has obtained interesting results In addition acertain number of tentative calculations concerning other domains where Planck-scalephysical effects could perhaps be observable have also been performed


A remarkable application of LQG is to early cosmology A direct treat-ment of semiclassical states in Kdiff representing cosmological solutionsof the Einstein equations is not yet available However it is possible toimpose homogeneity and isotropy on the basis states and operators of thetheory and in this way restrict the theory to a finite-dimensional systemdescribing a quantum version of the cosmological dynamics that can bestudied in detail

The result is different from the traditional WheelerndashDeWitt minisuper-space quantization of the dynamics of Friedmann models The key to thedifference is the fact that the system inherits certain physical aspects ofthe full theory In particular the quantization of the geometry These havea major effect on the dynamics of the early Universe The main resultsare the following

(i) Absence of singularities Dynamics is well defined at the Big Bangwith no singular behavior In particular the inverse scale factor isbounded In this sense the Universe has a minimal size

(ii) Semiclassical behavior Cosmological evolution approximates thestandard Friedmann dynamics for large values of the scale factora(t) but differs from it at small values of a(t)

296


(iii) Quantization of the scale factor The scale factor ndash and the volumeof the Universe ndash are quantized

(iv) Discrete cosmological evolution We can view the scale factor as acosmological time parameter Then we can say that cosmologicaltime is quantized Accordingly the WheelerndashDeWitt equation is adifference equation and not a differential equation in a

(v) Inflation Just after the Big Bang the Universe underwent an infla-tionary phase d2a(t)dt2 gt 0 This is driven not by a scalar inflatonfield but by quantum properties of the gravitational field itself

These are all remarkable results but of different kinds Results (i) and(ii) are what one would expect from a quantum theory of gravity giving aconsistent description of the early Universe Results (iii) and (iv) reflectthe most characteristic aspect of LQG the quantization of the geometryResult (v) came as a big surprise Let me briefly illustrate how theseresults are derived

Consider a homogeneous and isotropic Universe Its gravitational fieldis given by the well-known line element

ds2 = minusdt2 + a2(t)

(dr2

1 minus kr2+ r2(dθ2 + sin θ dφ2)

)

(81)

where a(t) is the scale factor and k is equal to zero or plusmn1 (see for instance[75]) The Einstein equations reduce to the Friedmann equation

(a

a

)2

=8πG

3ρminus k

a2 (82)

where ρ is the time-dependent matter energy density Let us represent thematter content of the Universe in terms of a single field that for simplicitycan be taken as a scalar field φ Homogeneity then demands that φ is afunction of the sole time coordinate The system is therefore describedby a(t) and φ(t) Assume for simplicity that φ(t) has a simple quadraticself-interaction (potential) term namely its hamiltonian is

Hφ =12(p2

φ + ω2φ2) (83)

where pφ is the momentum conjugate to φ Therefore

φ(t) = A sin(ωt + φ0) (84)

Do not confuse the field φ with the inflaton it has no inflationary po-tential The energy density is related to the conserved matter energy Hφ

by

ρ = aminus3Hφ = aminus3ρ0 =12aminus3ω2A2 (85)

298 Applications

The constant ρ0 is the density at a = 1 The Friedmann equation (82)can be derived from the hamiltonian

H = minus(p2a

8a+ 2ka

)

+ 16πGHφ (86)

by simply computing a = dHdpa and using H = 0 In the simplestspatially flat case k = 0 the Friedmann equation (82) reduces to

(a

a

)2

=8πG

3ρ0

a3 (87)

by taking a derivative we obtain

a = minus43πGρ0

1a2

(88)

which is precisely equation (2112) that we had obtained in the con-text of newtonian cosmology The equation is solved by the well-knownFriedmann evolution

a(t) = a0(tminus t0)23 (89)

Interpretation These equations are written in a particular gauge-choicefor the variable t but the full theory is invariant under reparametriza-tion in t The relativistic configuration space is coordinatized by a and φThe physical content of the theory is not in the dependence of these twoquantities on t but in their dependence on each other The proper mean-ing of (84)ndash(89) concerns the relation between φ and a For instancewe can interpret φ as a clock That is we can define its oscillations asisochronous This defines a physical time variable Then the scale factorgrows in this time variable as described by (89) Alternatively we canuse the scale factor as a measure of time In this cosmological time allmaterial physical processes slow down as in

φ(a) = A sin(ω a32 + φ0) (810)

In other words Friedmann evolution is the relative evolution of the rateof change of the material processes and the rate of change of the scalefactor it is the evolution of the ratio between the two rates of change Asolution of the Friedmann equation describes therefore the values thatthe scale factor can take for a given value φ of the matter variable orequivalently the values φ(a) that the matter variable can take at a givenvalue a of the scale factor (equation (810))

Notice that there is no need to think in terms of ldquoevolution in trdquo forthese relations to make sense As discussed in Section 34 time evolu-tion namely the idea of a physical ldquoflowrdquo of time with respect to which


a increases and φ oscillates may simply derive from the physics of ther-modynamical processes that happen in the presence of many (gravity andmatter) variables Therefore the question ldquoWhat happened before the BigBangrdquo might be as empty as the question ldquoWhat is there on the Earthrsquossurface one meter north of the North polerdquo

Traditional quantum cosmology In the traditional approach to quantumcosmology one introduces a wave function ψ(a φ) This is governed bythe WheelerndashDeWitt equation obtained from (86) Up to factor orderingthis can be written as

(h2

8apart

parta

part

parta+ 2ka

)

ψ(a φ) = 16πG H0φ ψ(a φ) (811)

where H0φ is the hamiltonian of an harmonic oscillator with angular fre-

quency ω This equation has semiclassical solutions which are wave pack-ets that approximate the Friedmann evolution for large a For small a onthe other hand the singular behavior of the classical theory persists

Loop quantum cosmology What changes if we use LQG The essentialnovelty is the quantization of the geometry Recall that up to a constant

a sim 3radicV (812)

where V is the volume of the compact universe But the volume has adiscrete spectrum in the theory Therefore we should expect a to have adiscrete spectrum In fact the detailed construction carried out in [232]shows that this is precisely the case The observable a has a discretespectrum with eigenstates |n〉 labeled by an integer n and eigenvalues

an = a1

radicn (813)

where the constant a1 is

a1 =radic

43γπhG (814)

Therefore in LQG the size of the universe is quantized If a is quantizedwe cannot represent states as functions ψ(a φ) (for the same reason thatwe do not represent states of the harmonic oscillators as continuous func-tions of the energy) Rather we can represent states in the form

ψn(φ) = 〈n φ|ψ〉 (815)

where |n φ〉 is an eigenstate of a and φ Accordingly the partial derivativeswith respect to a in the WheelerndashDeWitt equation (811) are replaced

300 Applications

in loop quantum cosmology with finite-difference operators In fact theWheelerndashDeWitt equation is explicitly derived in [232] It has the form

αnψn+4(φ) minus 2βnψn(φ) + γnψn+4(φ) = 16πG aminus3Hφψn(φ) (816)

where the constants αn βn γn are given in [232] Notice the volume den-sity factor aminus3 on the right-hand side It appears because the quantumconstraint must be obtained in LQG from the densitized hamiltonianaminus3H

The key point is now the meaning of the operator aminus3 in the right-hand side of this equation Recall that in the definition of the hamiltonianoperator of the full theory it was essential to use a proper definition ofthe inverse volume element 1detE in order to define a well-behavedoperator This was obtained by expressing it via a Poisson bracket inthe classical theory and via a commutator in the quantum theory Thisprocedure circumvents technical difficulties associated with the definitionof the inverse of the volume element operator The inverse scale factor aminus3

in (816) is what remains of that term in the cosmological theory But ifthe cosmological theory has to approximate the full theory we have tobetter define this operator in the same way the inverse volume elementwas defined in the full theory In fact it is not hard to do so writing

d = aminus3 (817)

as a commutator of well-defined quantum operators The resulting oper-ator d is well defined Its spectrum is however more complicated than thesimple inverse of the spectrum of a3

dn =

⎛

⎝ 12j(j + 1)(2j + 1)

sum

k=minusjj

kradicVn

⎞

⎠

6

(818)

In the definition of the operator d there is a quantization ambiguity Thisis because d is defined using a holonomy and this can be taken in anyrepresentation j This is precisely the ambiguity in the definition of thehamiltonian operator that was discussed above in Section 713 Remark-ably its spectrum turns out to be bounded In fact for large n we have

dn sim aminus3n (819)

but for small ndn sim a12

n (820)

There is a maximum value of dn whose value and location are determinedby the free quantization parameter j Using this operator in the WheelerndashDeWitt equation (816) yields a perfectly well-behaved evolution on and


around n = 0 A numerical study of this equation shows easily that it givesthe standard semiclassical behavior and therefore standard Friedmannevolution for large n

811 Inflation

The most surprising and intriguing aspect of LQC is the fact that itpredicts an inflationary phase in the expansion of the early UniverseThis can be seen by explicit numerical solutions of the WheelerndashDeWittequation (816) or more simply as follows For small n the behavior ofthe operator d is governed by (820) instead of (819) The correspond-ing cosmological evolution can therefore be effectively approximated bya modification of the Friedmann equation in which Hφ is proportional toa12 instead of aminus3 This yields an accelerated initial expansion of the form

a(t) sim (t0 minus t)minus29 (821)

A numerical solution of the WheelerndashDeWitt equation confirms this re-sult The initial acceleration subsequently decreases smoothly and con-verges to a standard decelerating Friedmann solution The duration ofthis inflationary expansion is governed by j

What goes on physically can be understood as follows The kinetic termof the matter hamiltonian contains effectively a coupling with gravityWhen the gravitational field is strong near the initial singularity thematter field feels the quantum structure of the gravitational field whichaffects its dynamics

This scenario deserves to be explored in more detail and better under-stood


The first hint that a black hole can have thermal properties came fromclassical GR In 1972 Hawking proved a theorem stating that the Einsteinequations imply that the area of the event horizon of a black hole cannotdecrease Shortly after Bardeen Carter and Hawking showed that in GRblack holes obey a set of laws that strongly resembles the principles ofthermodynamics impressed by this analogy Bekenstein suggested thatwe should associate an entropy

SBH = akB

hGA (822)

to a Schwarzschild black hole of surface area A (In this chapter wherethe connection does not appear the area is denoted A not A) Here a is a

302 Applications

constant of the order of unity kB the Boltzmann constant and the speedof light is taken to be 1 The reason for the appearance of h in this formulais essentially to get dimensions right Bekensteinrsquos suggestion was that thesecond law of thermodynamics should be extended in the presence of blackholes the total entropy that does not decrease in time is the sum of theordinary entropy with the black-hole entropy SBH Bekenstein presentedseveral physical arguments supporting this idea but the reaction of thephysics community was very cold mainly for the following reason Thearea A of a Schwarzschild black hole is related to its energy M by

M =

radicA

16πG2 (823)

If (822) was correct the standard thermodynamical relation Tminus1 =dSdE would imply the existence of a black-hole temperature

T =h

a32πkBGM (824)

and therefore a black hole would emit thermal radiation at this tempera-ture a consequence difficult to believe However shortly after Bekensteinrsquossuggestion Hawking derived precisely such a black-hole thermal emissionfrom a completely different perspective Using conventional methods ofquantum field theory in curved spacetime Hawking studied a quantumfield in a gravitational background in which a black hole forms (say a starcollapses) and found that if the quantum field is initially in the vacuumstate after the star collapse we find it in a state that has properties of athermal state This can be interpreted by saying that the black hole emitsthermal radiation Hawking computed the emission temperature to be

T =h

8πkBGM (825)

which beautifully supports Bekensteinrsquos speculation and fixes the con-stant a at

a =14 (826)

so that (822) becomes

SBH =kBA

4hG (827)

Since then the subscript BH in SBH does not mean ldquoblack holerdquo itmeans ldquoBekensteinndashHawkingrdquo Hawkingrsquos theoretical discovery of black-hole emission has since been rederived in a number of different ways andis today generally accepted as very credible1

1Although perhaps some doubts remain about its interpretation One can write a purequantum state in which the energy distribution of the quanta is planckian Is the


Hawkingrsquos beautiful result raises a number of questions First in Hawk-ingrsquos derivation the quantum properties of gravity are neglected Are thesegoing to affect the result Second we understand macroscopical entropyin statistical mechanical terms as an effect of the microscopical degrees offreedom What are the microscopical degrees of freedom responsible forthe entropy (822) Can we derive (822) from first principles Becauseof the appearance of h in (822) it is clear that the answer to these ques-tions requires a quantum theory of gravity The capability of answeringthese questions has since become a standard benchmark against which aquantum theory of gravity can be tested

A detailed description of black-hole thermodynamics has been devel-oped using LQG and research is active in this direction The major re-sult is the derivation of (822) from first principles for Schwarzschild andfor other black holes with a well-defined calculation where no infinitiesappear As far as I know LQG is the only detailed quantum theory ofgravity where this result can be achieved2

As I illustrate below the result of LQG calculations gives (822) with

a asymp 023754γ

(828)

where γ is the Immirzi parameter This agrees with Hawkingrsquos value (826)provided that the Immirzi parameter has the value

γ asymp 02375 (829)

In fact this is the way the value of γ is fixed in the theory nowadaysThe calculation can be performed for different kinds of black holes andthe same value of γ is found assuring consistency An independent wayof determining γ would make this result much stronger

In what follows I present the main ideas that underlie the derivationof this result


The degrees of freedom responsible for the entropy Consider a black holewith no charge and no angular momentum Its entropy (822) can originate

state of the quantum field after the collapse truly a thermal state or a pure state thathas the energy distribution of a thermal state Namely are the relative phases of thedifferent energy components truly random or are they fixed deterministically by theinitial state Do the components of the planckian distribution form a thermal or aquantum superposition In the second case the transition to a mixed state is just thenormal result of the difficulty of measuring hidden correlations

2So far string theory can only deal with the highly unphysical extreme or nearlyextreme black holes

304 Applications

from horizon microstates corresponding to a macrostate described bythe Schwarzschild metric Intuitively we can think of this as an effect offluctuations of the shape of the horizon

One can raise an immediate objection to this idea a black hole has ldquonohairrdquo namely a black hole with no charge and no angular momentum isnecessarily a spherically symmetric Schwarzschild black hole leaving nofree degrees of freedom to fluctuate

This objection however is not correct It is the consequence of a com-mon confusion about the meaning of the term ldquoblack holerdquo The confusionderives from the fact that the expression ldquoblack holerdquo is used with twodifferent meanings in the literature In its first meaning a ldquoblack holerdquo is aregion of spacetime hidden beyond an horizon such as a collapsed star Inits second meaning ldquoblack holerdquo is used as a synonym of ldquostationary blackholerdquo When one says that ldquoa black hole is uniquely characterized by massangular momentum and chargerdquo one refers to stationary black holes notto arbitrary black holes In particular a black hole with no charge and noangular momentum is not necessarily a Schwarzschild black hole and isnot necessarily spherical Its rich dynamics is illustrated for instance bythe beautiful images of the rapidly varying shapes of the horizon obtainedin numerical calculations of say the merging of two holes Generally ablack hole has a large number of degrees of freedom and its event horizoncan take arbitrary shapes These degrees of freedom of the horizon canbe the origin of the entropy

To be sure in the classical theory a realistic black hole with vanishingcharge and vanishing angular momentum evolves very rapidly towards theSchwarzschild solution by rapidly radiating away all excess energy Its os-cillations are strongly damped by the emission of gravitational radiationBut we cannot infer from this fact that the same is true in the quantumtheory or in a thermal context In the quantum theory the Heisenbergprinciple prevents the hole from converging exactly to a Schwarzschildmetric and fluctuations may remain In fact we will see that this is thecase

Recall that in the context of statistical mechanics we must distinguishbetween the macroscopic state of a system and its microstates Obviouslythe symmetry of the macrostate does not imply that the relevant micro-states are symmetric For instance in the statistical mechanics of a sphereof gas the individual motions of the gas molecules are certainly not con-fined by spherical symmetry When the macrostate is spherically symmet-ric and stationary the microstates are not necessarily spherically symmet-ric or stationary

When we study the thermodynamical behavior of a Schwarzschild blackhole it is therefore important to remember that the Schwarzschild sol-ution is just the macrostate Microstates can be nonstationary and


non-spherically symmetric Indeed trying to explain black-hole thermo-dynamics from properties of stationary or spherically symmetric metricsalone is a nonsense such as trying to derive the thermodynamics of anideal gas in a spherical box just from spherically symmetric motions ofthe molecules

Thermal fluctuations of the geometry To make the case concrete con-sider a realistic physical system containing a nonrotating and nonchargedblack hole as well as other physical components such as dust gas or radia-tion which I denote collectively as ldquomatterrdquo We are interested in the sta-tistical thermodynamics of such a system Because of Einsteinrsquos equationsat finite temperature the microscopic time-dependent inhomogeneities ofthe matter distribution due to its thermal motion must generate time-dependent microscopic thermal inhomogeneities in the gravitational fieldas well One usually safely disregards these ripples of the geometry Forinstance we say that the geometry over the Earthrsquos surface is given by theMinkowski metric (or the Schwarzschild metric due to the Earthrsquos grav-itational field) disregarding the inhomogeneous time-dependent gravita-tional field generated by each individual fast-moving air molecule TheMinkowski geometry is therefore a ldquomacroscopicrdquo coarse-grained aver-age of the microscopic gravitational field surrounding us These thermalfluctuations of the gravitational field are small and can be disregardedfor most purposes but not when we are interested in the statisticalndashthermodynamical properties of gravity these fluctuations are preciselythe sources of the thermal behavior of the gravitational field as is thecase for any other thermal behavior

In a thermal context the Schwarzschild metric represents therefore onlythe coarse-grained description of a microscopically fluctuating geometryMicroscopically the gravitational field is nonstationary (because it inter-acts with nonstationary matter) and nonspherically symmetric (becausematter distribution is spherically symmetric on average only and noton individual microstates) Its microstate therefore is not given by theSchwarzschild metric but by some complicated time-dependent nonsym-metric metric

Horizon fluctuations Let us make the considerations above slightly moreprecise Consider first the classical description of a system at finite tem-perature in which there is matter the gravitational field and a black holeFoliate spacetime into a family of spacelike surfaces Σt labeled with atime coordinate t The intersection ht between the spacelike surface Σt

and the event horizon (the boundary of the past of future null-infinity)defines the instantaneous microscopic configuration of the event horizonat coordinate time t I loosely call ht the surface of the hole or the hori-zon Thus ht is a closed 2d surface in Σt As argued above generally this

306 Applications

microscopic configuration of the event horizon is not spherically symmet-ric Denote by gt the intrinsic and extrinsic geometry of the horizon htLet M be the space of all possible (intrinsic and extrinsic) geometriesof a 2d surface As t changes the (microscopic) geometry of the horizonchanges Thus gt wanders in M as t changes

Since the Einstein evolution drives the black hole towards theSchwarzschild solution (we can choose the foliation in such a way that)gt will converge towards a point gA of M representing a sphere of a givenradius A However as mentioned before exact convergence may be for-bidden by quantum theory and quantum effects may keep gt oscillatingin a finite region around gA

Which microstates are responsible for SBH Let us assume that (822)represents a true thermodynamical entropy associated with the black holeThat is let us assume that heat exchanges between the hole and theexterior are governed by SBH Where are the microscopical degrees offreedom responsible for this entropy located The microstates that arerelevant for the entropy are only the ones that can affect energy exchangeswith the exterior That is only the ones that can be distinguished fromthe exterior If I have a system containing a perfectly isolated box theinternal states of the box do not contribute to the entropy of the systemas far as the heat exchange of the system with the exterior is concernedThe state of matter and gravity inside a black hole has no effect on theexterior Therefore the states of the interior of the black hole are irrelevantfor SBH

To put it vividly the black-hole interior may be in one out of an in-finite number of states indistinguishable from the outside For instancethe black-hole interior may in principle be given by an infinite Kruskalspacetime on the other side of the hole there may be billions of galaxiesthat do not affect the side detectable by us The potentially infinite num-ber of internal states does not affect the interaction of the hole with itssurroundings and is irrelevant here because it cannot affect the energeticexchanges between the hole and its exterior which are the ones that de-termine the entropy We are only interested in configurations of the holethat have distinct effects on the exterior of the hole

Observed from outside the hole is completely determined by the geo-metric properties of its surface Therefore the entropy (relevant for thethermodynamical description of the thermal interaction of the hole withits surroundings) is entirely determined by the geometry of the black-holesurface namely by gt

The statistical ensemble We have to determine the ensemble of the micro-states gt over which the hole may fluctuate In conventional statistical


thermodynamics the statistical ensemble is the region of phase space overwhich the system could wander if it were isolated namely if it did notexchange energy with its surroundings Can we translate this condition tothe case of a black hole The answer is yes because we know that in GRenergy exchanges of the black hole are accompanied by a change in itsarea Therefore we must define the statistical ensemble as the ensembleof gt with a given value A of the area

To support the choice of this ensemble consider the following3 Theensemble must contain reversible paths only In the classical theory re-versible paths conserve the area because of the Hawking theorem Quan-tum theory does not change this because it allows area decrease only byemitting energy (Hawking radiance) namely violating the (counterfac-tual) assumption that defines the statistical ensemble that the systemdoes not exchange energy

We can conclude that the entropy of a black hole is given by the numberN(A) of states of the geometry gt of a 2d surface ht of area A The quantityS(A) = kB lnN(A) is the entropy we should associate with the horizon inorder to describe its thermal interactions with its surroundings

Quantum theory This number N(A) is obviously infinite in the classicaltheory But not in the quantum theory The situation is similar to the caseof the entropy of the electromagnetic field in a cavity which is infiniteclassically and finite in quantum theory To compute it we have to countthe number of (orthogonal) quantum states of the geometry of a two-dimensional surface with total area A The problem is now well definedand can be translated into a direct computation

Two objections I have concluded that the entropy of a black hole is de-termined by the number of the possible states of a 2d surface with area AThe reader may wonder if something has got lost in the argument doesthis imply that any surface has an associated entropy just because it hasan area Where has the information about the fact that this is a blackhole gone And where has the information about the Einstein equationsgone These objections have often been raised to the argument aboveHere is the answer

The first objection can be answered as follows Given any arbitrarysurface we can of course ask the mathematical question of how manystates exist that have a given area But there is no reason generally tosay that there is an entropy associated with the surface In a general sit-uation energy or more generally information can flow across a surface

3In this context it is perhaps worthwhile recalling that difficulties to rigorously justify-ing a priori the choice of the ensemble plague conventional thermodynamics anyway

308 Applications

The surface may emit heat without changing its geometry Therefore ingeneral the geometry of the surface and the number of its states havenothing to do with heat exchange or with entropy But in the specialcase of a black hole the horizon screens us from the interior and any heatexchange that we can have with the hole must be entirely determined bythe geometry of the surface It is only in this case that the counting ismeaningful because it is only in this case that the number of states ofa geometry of a given area corresponds precisely to the number of statesof a region which are distinguishable from the exterior To put it moreprecisely the future evolution of the surface of a black hole is completelydetermined by its geometry and by the exterior this is not true for anarbitrary surface It is because of these special properties of the horizonthat the number of states of its geometry determine an entropy

You can find out how much money you own by summing up the numberswritten on your bank account This does not imply that if you sum upthe numbers written on an arbitrary piece of paper you get the amountof money you own The calculation may be the same but an arbitrarypiece of paper is not a bank account and only for a bank account doesthe result of the calculation have that meaning Similarly you can makethe same calculation for any surface but only for a black hole because ofits special properties is the result of the calculation an entropy

The second objection concerns the role of the Einstein equations thatis the role of the dynamics This objection has been raised often but Ihave never understood it The role of the Einstein equations is preciselythe usual role that the dynamical equations always play in statisticalmechanics Generally the only role of the dynamics is that of definingthe energy of the system which is the quantity which is conserved if thesystem is isolated and exchanged when heat is exchanged The statisticalensemble is then determined by the value of the energy In the case ofa black hole it is the Einstein equations that determine the fact thatthe area governs heat exchange with the exterior of the hole If it wasnrsquotfor the specific dynamics of general relativity the area would not increasefor an energy inflow or decrease for energy loss Thus it is the Einsteinequations that determine the statistical ensemble


Above we have found on physical grounds what the entropy of a blackhole should be It is given by

SBH = kB lnN(A) (830)

where N(A) is the number of states that the geometry of a surface witharea A can assume It is now time to compute it


Let the quantum state of the geometry of an equal-time spacelike 3dΣt be given by a state |s〉 determined by an s-knot s The horizon is a 2dsurface S immersed in Σt Its geometry is determined by its intersectionswith the s-knot s

Intersections can be of three types (a) an edge crosses the surface (b) avertex lies on the surface (c) a finite part of the s-knot lies on the surfaceHere we are interested in the geometry as seen from the exterior of thesurface therefore the geometry we consider is more properly the limitof the geometry of a surface surrounding S as this approaches S Thislimit cannot detect intersections of the type (b) and (c) and we thereforedisregard such intersections

Let i = 1 n label the intersections of the s-knot with the horizonS Let j1 jn be the spins of the links intersecting the surface Thearea of the horizon is

A = 8πγhGsum

i


The s-knot is cut into two parts by the horizon S Call sext the externalpart The s-knot sext has n open ends that end on the horizon Fromthe point of view of an external observer a possible geometry of thesurface is a possible way of ldquoendingrdquo the s-knot A possible ldquoendrdquo ofa link with spin j is simply a vector in the representation space Hj Therefore a possible end of the external s-knot is a vector in otimesiHji Thus seen from the exterior the degrees of freedom of the hole appear asa vector in this space In the limit in which the area is large any furtherconstraint on these vectors becomes irrelevant The possible states areobtained by considering all sets of ji that give the area A and for each setthe dimension of otimesiHji Let us first assume that the number of possiblestates is dominated by the case ji = 12 In this case the area of a singlelink is

A0 = 4πγhGradic

3 (832)

This is the first value that was derived for the Immirzi parameter fromback hole states counting Later Domagala and Lewandowski realized[233] that the assumption that the entropy is dominated by spin 12 iswrong and found a higher value that was evaluated by Meissner [234]giving (829)

Hence there are

n =A

A0=

A

4πγhGradic

3(833)

intersections and the dimension of H12 is 2 so the number of states ofthe black hole is

N = 2n = 2A4πγhGradic

3 (834)

310 Applications

and the entropy is

SBH = kB lnN =1γ

ln 24π

radic3

kB

hGA (835)

This is the BekensteinndashHawking entropy (822) The numerical factoragrees with the Hawking value (826) and we get (827) if the Immirziparameter is fixed at the value

γ =ln 2πradic

3 (836)

A far more detailed account of this derivation is given in [238] where the Hilbertspace of the states of the black-hole surface is carefully constructed by quantizing atheory that has an isolated horizon as a boundary

Following [238] however Thiemann [20] derives an equation that looks like (836)but in fact differs from it by a factor of 2 Why this discrepancy The reason is thatThiemann observes that if the horizon is a boundary then in the absence of the ldquootherside of the horizonrdquo in the formalism we have jd = 0 in (6125) With jiu = ji andjid = 0 we have by definition of jt j

it = ji and therefore (taking γ = 1) the area of a

link with spin j entering the horizon contributes a quantum of area

A = 4πGhradic

j(j + 1) (837)

which is one-half the contribution to the area of a link of spin j that cuts a surface inthe bulk What is happening is that the area operator in a sense counts the area of asurface by summing contributions of links entering the two sides in the absence of oneside this does not work Therefore the area of the boundary is one-half the area of abulk surface infinitesimally close to it This is not very convincing on physical groundsof course The authors of [238] correct this discrepancy by effectively doubling the areaof the horizon This gives the same final result as in this book Thiemann has observedthat the need of this correction ldquoby handrdquo is an inconvenience of a formalism thattreats the horizon as a boundary One way out is to define the horizon area as the limitof the area of a bulk surface approaching the horizon

Notice that the black hole turns out to carry one bit of informationper quantum of area A0 This is precisely the ldquoit from bitrdquo picture thatJohn Wheeler suggested should be at the basis of black-hole physics in[165 166]

One remark before concluding The reader may object to the derivationabove (and the one in [238]) as follows the states that we have countedare transformed into each other by a gauge transformation Why thendo I consider them distinct in the entropy counting The answer to thisobjection is the following When we break the system into componentsgauge degrees of freedom may become physical degrees of freedom on theboundary The reason is that if we let the gauge group act independentlyon the two components it will act twice on the boundary A holonomyof a connection across the boundary for instance will become ill definedTherefore there are degrees of freedom on the boundary that are notgauge they tie the two sides to each other so to say


To illustrate this point let us consider two sets A and B and a groupG that acts (freely) on A and on B Then G acts on AtimesB What is thespace (AtimesB)G One might be tempted to say that it is (isomorphicto) AGtimesBG but a moment of reflection shows that this is not correctand the correct answer is

AtimesB

Gsim A

GtimesB (838)

(If G does not act freely over A we have to divide B by the stability groupsof the elements of A) Now imagine that A is the space of the states ofthe exterior of the black hole B the space of the states of the black holeand G the gauge group of the theory Then we see that we must not divideB by the gauge group of the surface but only by those internal gaugesand diffeomorphisms that leave the rest of the spin network invariant4


Is there a way of understanding the peculiar numerical value (836) Theminimal quantum of area that plays a central role in black-hole thermo-dynamics is the one with spin j = 12 which using (836) is

A12 = 8πhGγ

radic12

(12 + 1

)= 4 ln 2 hG (839)

Because of (823) a change of one such quantum of area implies a changeof energy

ΔE = ΔM =A12

32πGM=

ln 2 h

8πM (840)

If we use the Bohr relation ΔE = hω to interpret this as a quantumemitted by an oscillator with angular frequency ω then the quantumgravity theory indicates that in the system there should be somethingoscillating with a proper frequency

ω =ln 2

8πM (841)

As far as I know this frequency plays no role in the classical theoryHowever suppose that for some reason the minimal quantum of area thatplays a central role in black-hole thermodynamics was due not to j=12spins as above but to j = 1 spins Then the calculation above would beslightly different The minimal area is

A1 = 8πhGγradic

1(1 + 1) (842)

4Actually in [239] only the boundary degrees of freedom due to diff invariance weretaken into account while in [238] and here only the boundary degrees of freedomdue to internal gauge invariance are taken into account Perhaps by taking both intoaccount (836) could change

312 Applications

and since the spin-1 representation has dimension 3 the entropy is

S = ln(3AA1) =ln 3

8πhGγradic

2A (843)

This agrees with the BekensteinndashHawking entropy if

γ =ln 3

2πradic

2 (844)

which in turn fixes the minimal relevant quantum of area to be

A1 = 4 ln 3hG (845)

Using (823) and the Bohr relation we obtain the proper frequency

ω =ln 3

8πM (846)

Now very remarkably there is something oscillating precisely with thisfrequency in a classical Schwarzschild black hole In fact the frequency(846) is precisely the frequency of the most damped ringing mode ofa Schwarzschild black hole The calculation of this frequency from theEinstein equations is complicated It was first computed numerically thenguessed to be (846) on the basis of the numerical value and only recentlyderived analytically Quite remarkably the quantum theory of gravityappears to know rather directly about this frequency hidden inside thenonlinearity of the Einstein equations

This fact seems to support the idea that the ringing modes of the blackhole are at the roots of its thermodynamics On the other hand it isnot clear why we should not consider the spin j = 12 Several possi-bilities have been suggested including dynamical selection rules and anequispaced area spectrum Overall this intriguing observation raises morequestions than providing fully satisfactory answers


In 1995 Bekenstein and Mukhanov suggested that the thermal nature ofHawkingrsquos radiation may be affected by quantum properties of gravityquite dramatically They observed that in some approaches to quantumgravity the area can take only quantized values Since the area of theblack-hole surface is connected to the black-hole energy the latter is likelyto be quantized as well The energy of the black hole decreases whenradiation is emitted Therefore emission happens when the black holemakes a quantum leap from one quantized value of the energy to a lowerquantized value very much as atoms do A consequence of this picture


is that radiation is emitted at quantized frequencies corresponding tothe differences between energy levels Thus quantum gravity implies adiscrete emission spectrum for the black-hole radiation

This result is not physically in contradiction with Hawkingrsquos predictionof an effectively continuous thermal spectrum To understand this con-sider the black-body radiation of a gas in a cavity at high temperatureThis radiation has a thermal planckian emission spectrum essentially con-tinuous However radiation is emitted by elementary quantum emissionprocesses yielding a discrete spectrum The solution of the apparent con-tradiction is that the spectral lines are so dense in the range of frequenciesof interest that they give rise ndash effectively ndash to a continuous spectrum

However Bekenstein and Mukhanov suggest that the case of a blackhole may be quite different from the case of the radiation of a cavityThey consider a simple ansatz for the spectrum of the area that the areais quantized in multiple integers of an elementary area A0 Namely thatthe area can take the values

An = nA0 (847)

where n is a positive integer and A0 is an elementary area of the orderof the Planck area

A0 = αhG (848)

where α is a number of the order of unity The ansatz (847) agrees withthe idea of a quantum picture of a geometry made up of elementaryldquoquanta of areardquo Since the black-hole mass (energy) is related to thearea by (823) it follows from this relation and the ansatz (847) that theenergy spectrum of the black hole is given by

Mn =

radicnαh

16πG (849)

Consider an emission process in which the emitted energy is much smallerthan the mass M of the black hole From (849) the spacing between theenergy levels is

ΔM =αh

32πGM (850)

From the quantum mechanical relation E = hω we conclude that energyis emitted in frequencies that are integer multiples of the fundamentalemission frequency

ω =α

32πGM (851)

This is the fundamental emission frequency of Bekenstein and MukhanovLet us now assume that the emission amplitude is correctly given by

314 Applications

Hawkingrsquos thermal spectrum Then the full emission spectrum is givenby spectral lines at frequencies that are multiples of ω whose envelopeis Hawkingrsquos thermal spectrum Now this spectrum is drastically differ-ent than the Hawking spectrum Indeed the maximum of the planckianemission spectrum of Hawkingrsquos thermal radiation is around

ωH sim 282kBTH

h=

2828πGM

=282 times 4

αω asymp ω (852)

The fundamental emission frequency ω is of the same order of magnitudeas the maximum of the Planck distribution of the emitted radiation Itfollows that there are only a few spectral lines in the regions where emis-sion is appreciable The BekensteinndashMukhanov spectrum and the Hawk-ing spectrum have the same envelope but while the Hawking spectrum iscontinuous the BekensteinndashMukhanov spectrum is formed by just a fewlines in the interval of frequencies where emission is appreciable This isthe BekensteinndashMukhanov effect

Is this BekensteinndashMukhanov effect truly realized in LQG At firstsight one is tempted to say yes since the spectrum of the area in LQGgiven in (678) is quite similar to the ansatz (847) If we disregard the+1 under the square root in (678) we obtain the ansatz (847) and thusthe BekensteinndashMukhanov effect But the +1 is there and the differenceturns out to be crucial

Let us study the consequences of the presence of the +1 Consider asurface Σ in the present case the event horizon of the black hole Thearea of Σ can take only a set of quantized values These quantized valuesare labeled by unordered n-tuples of positive half-integers j = (j1 jn)of arbitrary length n

We estimate the number of area eigenvalues between the value A hG and the value A + dA of the area where we take dA much smallerthan A but still much larger than hG Since the +1 in (678) affects ina considerable way only the terms with low spin ji we can neglect it fora rough estimate It is more convenient to use integers rather than half-integers Let us therefore define pi = 2ji We must estimate the numberof unordered strings of integers p = (p1 pn) such that

sum

i=1n

pi =A

8πγhG 1 (853)

This is a well-known problem in number theory It is called the partitionproblem It is the problem of computing the number N of ways in whichan integer I can be written as a sum of other integers The solution forlarge I is a classic result by Hardy and Ramanujan [240] According tothe HardyndashRamanujan formula N grows as the exponent of the square


root of I More precisely we have for large I that

N(I) sim 14radic

3Ieπ

radic23I (854)

Applying this result in our case we have that the number of eigenvaluesbetween A and A + dA is

ρ(A) asymp e

radicπA

12γhG (855)

Using (823) we have that the density of the states is

ρ(M) asymp e

radic4G3γh

πM (856)

Now without the +1 term there is a high degeneracy due to the fact thatall states with the same value of

sumn pn are degenerate The presence of the

+1 term kills this degeneracy and eigenvalues can overlap only acciden-tally generically all eigenvalues will be distinct Therefore the averagespacing between eigenvalues will be the inverse of the density of statesand will decrease exponentially with the inverse of the square of the areaThis result is to be contrasted with the fact that this spacing is constantand of the order of the Planck area in the case of the ansatz (847) Itfollows that for a macroscopic black hole the spacing between energy lev-els is infinitesimal and the spectral lines are virtually dense in frequencyWe effectively recover in this way Hawkingrsquos thermal spectrum5

The conclusion is that the BekensteinndashMukhanov effect disappears if wereplace the naive ansatz (847) with the area spectrum (678) computedfrom LQG


Possible low-energy effects of LQG have been studied using a semiclassicalapproximation Gambini and Pullin have introduced the idea to study thepropagation of matter fields over a weave state taking expectation valuesof smeared geometrical operators For suitable weave states this may leadto the possibility of having quantum gravitational effects on the dispersionrelations In particular they have studied light propagation and pointedout the possibility of an intriguing birefringence effect Alfaro Morales-Tecotl and Urrita have developed this technique In particular for a

5Mukhanov subsequently suggested that discretization could still occur as a conse-quence of dynamics For instance transitions in which a single Planck unit of area islost could be strongly favored by the dynamics

316 Applications

fermion of mass m they derived dispersion relations between energy Eand momentum p of the general form

E2 = p2 + m2 + f(p lP) (857)

where the last Lorentz-violating term may be helicity dependent Thepossibility that LQG could yield observable effects indeed has raised muchinterest Suggestions that these effects may be connected to observableor even already observed effects have been put forward These regardcosmic-ray energy thresholds gamma-ray bursts pulsar velocities andothers

In fact the old idea that quantum gravitational effects are certainly un-observable at present has been strongly questioned in recent years Somehave even expressed the hope that we could be ldquoat the dawn of quantumgravity phenomenologyrdquo [241] It is too soon to understand if these hopeswill be realized but the possibility is fascinating and the developmentof LQG calculations that could relate to these possible observations is animportant direction of development

Lorentz invariance in LQG Lorentz-violating effects might not bepresent in LQG There are two reasons for expecting Lorentz violations inLQG One is that the short-scale structure of a macroscopically Lorentz-invariant weave might break Lorentz invariance However it is not clearwhether all weave states break Lorentz invariance or not A single spinnetwork state cannot be Lorentz invariant but this does not imply thata state which is a quantum superposition of spin network states cannoteither

The second reason which is often mentioned is the observation that aminimal length (or a minimal area) necessarily breaks Lorentz invarianceThe reason would be the following if an observer measures the minimallength 13P then a boosted observer will observe the Lorentz-contractedlength 13prime = γminus113P which is shorter than 13P and therefore 13P cannotbe a minimal length Here γ = 1

radic1 minus v2c2 is the LorentzndashFitzgerald

contraction factor This observation is wrong because it ignores quantummechanics

Length area and volume are not classical quantities They are quantumobservables If an observer measures the length 13P of some system thismeans that the system is in an eigenstate of the length operator A boostedobserver who measures the length of the same system is measuring adifferent observable Lprime which generally does not commute with L If thesystem is in an eigenstate of L generally it will not be in an eigenstateof Lprime Therefore there will be a distribution of probabilities of observingdifferent eigenvalues of Lprime The eigenvalues of Lprime will be the same as the


Σ

Σprime

Fig 81 The grey region represents the world-history of an object The twoarrows represent the worldlines of two observers The bold segments Σ and Σprime

are the intersections between the object world-history and the observersrsquo equal-time surfaces Two observers in relative motion measuring the lengths of thesame object measure the gravitational field on these two distinct surfaces Thegravitational field on Σ does not commute with the gravitational field on ΣprimeHence the two lengths do not commute

eigenvalues of L it is the expectation value of Lprime that will be Lorentzcontracted

The situation is the same as for the Lz component of angular momen-tum Consider a quantum system with total spin = 1 Say an observermeasures Lz and obtains the eigenvalue Lz = h Does this mean thata second observer rotated by an angle α will observe the eigenvalueLprimez = cosα h Of course not The second observer will still measure

Lprimez = 0plusmnh with a probability distribution such that the mean value

is Lprimez = cosα h States and mean values transform continuously with a

rotation but eigenvalues stay the same In the same fashion in a Lorentzboost states and values transform continuously while the eigenvalues staythe same

To understand why a length L and boosted length Lprime do not commuteconsider Figure 81 It represents the world-history of an object and thetwo lengths measured by two observers in relative motion Notice that thetwo observers measure the gravitational field on two distinct segments Σ

318 Applications

and Σprime with a time separation Since no quantum field operator com-mutes with itself at timelike separations clearly the two functions of thegravitational field e(x)

L =int

Σ

radic|e| and Lprime =

int

Σprime

radic|e| (858)

do not commute For a detailed discussion of this point and the actualconstruction of boosted geometrical operators see [242]

mdashmdash


Loop quantum cosmology has been mostly developed by Martin BojowaldThe absence of the initial singularity was derived in [243] and the sug-gestion about the possibility of a quantum-driven inflation presented in[244] For a review see [232] Loop cosmology is rapidly developing arecent introduction with up-to-date (2007) results and references is [235]

Another powerful recent application of loop quantum gravity is to re-solve the r = 0 singularity at the center of a black hole The idea hasbeen proposed by Leonardo Modesto [236] and has been independentlyconsidered and developed by Ashtekar and Bojowald [237]

The Hawking theorem on the growth of the black-hole area is in [245]Bardeen Carter and Hawking presented their ldquofour laws of black-hole mechanicsrdquo in [246] Bekenstein entropy was presented in [247] andHawking black-hole radiance in [248] For a review of the field see [28]

The first suggestions on the possibility of using the counting of thequanta of area of LQG to describe black-hole thermodynamics were pro-posed by Kirill Krasnov [249] For discussions on black-hole entropy inLQG I have followed here [239] The derivation with the horizon as aboundary is in [238] The idea that black-hole entropy originates fromthe fluctuations of the shape of the horizon was suggested by York [250]The relevance of the horizon surface degrees of freedom for the entropyhas since been emphasized from different perspectives see for instance[251] On the notion of an isolated horizon see [252] The relevance of theChernndashSimon boundary theory for the description of the horizon surfacedegrees of freedom was noticed in [253] the importance of the gauge de-grees of freedom on the boundary has been emphasized in [254] A recentreview including the new value of the Immirzi parameter is []

The relation between the ringing-mode frequencies and the spectrumhas been studied by S Hod in [255] and the fact that spin-1 area excitationsin LQG are related to the ringing-mode frequency has been pointed out


by Olaf Dreyer in [256] On the reason why spin-1 could contribute tothe horizon area more than spin-12 see [257] for a possible dynamicalselection rule and [227] for the role of the ordering of the area operator

The BekensteinndashMukhanov effect was presented in [258] (with α =4 ln 2) For a review of earlier suggestions in this direction see [259] Theargument presented here on the absence of this effect in loop gravityappeared in [260] The same result was derived in [173]

The possibility of a semiclassical description of the propagation over aweave that leads to estimates of quantum gravity effects was introducedin [261] for photons and developed in [262] for photons and neutrinosFor a review see [232] On the fact that violations of Lorentz invarianceare not necessarily implied by LQG see [242] and [263] On the difficultiesimplied by the breaking of Lorentz invariance in quantum gravity see also[264] On the possibility that Planck-scale observation might be withinobservation reach see [241 265 266]

9Quantum spacetime spinfoams

Classical mechanics admits two different kinds of formulations hamiltonian and la-grangian (I never understood why) Feynman realized that so does quantum mechanicsit can be formulated canonically with Hilbert spaces and operators or covariantly asa sum-over-paths The two formulations have different virtues and calculations thatare simple in one can be hard in the other Generically the lagrangian formalism issimpler more transparent and intuitive and keeps symmetries and covariance manifestThe hamiltonian formalism is more general more powerful and in the case of quantumtheory far more rigorous The ideal situation of course is to be able to master a theoryin both formalisms

So far I have discussed the hamiltonian formulation of LQG In this chapter I discussthe possibility of a lagrangian sum-over-paths formulation of the same theory Thisformulation has been variously called sum-over-surfaces state-sum and goes todayunder the name of ldquospinfoam formalismrdquo The spinfoam formalism can be viewed as amathematically well-defined and possibly divergence-free version of Stephen Hawkingrsquosformulation of quantum gravity as a sum-over-geometries

The spinfoam formalism is less developed than the hamiltonian version of loop the-ory Also while the general structure of the spinfoam models matches nicely with thehamiltonian loop theory the precise relation between the two formalisms (each of whichexists in several versions) has been rigorously established only in 3d But research iscurrently moving fast in this direction

The aim of the spinfoam formalism is to provide an explicit tool to compute tran-sition amplitudes in quantum gravity These are expressed as a sum-over-paths Theldquopathsrdquo summed over are ldquospinfoamsrdquo A spinfoam can be thought of as the worldsur-face swept out by a spin network Spinfoams are background-independent combinatorialobjects and do not need a spacetime to live in A spinfoam itself represents a spacetimein the same sense in which a spin network represents a space

The most remarkable aspect of the spinfoam approach is the fact that a surprisingnumber of independent research directions converge towards the same formalism Iillustrate some of these converging research paths below This convergence seems toindicate that the spinfoam formalism is a sort of natural general language for sum-over-paths formulations of general-relativistic quantum field theories

There are several good review articles on this subject ndash see the bibliographical notesat the end of the chapter I do not repeat here what is done in detail in other reviewsand I recommend the studious reader to refer to these reviews to complement the

320


introduction provided here since this is a subject which can be approached from avariety of points of view Here I focus on the overall significance of these models forour quest for a complete theory of quantum gravity


Sum-over-paths Consider a nonrelativistic one-dimensional quantum sys-tem Let x be its dynamical variable The propagator W (x t xprime tprime) of thesystem is defined in (510) As emphasized by Feynman W (x t xprime tprime)contains the full dynamical information about the quantum system It issimply related to the exponential of the Hamilton function S(x t xprime tprime)which as illustrated in Chapter 3 codes the classical dynamics of the sys-tem As illustrated in Chapter 5 in the relativistic formalism W (x t xprime tprime)can be obtained as the matrix element of the projection operator P be-tween states |x t〉 that are eigenstates of the operators corresponding tothe partial observables x and t

W (x txprime tprime) = 〈x t|P |xprime tprime〉K (91)

where K = L2[R2 dxdt] is the kinematical Hilbert space on which theoperators corresponding to the partial observables x and t are defined

A key intuition of Richard Feynman was that the propagator can beexpressed as a path integral

W (x t xprime tprime) simint

x(t)=x

x(tprime)=xprimeD[x(t)] eiS[x] (92)

in which the sum is over the paths x(t) that start at (xprime tprime) and endat (x t) and S[x] =

int ttprime L(x(t) x(t))dt is the action of this path There

are several techniques to define and manipulate this integral Feynmanand many after him have suggested that (92) can be taken as the basicdefinition of the quantum formalism This can therefore be based on asum of complex amplitudes eiS[x] over the paths x(t)

The idea of utilizing a sum-over-paths formalism in quantum gravity isold (see Appendix B) and has been studied extensively The idea is to tryto define a path integral over 4d metrics

intD[gμν(x)] eiSGR[g] (93)

In particular we can consider 3d metric gprime and a final 3d metric g andstudy the transition amplitude between these defined by

W [g gprime] =int

g|t=1=g

g|t=0=gprimeD[gμν(x)] eiSGR[g] (94)


where SGR is the action of the strip between t = 0 and t = 1 The specificvalue chosen for the coordinate t is irrelevant if the functional integralis defined in a diffeomorphism-invariant manner The techniques used inquantum mechanics and quantum field theory to give meaning to the func-tional integral however fail in gravity and the effectiveness of the pathintegral approach has long remained confined to crude approximationsThe reason is that nonperturbative definitions of the measure D[gμν(x)]leading to a complete theory are not known and perturbative definitionsaround a background metric lead to nonrenormalizable divergences

The situation has changed with loop gravity because of the discoveryof the discreteness of physical space To understand why a few generalconsiderations are in order

From the hamiltonian theory to the sum-over-paths Generally we do nothave a very good control of the direct definition of the Feynman sum-over-paths For many issues such as the choice of the measure the canonicaltheory often provides the best route toward the correct definition of the in-tegral Feynman of course derived the functional integral from the canon-ical theory in the first place He did so by writing the evolution operatoreminusiH0t as a product of small time-step evolution operators inserting res-olutions of the identity 1 =

intdx |x〉〈x| and taking the limit for the time

interval dt = (tminus tprime)N rarr 0 The functional integral is then defined asint

x(t)=x

x(tprime)=xprimeD[x(t)] eiS[x] equiv lim

Nrarrinfin

intdx1 dxNminus1

〈x| eminusiH0(tminustprime)

N |xNminus1〉〈xNminus1| eminusiH0(tminustprime)

N |xNminus2〉

〈x2|eminusiH0(tminustprime)

N |x1〉〈x1| eminusiH0(tminustprime)

N |xprime〉(95)

The canonical theory can therefore be used to construct a sum-over-pathsCan this be done in quantum gravity The problem is to obtain in quan-tum gravity the analog of the FeynmanndashKac formula which provides aprecise definition of the path integral So far this problem is unsolvedand a sum-over-paths formalism has not yet been rigorously derived fromthe canonical theory However we can still proceed in this direction usingthe following strategy

First we may write formal equations that indicate the general structurethat a sum-over-paths in quantum gravity should have That is we canderive the basics of the gravitational sum-over-paths formalism from thecanonical theory Indeed as we shall see in a moment a few loose for-mal manipulations show immediately that a generally covariant sum-over-paths formalism for quantum gravity must have quite peculiar properties


because of the discreteness of space This is done below Second we canstudy specific theoretical models having this general structure This isdone in the subsequent sections

Transition amplitudes are between spin networks A key observation isthat the quantity x in the argument of the propagator is not the classicalvariable but rather a label of an eigenstate of this variable The differenceis irrelevant so long as x is an observable with a continuous spectrum suchas the position But it becomes relevant if the spectrum of x is nontrivialFor instance consider a harmonic oscillator subjected to an external forceor to a small nonlinear perturbation Instead of asking for the amplitudeW (x txprime tprime) of measuring x given xprime ask for the probability amplitudeW (E t Eprime tprime) of measuring the (unperturbed) energy E This is given by

W (E t Eprime tprime) = 〈E|eminusiH0(tminustprime)|Eprime〉 (96)

where |E〉 is the eigenstate of the unperturbed energy with eigenvalueE (H0 is the nonrelativistic hamiltonian not the unperturbed one) Butthe eigenvalues E are quantized E = En and |n〉 = |En〉 ThereforeW (E t Eprime tprime) is defined only for the values of E = En in the spectrumThe argument of the propagator must be discrete energy levels not clas-sical energies Thus (96) only makes sense in the special form

W (n t nprime tprime) equiv W (En t Enprime tprime) = 〈n|eminusiH0(tminustprime)|nprime〉 (97)

Consider now the integral (94) This must define transition amplitudesbetween eigenstates of the 3-geometry We have seen in Chapter 6 that theeigenvalues of the 3-geometry are not 3d continuous metrics Rather theyhave a discrete structure and are labeled by spin networks Therefore thepropagator in quantum gravity must be a function of spin networks Forinstance we should study instead of (94) a quantity W (s sprime) giving theprobability amplitude of measuring the quantized 3-geometry describedby the spin network s if the quantized 3-geometry described by the spinnetwork sprime has been measured

Histories of spin networks Consider the propagator W (s sprime) As dis-cussed at the end of Section 74 we can express W (s sprime) in the form

W (s sprime) = 〈s|P |sprime〉K (98)

I now write some formal expressions that may indicate the way to expressW (s sprime) as a sum-over-paths Loosely speaking the operator P is theprojector on the kernel of the hamiltonian operator H If we tentatively


assume that H has a nonnegative spectrum we can formally write thisprojector as

P = limtrarrinfin

eminusHt (99)

because if |n〉 is a basis that diagonalizes H and En are the correspondingeigenvalues then

P = limtrarrinfin

sum

n

|n〉eminusEnt〈n| =sum

n

δ0En |n〉〈n| (910)

Since H is a function of a spatial coordinate x we can formally write

P = limtrarrinfin

prod

x

eminusH(x)t = limtrarrinfin

eminusint

d3xH(x)t (911)

Hence

W (s sprime) = limtrarrinfin

〈s|eminusint

d3xH(x)t|sprime〉K (912)

If we can define the 4d propagation generated by H in a diff-invariantmanner the limit is irrelevant and we can write again quite loosely

W (s sprime) = 〈s|eminusint 1

0dt

intd3xH(x)|sprime〉K (913)

We can now expand this expression in the same manner as in the right-hand side of (95) Inserting resolutions of the identity 1 =

sums |s〉〈s| we

obtain an expression of the form

W (s sprime) = limNrarrinfin

sum

s1sN

〈s|eminusint

d3xH(x) dt|sN 〉K 〈sN |eminusint

d3x H(x) dt|sNminus1〉K

〈s1|eminusint

d3xH(x) dt|sprime〉K (914)

For small dt we can expand the exponentials At fixed N the first termin the expansion (ex = 1 + 0(x)) produces a term equivalent to historieswith lower N Therefore we can view the sum as a sum-over-sequences ofspin networks where the sequences can have arbitrary lengths The resultis that the transition amplitude is not expressed as an integral over 4dfields but rather as a discrete sum-over-histories σ of spin networks

W (s sprime) =sum

σ

A(σ) (915)

A history is a discrete sequence σ = (s sN s1 sprime) of spin networks

The amplitude associated with a single history is a product of terms

A(σ) =prod

v

Av(σ) (916)



where v labels the steps of the history and Av(σ) is determined by thematrix elements

〈sn+1 |eminusint

d3xH(x)dt|sn〉K (917)

For small dt we can keep only the linear term in H in these matrix el-ements as Feynman did in (95) Now the action of the hamiltonianoperator H is given in (721) It is a sum over individual terms actingon the nodes of the spin network Therefore it has nonvanishing matrixelements only between spin networks sn+1 and sn that differ at one nodeby the action of H A typical term in the action (for instance) of H ona trivalent node is illustrated in Figure 72 or more schematically inFigure 91

Summarizing we can write W (s sprime) as a sum-over-paths of spinnetworks the paths are generated by individual steps such as the oneillustrated in Figure 72 the amplitude of each step is determined by thecorresponding matrix element of H the amplitude of the history is theproduct of the individual amplitudes of the steps

Spinfoams A history σ = (s sN s1 sprime) of spin networks is called a

spinfoam A more precise definition is given below A spinfoam admits anatural representation as follows Imagine a 4d space (representing coor-dinate spacetime) in which the graph of a spin network s is embeddedNow imagine that this graph moves upward along a ldquotimerdquo coordinateof the 4d space sweeping a worldsheet changing at each step under theaction of H Call ldquofacesrdquo the worldsurfaces of the links of the graph anddenote them f Call ldquoedgesrdquo the worldlines of the nodes of the graph anddenote them as e Figure 92 illustrates the worldsheet of a theta-shapedspin network

Since the hamiltonian acts on nodes the individual steps in the historyof a spin network can be represented as the branching of the edges whichlocally changes the number of nodes We call ldquoverticesrdquo the points whereedges branch and denote them as v

For instance an edge can branch to form three edges as in Figure 93representing the action of the hamiltonian constraint illustrated in Figure91


i

fs

is

f

Σ

Σ

Fig 92 The worldsheet of a spin network si on an initial surface Σi evolvingwithout intersections into the spin networks sf on the final surface Σf forminga spinfoam

Fig 93 A vertex of a spinfoam

The resulting worldsheet is illustrated in Figure 94 which represents aspinfoam with a single vertex A spinfoam with two vertices is representedin Figure 95 What we obtain in this manner is a collection of faces f meeting at edges e which in turn meet at vertices v The combinatorialobject Γ defined by the set of these elements and their adjacency relationsis called a ldquotwo-complexrdquo

A spin network is not defined solely by its graph but also by the coloringof its links (representations) and nodes (intertwiners) Accordingly thetwo-complex Γ determined by a sequence of spin networks is colored withirreducible representations jf associated with faces and intertwiners ieassociated with edges A spinfoam σ = (Γ jf ie) is a two-complex Γ withcolored faces and edges That is it is a two-complex with a representationjf associated with each face f and an intertwiner ie associated with eachedge e

92 Spinfoam formalism 327

v

si

sf

Σi

Σf

Fig 94 A spinfoam with one vertex

v2

v1

5

56

7

8

8

1

3

7

63

3

si

sf

s1

Σi

Σf

Fig 95 A spinfoam with two vertices


We are now ready to give a general definition of a spinfoam theory Thediscussion in the previous section indicates that a sum-over-paths formu-lation of quantum gravity can be cast in the form of a sum-over-spinfoamsof amplitudes given by products of individual vertex amplitudes


Consider a sum defined as follows

Z =sum

Γ

w(Γ)sum

jf ie

prod

v

Av(jf ie) (918)

The sum is over a set of two-complexes Γ and a set of representations andintertwiners j and i The function Av(jf ie) called the vertex amplitudeis an amplitude associated with each vertex v It is a function of the colorsadjacent to that vertex w(Γ) is a weight factor that depends only on thetwo-complex

It is often convenient to rewrite expression (918) in the extended form

Z =sum

Γ

w(Γ)sum

jf ie

prod

f

Af (jf )prod

e

Ae(jf ie)prod

v

Av(jf ie) (919)

with amplitudes Af and Ae associated to faces and edges as well al-though these can in principle be included in a redefinition of Av Formost models so far considered Af (jf ) is simply the dimension dim(jf ) ofthe representation jf Hence we have (using the notation σ = (Γ jf ie))

Z =sum

σ

w(Γ(σ))prod

f

dim(jf )prod

e

Ae(jf ie)prod

v

Av(jf ie) (920)

This is the general expression that we take as the definition of the spin-foam formalism

A choice of(i) a set of two-complexes Γ and associated weight w(Γ)(ii) a set of representations and intertwiners j and i(iii) a vertex amplitude Av(jf ie) and an edge amplitude Ae(jf ie)

defines a ldquospinfoam modelrdquo We shall study several of these models andtheir relation with gravity in the next section Generally speaking thechoice (iii) of the vertex amplitude corresponds to the choice of a specificform of the hamiltonian operator in the canonical theory

What is remarkable about expression (920) is that many very differentapproaches and techniques have converged precisely to this formula aswe shall see later on Perhaps an expression of this sort can be taken as ageneral definition of a background-independent covariant QFT formalism

In Table 91 I have summarized the terminology used to denote theelements of spin networks spinfoams and triangulations (which play arole later on)

921 Boundaries

The boundary of a spinfoam σ is a spin network s This follows easilyfrom the very way we have constructed spinfoams If σ is bounded by

93 Models 329

Table 91 Terminology

0d 1d 2d 3d 4d

spin network node link

spinfoam vertex edge face

triangulation point segment triangle tetrahedron four-simplex

the spin network s we write this as partσ = s The relation between (920)and transition amplitudes is obtained by summing over spinfoams with agiven boundary

W (s) =sum

partσ=s

w(Γ(σ))prod

f

dim(jf )prod

e

Ae(jf ie)prod

v

Av(jf ie) (921)

In particular if the spin network s is connected it can be interpreted asthe state of the gravitational field on the connected boundary of a space-time region For instance the boundary of a finite region of spacetime

If the boundary spin network is composed of two connected componentss and sprime we write

W (s sprime) =sum

partσ=scupsprimew(Γ(σ))

prod

f

dim(jf )prod

e

Ae(jf ie)prod

v

Av(jf ie) (922)

and we interpret the spinfoam model sum as a sum-over-paths definition ofthe transition amplitude between two quantum states of the gravitationalfield in analogy with (92)

So far the situation is not that we can compute W (s sprime) in the two for-malisms and prove the two to be equal In the spinfoam framework thereis uncertainty in the definition of the model but as we shall see tran-sition amplitudes can be computed (order by order) In the hamiltonianformalism on the other hand even disregarding the uncertainty in thedefinition of the hamiltonian we are not yet able to compute transitionamplitudes

93 Models

I shall now illustrate a few key examples of spinfoam models Each of theseis a realization of equation (920) Namely each of these is obtained from(920) by choosing a set of two-complexes a set of representations and in-tertwiners and vertex and edge amplitudes These models are relatedto different theories general relativity without and with cosmological


Table 92 Spinfoam models (λ cosmological constant Tr triangulation)

Model Class theory Two-complexes Representation Vertex

PonzanondashRegge 3d GR fixed dual 3d Tr SU(2) 6j

TuraevndashViro 3d GR +λ fixed dual 3d Tr SU(2)q 6jq

Ooguri (TOCY) 4d BF fixed dual 4d Tr SO(4) 15j

CranendashYetter 4d BF +λ fixed dual 4d Tr SO(4)q 15jq

BarrettndashCrane A cut-off 4d GR fixed dual 4d Tr simple SO(4) 15j

BarrettndashCrane B cut-off 4d GR fixed dual 4d Tr simple SO(4) 10jBC

GFT A 4d GR Feynman graphs simple SO(4) 15j

GFT B 4d GR Feynman graphs simple SO(4) 10jBC

constant in 3d and 4d and in BF theory (see Section 932) They arelisted in Table 92

These models form a natural sequence that has historically led to thepresent formulation of spinfoam quantum gravity This is represented bythe last of them constructed using an auxiliary field theory defined ona group manifold This represents a complete tentative sum-over-pathsformalism for quantum gravity in 4d The sequence of these models doesnot have just historical interest rather it allows me to introduce stepby step the ingredients that enter the complete model The models haveincreasing complexity Each of them has introduced and illustrates animportant peculiar aspect of the complete theory Here is a condensedpreview of the way each model has contributed to the construction ofthe formalism (terms and concepts will be clarified in the course of thechapter)

(i) The PonzanondashRegge theory is a quantization of gravity in 3d Itillustrates how a sum-over-paths in quantum gravity naturally takesthe form of a spinfoam model and why the gravitational vertexamplitude can be expressed in terms of simple invariant objects fromgroup representation theory For a long time it was assumed thatthese simple features were characteristic of 3d andor reflected thefact that the theory has no local degrees of freedom (is topological)The other models show that this assumption was wrong

93 Models 331

(ii) The Ooguri or TOCY (TuraevndashOogurindashCranendashYetter) model ex-tends the formalism to 4d

(iii) The BarrettndashCrane models extend the formalism to a theory withlocal degrees of freedom The key for doing this is the realizationthat GR can be obtained from a topological theory by adding certainconstraints and these constraints can be implemented in the modelas a restriction on the set of representations summed over

(iv) As soon as the model is no longer topological the sum over two-complexes becomes nontrivial A way to implement it is providedby the Group Field Theory (GFT )


3d GR Consider riemannian general relativity in three dimensions Thiscan be defined by minor modifications of the definition of 4d GR given atthe beginning of Chapter 2 In 3d the gravitational field e is a one-form

ei(x) = eia(x)dxa (923)

with values in R3 The spin connection ω is a one-form with values in theso(3) Lie algebra

ωi(x) = ωia(x)dxa (924)

and we denote its curvature two-form as Ri The action that defines thetheory is

S[e ω] =int

ei andR[ω]i (925)

Varying ω gives the Cartan structure equation De = 0 Varying e givesthe equation of motion R = 0 which implies that spacetime is flat1 Thatis locally in spacetime there is a single solution of the equations of motionup to gauge

The theory is nevertheless nontrivial if the space manifold has nontrivialglobal topology For instance there are distinct nonisometric flat tori Aflat torus is characterized say by its volume and its two radii which areglobal variables The dynamics of 3d general relativity is reduced to thedynamics of this kind of global variables A theory of this sort that has

1The EinsteinndashHilbert action S[g] =int

d3xradicgR where gab is the 3d metric and R its

Ricci scalar gives the same equations of motion as (925) but differs from (925) by asign when the determinant of eia is negative Hence quantizations of the two actionsmight lead to inequivalent theories Which of the two is to be called 3d GR is a matterof taste as they have the same classical solutions


Table 93 Relation between a triangulation Δ and its dual Δ in 3d (left) and4d (right) In italic the two-complex In parentheses adjacent elements

Δ3 Δlowast3

tetrahedron vertex (4 edges 6 faces)

triangle edge (3 faces)

segment facepoint 3d region

Δ4 Δlowast4

4-simplex vertex (5 edges 10 faces)

tetrahedron edge (4 faces)

triangle facesegment 3d regionpoint 4d region

no local degrees of freedom but only global ones is called a topologicaltheory2

Discretization I now give a concrete definition of the functional integral(93) for 3d GR by discretizing the theory To this end fix a triangulationΔ of the spacetime manifold It is more convenient to work with the dualΔlowast of the triangulation and in particular with the two-skeleton of ΔlowastThese are defined as follows see Table 93

To obtain Δlowast we place a vertex v inside each tetrahedron of Δ if two tetrahedrabound the same triangle e we connect the two corresponding vertices by an edge edual to the triangle e for each segment f of the triangulation we have a face f of Δlowastbounded by the edges corresponding to the triangles of Δ that are bounded by thesegment f finally to each point of Δ we have a 3d region of Δlowast bounded by the facesdual to the segments that are bounded by the point In 4d Δlowast is obtained by placinga vertex in each four-simplex and so on The collection of the sole vertices edges andfaces of Δlowast (with their boundary relations) is called the two-skeleton of Δlowast and isprecisely a two-complex

Let ge be the holonomy of ω along each edge e of Δlowast (The connectionω is in the algebra and the algebra of SO(3) is the same as the algebra ofSU(2) In defining the holonomy we have to decide whether we interpretω as an SO(3) or an SU(2) connection Let us choose the SU(2) inter-pretation That is we define ge = P expinte ωiτi isin SU(2) where τi arePauli matrices)

Let lif be the line integral of ei along the segment f of Δ Wechoose these as basic variables for the discretization The variables of the

2The expression ldquotopological field theoryrdquo is used with different meanings in the litera-ture In [267] for instance it is used to designate any diffeomorphism-invariant theorywith a finite or infinite number of degrees of freedom This is done to emphasize thesimilarities among all these theories and their difference from QFT on a backgroundHere there is no risk of underemphasizing this difference which is stressed throughoutthis book and I prefer to follow the common usage

93 Models 333

discretized theory will therefore be an SU(2) group element ge associatedwith each edge e of Δlowast and a variable lif in R3 associated with each seg-ment of Δ or equivalently to each face f of Δlowast Accordingly we candiscretize the action as

S[lf ge] =sum

f

lif tr[gfτi] (926)

wheregf = g

ef1 g

efn(927)

is the product of the group elements associated with the edges ef1 efn

that bound the face f If we vary this action with respect to lif we obtainthe equation of motion gf = 1 namely the lattice connection is flat Usingthis if we vary this action with respect to ge we obtain the equation ofmotion lif1 + lif2 + lif3 = 0 for the three sides f1 f2 f3 of each triangle Thisis the discretized version of the Cartan structure equation De = 0

Path integral Using this discretization we can define the path integralas

Z =int

dlif dge eiS[lf ge] (928)

where the measure on SU(2) is the invariant Haar measure The integralover lf gives immediately (up to an overall normalization factor that weabsorb in the definition of the measure)

Z =int

dgeprod

f

δ(gef1

gefn

) (929)

We can now expand the delta function over the group manifold using theexpansion

δ(g) =sum

j

dim(j) trRj(g) (930)

where the sum is over all unitary irreducible representations of SU(2)Inserting this in (929) and exchanging the sum and the product wehave

Z =sum

j1jN

prod

f

dim(jf )int

dgeprod

f

trRjf (gef1

gefn

) (931)

It is not difficult to perform the integrations over the group There isone integral per edge Since every edge bounds precisely three faces eachintegral is of the form


j3(U)γγprime = vαβγ vαprimeβprimeγprime (932)


where vαβγ is the (unique) normalized intertwiner between the represen-tations of spin j1 j2 j3 The reader should not confuse the symbol v thatdenotes vertices with the tensor vα1middotmiddotmiddotαn used to denote intertwiners

Each of the two invariant tensors on the right-hand side is associatedwith one of the two vertices that bound the edge (whose group elementis integrated over) Its indices get contracted with those coming from theother edges at this vertex A moment of reflection shows that at eachvertex we have four of these tensors that contract giving a function ofthe six spins associated with the six faces that bound the vertex

6j equiv(j1 j2 j6j4 j3 j5

)equiv

sum

α1α6

vα3α6α2 vα2α1α5 vα6α4α1 vα4α3α5 (933)

The pattern of the contraction of the indices reproduces the structure ofa tetrahedron if we (i) represent each 3-tensor vα1α2α3 using the repre-sentation (686) namely we write it as a trivalent vertex (ii) representindex contraction by joining legs (open ends) and (iii) indicate the rep-resentation to which the index belongs the 6j symbol is represented as

j1

j2 j3

j4 j6

j5

(934)

This function denoted 6j is a well-known function in the representa-tion theory of SU(2) It is a natural object that one can construct givensix irreducible representations It is called the Wigner 6j symbol see Ap-pendix A1

Bringing all this together we obtain the following form for the partitionfunction of 3d GR

ZPR =sum

j1jN

prod

f

dim(jf )prod

v

6jv (935)

This is the PonzanondashRegge spinfoam model Using the notation (686)we can write it in the form

ZPR =

sum

j

prod

f

dim(jf )prod

v

j1

j2

j3

j4 j6j5

(936)

93 Models 335

This expression has the general form (921) with the following choicesbull The set of two-complexes summed over is formed by a single two-

complex This is chosen to be the two-skeleton of the dual of a 3dtriangulation

bull The representations summed over are the unitary irreducibles ofSU(2) The intertwiners are trivial

bull The vertex amplitude is Av = 6jRemarkably these simple choices define the PonzanondashRegge quantizationof 3d GR

The fact that the vertex amplitude is simply the Wigner 6j symbol isperhaps surprising The Wigner 6j symbol is a simple algebraic constructof SU(2) representation theory The action of general relativity is a com-plicated expression coding the complexity of the gravitational interactionEven more remarkable is the fact that as we shall see this is not a strangecoincidence of the particularly simple form of GR in 3d rather the sameconnection between simple algebraic group theoretical quantities and thegravitational action holds in four dimensions as well This connection isone of the ldquomiraclesrdquo that nurtures the interest in the spinfoam approach

The original derivation the PonzanondashRegge ansatz The derivation aboveis not the original one of Regge and Ponzano It is instructive to men-tion also the general lines of the original derivation Regge introduced adiscretization of classical general relativity called Regge calculus definedover a fixed triangulation Consider the triangulation Δ of the spacetimemanifold and denote as f its segments (the choice of the letter f will beclear below) A gravitational field associates length lf to each segment f In turn these lengths lf can be taken as a discrete set of variables thatreplace the continuous metric The action of a given gravitational fieldcan be approximated by an action functional SRegge(lf ) of these lengthsAs the continuous action is an integral over spacetime the Regge action isa sum over the n-simplices v of the triangulation of the action of a singlen-simplex

S =sum

v

Sv (937)

We can then write a discretized version of (93) in the form

Z =int

dl1 dlN eiSRegge(lf ) (938)

Hence we can write

Z =int

dl1 dlNprod

v

eiSv(lf ) (939)

where Sv(lf ) is the Regge action of an individual n-simplex


Ponzano and Regge studied the integral (939) in the case of generalrelativity in 3d under one additional assumption that the length of eachlink can take only the discrete values

ln = jn jn =12 1

32 2 (940)

in units in which the Planck length lP = 1 This assumption is calledthe PonzanondashRegge ansatz Ponzano and Regge did not provide any jus-tification for it They introduced it just as discretization of the multipleintegral over lengths Notice that this is a second discretization in addi-tion to the triangulation of spacetime Its physical meaning was clarifiedmuch later and I will come back to it later on Under the PonzanondashReggeansatz (940) equation (939) becomes

ZPR =

sum

j1jN

prod

v

eiSv(jn) (941)

In 3d the Regge action of a 3-simplex (a tetrahedron) v can be writtenas a sum over the segments f in v as

Sv =sum

f

lfθf (lf ) (942)

where θf is the dihedral angle of the segment f that is the angle betweenthe outward normals of the triangles incident to the segment One canshow that this action is an approximation to the integral of the Ricciscalar curvature Under the PonzanondashRegge ansatz therefore Sv is afunction Sv(jn) of six spins j1 j6 (A tetrahedron has six edges)

The ldquomiraclerdquo GR dynamics in a symbol The surprising discovery ofPonzano and Regge was that the Wigner 6j symbol approximates theaction of general relativity More precisely they were able to show thatin the limit of large js we have the asymptotic formula

6j sim(eiSv(jn) + eminusiSv(jn)

)+

π

4 (943)

The term π4 does not affect classical dynamics The two exponentialterms in (943) are analogous to the two terms that we found in Sec-tion 523 see in particular the discussion at the end of that section Theclassical theory does not distinguish between forward and backward prop-agation in coordinate time and the path integral sums over the two Thetwo terms in (943) correspond to these two propagations Inserting (943)in (941) and fixing the normalization factors by imposing triangulationindependence (see below) we get (935) which is the expression that Pon-zano and Regge proposed as a discretized path integral for 3d quantumgravity

93 Models 337

Physical meaning of the PonzanondashRegge ansatz As noted at the begin-ning of this chapter the path integral defines transition amplitudes be-tween eigenstates of field operators not between classical fields ThePonzanondashRegge path integral (935) defines transition amplitudes betweentriangulated 2d surfaces where the links of the 2d triangulation have alength lf These lengths under the PonzanondashRegge ansatz are quan-tized Therefore the PonzanondashRegge ansatz is equivalent to the physicalassumption that length is quantized in 3d quantum gravity Now thislength quantization is not an ansatz but a result in loop quantum gravityIndeed it is not hard to see that in 3d the result that the area is quantizeddescribed in Chapter 6 translates into the quantization of length There-fore the key additional input provided by the PonzanondashRegge ansatz isphysically justified by loop gravity

In the previous derivation the discretization of the length derives fromthe following manipulation that we did over the integral First we inte-grated over the continuous variable lif obtaining a delta function (equa-tion (929)) Then second we expressed this delta function as a sum inequation (930) To understand the sense of this back and forward trans-formation consider the following example Let x be a variable in theinterval (minusπ π) We have

intdp eipx = 2πδ(x) (944)

and also sum

n

eipnx = 2πδ(x) (945)

where pn = n Here the distribution δ(x) is over functions on the compact(minusπ π) interval Therefore as long as we deal with functions on thiscompact interval we can replace an integral with a sum This is preciselywhat we did between (929) and (931) The compact space is the groupmanifold over which the holonomy takes values

The mathematical fact that the ldquoFourier componentsrdquo over a compactinterval are discrete is of course strictly connected to the physical factthat quantities conjugate to variables that take value in a compact spaceare quantized In fact this is the origin of the quantization of the area in4d and the quantization of length in 3d

Indeed we can view pn in (945) as the quantized value of the continuousvariable p in (944) conjugate to the compact variable x Similarly therepresentation je can be considered as quantization of the continuousvariable le More precisely we can identify lif with the generator of thespin-jf representation and identify the length |lf | of the segment f withthe square root of the quadratic Casimir operator of the representationjf


Sum-over-surfaces The sum over representations in (935) has a nice in-terpretation as a sum-over-surfaces Consider a triangulation Δ and aspecific assignment of representations jf to the faces of Δlowast Consider anedge e that bounds the three faces f f prime f primeprime The intertwiner on e is non-vanishing (and therefore the amplitude of the spinfoam is nonzero) onlyif the three representations jf jf prime jf primeprime satisfy the ClebschndashGordan rela-tions (A10)ndash(A11) Assuming this is the case associate 2jf elementaryparallel surfaces with each face f Join these elementary surfaces acrosseach edge The constraint (A10)ndash(A11) is precisely the condition underwhich the surfaces can be joined and there is only one way of joiningthem across each edge (There are a b and c surfaces crossing over fromjf to jf prime from jf prime jf primeprime and from jf to jf primeprime respectively and

2 jf = a + c 2 jf prime = a + b 2 jf primeprime = b + c) (946)

In this way we obtain surfaces without boundaries that wrap around thetwo-skeleton of Δlowast Each such surface carries a spin = 12

The sum over representations in (935) can therefore be viewed as asum over all the ways of wrapping these spin-12 surfaces around thetwo-skeleton of Δlowast The coloring jf of the face f is the total spin on theface that is half the number of spin-12 surfaces passing on f

Divergences bubbles The PonzanondashRegge model suffers from infrared di-vergences These have a peculiar structure which is reproduced in allspinfoam models

In the Feynman diagrams of ordinary QFT divergences are associatedwith loops that is closed curves within the Feynman diagram In theabsence of loops divergences do not appear because momentum conser-vation at the vertices constrains the value of the momenta on the internalpropagators In a spinfoam model the role of the internal momenta (in-tegrated over) is played by the representations (summed over) These areconstrained at edges by the ClebschndashGordan conditions at the edge whichplay the role of momentum conservation Accordingly divergences are notassociated with loops as in ordinary Feynman diagrams but rather withldquobubblesrdquo A bubble is a collection of faces f in Δlowast that form a closedtwo-surface If we increase each of the representations associated with thefaces forming the bubble by the same amount j then the relations (A10)ndash(A11) remain satisfied because if jf jf prime jf primeprime satisfy (A10)ndash(A11) so dojf + j jf prime + j jf primeprime

The sum-over-surfaces picture described in the previous section givesus a clear understanding of the structure of the resulting divergences wecan always add an arbitrary number of spin-12 surfaces wrapped aroundthe bubble

93 Models 339

The minimal bubble configuration is the ldquoelementary bubblerdquo This isformed by four triangular faces connected to each other as in a tetrahe-dron This elementary bubble appears if four vertices are connected toone another in Δlowast and equivalently if four tetrahedra are connectedto one another in Δ This configuration can be obtained for instance bydecomposing a single tetrahedron into four tetrahedra by picking an inte-rior point and connecting it to the four vertices Within the representation(929) this configuration gives the contribution

Abubble =int

dg1 dg6 δ(g1g2gminus16 )δ(g3g4g6)δ(g4g1g

minus15 )δ(g2g3g5) (947)

to Z (see (934)) Integration is immediate giving the divergent expression

Abubble = δ(0) (948)

The same result can be obtained in the sum over representations Forinstance assume the spins of all faces connected to the elementary bubblevanish Then

Abubble =sum

j

(dim(j))4 (6j(j j j 0 0 0))4 (949)

From the definition (933)

6j(j j j 0 0 0) = vα1α2 vα1α3 vα2α3 (950)

The normalized intertwiner vα1α2 is vα1α2 = δα1α2(dim(j))12 giving

6j(j j j 0 0 0) = dim(j)minus32 dim(j) = dim(j)minus12 (951)

Inserting this in (949) yields

Abubble =sum

j

(dim(j))2 (952)

which is equal to (948) as is clear from (930)In the Regge triangulation picture this divergence corresponds to the

case in which there is a point of the Regge lattice connected to the restof the lattice by four segments We can then make the lengths of thesefour segments arbitrarily large Geometrically this describes a long andnarrow ldquospikerdquo emerging from the 3d (discretized) manifold

Notice that this is an infrared divergence since it regards large jf snamely large lengths It is not related to ultraviolet divergences absentin the theory Ponzano and Regge have developed a renormalization pro-cedure to divide away this divergence


TuraevndashViro An appealing way to get rid of the divergence of the modelis to replace the representation theory of the group SU(2) with the repre-sentation theory of the quantum group SU(2)q with q chosen to be a rootof unity Both the dimension and the Wigner 6j symbol are well definedfor this quantum group The irreducible representations of this quantumgroup are finite in number and therefore the sum is finite This sum iscalled the TuraevndashViro invariant Furthermore it can be argued that thedeformation of the group from SU(2) to SU(2)q corresponds simply tothe addition of a cosmological term to the classical action Namely to theaction

S[e ω] =int

εijk ei and (R[ω]jk minus λ

3ej and ek) (953)

A good discussion on this issue can be found in [231]

Triangulation independence A remarkable result by Ponzano and Reggeis triangulation independence the quantity Z defined by (935) dependson the global topology of the 3-manifold chosen but not on Δ namelynot on the way the manifold is triangulated In particular whether wechoose a minimal triangulation or a very fine triangulation the partitionfunction does not change

The proof of triangulation independence is tricky in the PonzanondashReggecase where a renormalization procedure is needed On the other handit is a clean theorem in the TuraevndashViro case where everything is finiteThe TuraevndashViro sum is defined in terms of a triangulation of a compact3-manifold but it is a well-defined 3-manifold invariant

Triangulation independence is the consequence of the fact that 3d GRis a topological theory Since the theory has no local degrees of freedomwe are not really losing degrees of freedom in the discretization Usuallydiscretization loses the short-scale degrees of freedom but there are noshort-scale degrees of freedom in this theory Hence the triangulated ver-sion of the theory has the same number of degrees of freedom as the fullfield theory and refining the triangulation does not change this number

The triangulation independence of the expression (935) is an importantmathematical property It has inspired much mathematical work But wedo not expect triangulation independence to hold for 4d GR which isnot a topological theory since it has local degrees of freedom Thereforefrom the point of view of the problem of quantum gravity triangulationindependence is a less interesting aspect of the PonzanondashRegge theory Itwill not survive the generalization that we will study later on

932 BF theory

Let us extend the above construction to four dimensions As a first stepwe do not consider GR but a much simpler 4d theory called BF theory

93 Models 341

which is topological and is a simple extension to 4d of the topological3d GR Consider BF theory for the group SO(4) This is defined by twofields a two-form BIJ with values in the Lie algebra of SO(4) and anSO(4) connection ωIJ The action is a direct generalization of (925)

S[Bω] =intBIJ and F IJ [ω] (954)

where F is the curvature two-form of ω I use here the notation F insteadof R as this is standard in this context (and is the origin of the nameldquoBFrdquo of theory)

We can discretize this theory and define a path integral following thevery same steps we took for 3d GR We obtain precisely equation (929)again and from this equation (931) with the sole difference that the sumis over irreducible representations of SO(4) and that the two-complex isthe two-skeleton of the dual of a four-dimensional triangulation

Again it is not difficult to perform the integrations over the group Butnow every edge bounds four faces not three We have then instead of(932) the integral


j3(U)γγprime Rj4(U)δδprime =

sum

i

vαβγδi viαprimeβprimeγprimeδprime (955)

Here the index i labels the orthonormal basis vαβγδi in the space of the in-tertwiners between the representations of spin j1 j2 j3 j4 We have there-fore a sum over intertwiners for each edge in addition to the sums overrepresentations for each face At each vertex we have now ten represen-tations (because the vertex bounds ten faces) and five intertwiners Thesedefine the function

15j equiv A(j1 j10 i1 i5)

equivsum

α1α10

vα1α6α9α5i1

vα2α7α10α1i2

vα3α7α8α2i3

vα4α9α7α3i4

vα5α10α8α4i5

(956)

where the indices an are in the representation jn The pattern of thecontraction of the indices reproduces the structure of (the one-skeletonof ) a four-simplex

131313

vi1

vi2

vi3

vi5

vi4

α1

α2

α3

α4

α5

α6

α8

α9 α7

α10

(957)


The function (956) is denoted 15j The name comes from the factthat if the group is SU(2) the intertwiners can be labeled with the repre-sentation of the internal virtual link hence this function depends on 15spins

Combining everything we obtain the following form for the partitionfunction of 4d BF theory

ZTOCY =sum

jf ie

prod

f

dim(jf )prod

v

15jv (958)

which we can write as

ZTOCY =sum

jf ie

prod

f

dim(jf )prod

v

1313

i1

i2

i3

i5

i4

j1

j2

j3

j4

j5

j6j8j9 j7

j10(959)

This expression is called the TOCY model (from Turaev Ooguri Craneand Yetter) or the Ooguri model It has the general form (921) with thefollowing choices


bull The representations summed over are the unitary irreducibles ofSO(4)

bull The vertex amplitude is Av = 15jThese choices define the quantization of 4d BF theory

Divergences and CranendashYetter model As for the PonzanondashRegge modelthe sum (958) suffers from infrared divergences The typical divergence isagain associated with a ldquobubblerdquo The elementary bubble is now formedby five vertices of Δlowast connected to each other In Δ this corresponds tofive four-simplices connected to each other Namely the configuration ob-tained by subdividing a single four-simplex into five four-simplices addinga single point inside and connecting it to the vertices We can computethe degree of this divergence as we did for the PonzanondashRegge modelstarting from the expression (929) In the present case the pattern ofthe integration variables and the delta functions are given by (957) Thisgives

Abubble =int

dg1 dg10 δ(g1g2gminus16 )δ(g2g3g

minus17 )δ(g3g4g

minus18 )δ(g4g5g

minus19 )

times δ(g5g1gminus110 )δ(g1g7g8)δ(g2g8g9)δ(g3g9g10)δ(g4g10g7)δ(g5g6g8)

(960)

93 Models 343

Integration is immediate giving the divergent expression

Abubble = δ4(0) (961)

The divergences can be cured by passing to the quantum group SO(4)qThe definition of the quantum 15j symbol requires care but can begiven The resulting model is finite and triangulation independent It iscalled the CranendashYetter model Its classical limit can be shown to berelated to BF theory plus a cosmological term


There is a surprising duality between the PonzanondashRegge and TOCYmodels on the one hand and certain peculiar QFTs defined over a group(Group Field Theory or GFT) on the other This duality will play animportant role in what follows I illustrate it here in the 4d case

Consider a real field φ(g1 g2 g3 g4) over the cartesian product of fourcopies of G = SO(4) Require that φ is symmetric and SO(4) invariantin the sense

φ(g1 g2 g3 g4) = φ(g1g g2g g3g g4g) (forall g isin SO(4)) (962)

Consider the QFT defined by the action

S[φ] =12

int 4prod

i=1

dgi φ2(g1 g2 g3 g4)

+λ

5

int 10prod

i=1

dgi φ(g1 g2 g3 g4)φ(g4 g5 g6 g7)φ(g7 g3 g8 g9)

timesφ(g9 g6 g2 g10)φ(g10 g8 g5 g1) (963)

The potential (fifth-order) term has the structure of a 4-simplex if werepresent each of the five fields in the product as a node with 4 legs ndashone for each gi ndash and connect pairs of legs corresponding to the sameargument we obtain (the one-skeleton of) a 4-simplex see Figure 96

The remarkable fact about this field theory is the following The Feyn-man expansion of the partition function of the GFT

Z =int

Dφ eminusS[φ] (964)

turns out to be given by a sum over Feynman graphs

Z =sum

Γ

λv[Γ]

sym[Γ]Z[Γ] (965)

where the amplitude of a Feynman graph is

Z[Γ] =sum

jf ie

prod

f

dim(jf )prod

v

15jv (966)


6

g

g

g10 8

9

g2g3

1

5

7

f

g

f

g2

g3

4g

4g

1

g g

f

f

g

f f

f

g

Fig 96 The structure of the kinetic and potential terms in the action

Here Γ is a Feynman graph v[Γ] the number of its vertices and sym[Γ]its symmetry factor The Feynman graphs Γ of the theory have a naturaladditional structure as two-complexes The Feynman integrals over mo-menta are discrete sums (because the space on which the QFT is definedis discrete) over SO(4) representations jf and over intertwiners ie associ-ated with faces and edges of the two-complex Furthermore for each giventwo-complex Γ the Feynman sum over momenta is precisely the TOCYmodel defined on that two-complex Indeed the right-hand side of (966)is equal to the right-hand side of (958) That is

Z[Γ] = ZTOCY (967)

The proof of these results is a straightforward application of perturbativeexpansion methods in QFT and the use of the PeterndashWeyl theorem thatallows us to mode-expand functions on a group in terms of a basis givenby the unitary irreducible representations of the group This is done insome detail below

Mode expansion First expand the field φ(g1 g2 g3 g4) into modes and rewrite theaction in terms of these modes (in ldquomomentum spacerdquo) Consider a square integrablefunction φ(g) over SO(4) The PeterndashWeyl theorem tells us that we can expand this

function in the matrix elements R(j)αβ(g) of the unitary irreducible representations j

φ(g) =sum

j

φjαβ R

(j)αβ(g) (968)

The indices α β label basis vectors in the corresponding representation space Accord-ingly the field can be expanded into modes as

φ(g1 g4) =sum

j1j4

φj1j4α1β1α4β4

R(j1)α1β1

(g1) R(j4)α4β4

(g4) (969)

Using the invariance (962) under the left group action we can write

φ(g1 g4) =

int

SO(4)

dg φ(gg1 gg4) (970)

93 Models 345

Substituting here the mode expansion (969) and using the expression (955) for theintegral of the product of four group elements we can write

φ(g1 g4) =sum

j1j4

φj1j4α1α4i

R(j1)α1β1

(g1) R(j4)α4β4

(g4) viβ1β4 (971)

where we have definedφj1j4α1α4i

= φj1j4α1β1α4β4

viβ1β4 (972)

We use the quantities φj1j4α1α4i

as the Fourier components of the field Written in termsof these the kinetic term of the action reads

1

2

int 4prod

i=1

dgi φ2(g1 g4) =

1

2

sum

j1j4

sum

i

φj1j4 iφj1j4 i (973)

The interaction term becomes

λ

5

int 10prod

i=1

dgiφ(g1 g2 g3 g4)φ(g4 g5 g6 g7)φ(g7 g3 g8 g9)

timesφ(g9 g6 g2 g10)φ(g10 g8 g5 g1)

=λ

5

sum

j1j10

sum

i1i5

φj1j2j3j4 i1φj4j5j6j7 i2φj7j3j8j9 i3φj9j6j2j10 i4φj10j8j5j1 i5

timesA(j1 j10 i1 i5) (974)

where A is given in (956)

Feynman graphs The partition function is given by the integral over modes

Z =

int [Dφj1j4 i

]eminusS[φA] (975)

We expand Z in powers of λ The gaussian integrals are easily computed giving thepropagator

P j1j4 i jprime1jprime4 iprime equiv 〈φj1j4 i φjprime1j

prime4 iprime〉 =

1

4

sum

σ

δj1j

primeσ(1) δ

j4jprimeσ(4) δiiprime (976)

where σ are the permutations of 1 2 3 4 There is a single vertex of order five whichis

〈φj1j2j3j4 i1 φj10j8j5j1 i5〉 = λ A(j1 j10 i1 i5) (977)

The set of Feynman rules one gets is as follows First we obtain the usual overallfactor λv[Γ]sym[Γ] (see for instance [268] page 93) Second we represent each of theterms in the right-hand side of the definition (976) of the propagator by four parallelstrands as in Figure 97 carrying the indices at their ends We can represent thepropagator itself by the symmetrization of the four strands In addition edges e arelabeled by a representation je

The Feynman graphs we get are all possible ldquo4-strandrdquo five-valent graphs wherea ldquo4-strand graphrdquo is a graph whose edges are collections of four strands and whosevertices are those shown in Figure 98 Each strand of the propagator can be connectedto a single strand in each of the five ldquoopeningsrdquo of the vertex Orientations in thevertex and in the propagators should match (this can always be achieved by changinga representation to its conjugate) Each strand of the 4-strand graph goes throughseveral vertices and several propagators and then closes forming a cycle A particular


Fig 97 The propagator can be represented by a collection of four strands eachcarrying a representation

Fig 98 The structure of the vertex generated by the Feynman expansion

strand can go through a particular edge of the 4-strand graph more than once Cyclesget labeled by the simple representations of the indices For each graph the abstractset formed by the vertices the edges and the cycles forms a two-complex in whichthe faces are the cycles The labeling of the cycles by simple representations of SO(4)determines a coloring of the faces by spins Thus we obtain a colored two-simplexnamely a spinfoam

Edges e are labeled by an intertwiner with index ie Vertices v contribute a factorλ times A which depends on the ten simple representations labeling the cycles thatgo through the vertex and on the five intertwiners basis elements in Kie

labeling theedges that meet at v The weight of two-complex Γ is then given by (966)3

3The following remarks may be useful for a reader who wants to compare the formulasgiven with others in the literature For each given permutation namely for each two-complex and for each edge e I have chosen a fixed orthonormal basis in the spaceof the intertwiners associated with the edge e Alternatively one can choose a basisassociated with a decomposition of the four faces adjacent to e in two couples If wedo so the propagator (976) contains also a matrix of change of basis It reads

P j1j4 i jprime1jprime4 iprime equiv 〈φj1j4 Λ φjprime1j

prime4 iprime〉 =

1

4

sum

σ

δj1j

primeσ(1) δ

j4jprimeσ(4) M j1j4

σiiprime

(978)

where the matrix Mσ is given by a 6j symbol Each edge contracts two vertices sayv and vprime and contributes a matrix Mσ This is the matrix of the change of basis fromthe intertwiner basis used at v and that used at vprime Since here I have fixed a basis ofintertwiners for every e once and for all for each fixed two-complex the matrix Mσ isautomatically included in the vertex amplitude and the propagator is the identity

93 Models 347

Transition amplitudes Next consider SO(4)-invariant transition ampli-tudes in the GFT That is let f [φ] be an SO(4)-invariant polynomialfunctional of the field and consider the amplitude

W (f) =int

Dφ f [φ] eminusS[φ] (979)

and its expansion in Feynman graphs

W (f) =sum

Γ

λv[Γ]

sym[Γ]Zf [Γ] (980)

It is simplest to construct all SO(4)-invariant polynomial functionals ofthe field in momentum space namely as functions of the Fourier modesφj1j4α1α4i

defined in (969)ndash(971) To obtain an SO(4) scalar we must con-tract the indices αn We start with n field variables φj1j4

α1α4i and contract

the indices pairwise in all possible manners The resulting functional isdetermined by a four-valent graph Γ giving the pattern of the indices con-traction colored with representations jl on the links and the intertwinersin on the nodes The set of data s = (Γ jl in) forms precisely a spinnetwork In other words the SO(4)-invariant observables of the GFT arelabeled by spin networks

Writing n1 n4 to indicate four links adjacent to the node n wehave

fs[φ] =prod

n

φjn1 jn4αn1 αn4 in

prod

l

δl1l2 (981)

where ni = l1 (or ni = l2) if the ith link of the node n is the outgoing (oringoing) link l

For instance the spin network s = (Γ j1 j4 i1i2) on a graph with two nodesconnected by four links determines the function of the field

fs[φ] =sum

α1α4

φj1j4α1α4i1

φj1j4α1α4i2

(982)

I leave to the reader the simple exercise to show that expressions of this kind correspondto coordinate space expressions such as

fs[φ] =

int prod

l

dglprod

n

φ(gn1 gn4)fs(gni) (983)

where the spin network function is

fs(gni) =prod

n

vαn1 αn4in

prod

l

Rjl(gl)αl1αl2 (984)


All transition amplitudes of the GFT can therefore be expressed interms of the spin network amplitudes

W (s) =int

Dφ fs[φ] eminusS[φ] (985)

Consider the Feynman expansion of these As usual in feynmanology theexpectation value of a polynomial of order n in the fields has n externallegs In the Feynman expansion of the GFT we have in addition toconsider the faces I leave to the reader the simple and instructive exerciseto show that these turn out to be bounded precisely by the links of thespin network In other words the Feynman expansion of W (s) is given by

W (s) =sum

partΓ=s

λv[Γ]

sym[Γ]Zs[Γ] (986)

where the sum is over all two-complexes bounded by s and the amplitudeof the Feynman graph is

Zs[Γ] =sum

jf ie

prod

f

dim(jf )prod

v

15jv (987)

The coloring on the external nodes and links is determined by s and notsummed over

Expressing this the other way around the spinfoam sum at a fixed spinnetwork boundary s is determined by the GFT expectation value (985)

As far as the TOCY model is concerned the duality I have just illus-trated is not particularly useful BF theory has a large invariance groupthat implies that the theory is topological This implies that the corre-sponding spinfoam model is triangulation invariant up to a divergentfactor Therefore the GFT amplitudes are given by divergent sums ofequal terms On the other hand the spinfoamGFT duality will play acrucial role in the context of the BC models

934 BC models

It is time to begin the return towards 4d GR There is a strict relationbetween SO(4) BF theory and euclidean GR If we replace BIJ in (954)by

BIJ = εIJKL eK and eL (988)

we get precisely the GR action We can therefore identify the B field ofBF theory with the gravitational field e and e The constraint (988) on Bsometimes called the Plebanski constraint transforms BF theory into GRCan we implement the constraint (988) directly in the quantum theory

93 Models 349

An immediate consequence of (988) is

εIJKL BIJBKL = 0 (989)

In 3d the continuous variable lif can be identified with the generators ofthe representation jf In 4d it is the variable BIJ

f that we can identifywith a generator of the SO(4) representation If we do so (989) becomessimply a restriction on the representations summed over

Recall that the Lie algebra of SO(4) is sim su(2)oplussu(2) The irreduciblerepresentations of SO(4) are therefore labeled by couples of representa-tions of SU(2) namely by two spins j = (j+ jminus) If BIJ is the generatorof SO(4) the generators of the two SU(2) groups are

Biplusmn = P i

plusmnIJBIJ (990)

where the projectors P iplusmnIJ are the euclidean analogs of the projectors

defined in (217) That is

P iplusmnjk =

12εijk P i

plusmn0j = minusP iplusmnj0 = plusmn1

2δij (991)

SO(4) has two Casimirs the scalar Casimir

C = BIJBIJ = |B|2 (992)

and the pseudo-scalar Casimir

C = εIJKL BIJBKL (993)

This last one is precisely the quantity which is constrained to zero by(989) The SO(4) representations in which the pseudo-scalar Casimir(989) vanishes are called ldquosimplerdquo or ldquobalancedrdquo The value of C in therepresentation (j+ jminus) is easy to compute because

εIJKL BIJBKL = Bi+B+i minusBi

minusBminusi = j+(j+ + 1) minus jminus(jminus + 1) (994)

From (989) and (994) we have j+ = jminus The representations that satisfythis constraint namely those of the kind (j+ jminus) = (j j) are the simplerepresentations they are labeled by a single spin j Some mathematicalfacts about representation theory of SO(4) and about simple representa-tions are included in Appendix A3

This suggests that quantum GR can be obtained by restricting thesum over representations in (958) to the simple representations Thisprocedure defines a class of models denoted the BC models (from JohnBarrett and Louis Crane who introduced the use of simple representationsin the spinfoam formalism)


Relation with loop gravity In the discretization of BF theory we havediscretized the B field which is a two-form by assigning a variable Bf

to each triangle f of the triangulation Bf can be taken to be the surfaceintegral of B on f We can discretize the gravitational field e which is aone-form by assigning a variable es to each segment s of the triangulationes can be taken to be the line integral of e along the segment s Equation(988) then relates the variable Bf on a triangle with the variables es ontwo sides of the triangle The scalar Casimir (992) can be expressed interms of the gravitational field using (988) obtaining

C = |e and e|2 (995)

which is the square of the area of the triangle Hence the Casimir of therepresentation jf gives the area of the triangle f The spin jf can thereforebe interpreted as the quantum number of the area of the triangle f

Consider the case in which the triangulated manifold has a boundaryand the triangle f belongs to the boundary In the two-complex picture fis a face dual to the f triangle and jf is associated with this face The facef cuts the boundary along a link which is one of the links of the boundaryspin network The color of the link is jf This link intersects once and onlyonce the triangle f Hence we conclude that the representation associatedwith the link that intersects the triangle f is the quantum number thatdetermines the area of the triangle f But this is precisely the result thatwe have obtained in hamiltonian theory in Chapter 6

The boundary states of a BC spinfoam model is a spin network whoselinks carry quanta of area labeled by a spin j This is precisely the struc-ture of the states of hamiltonian loop quantum gravity Spinfoam modelsand hamiltonian LQG ldquotalkrdquo very nicely to each other

The other constraints Equation (989) does not imply (988) On theother hand the system formed by (989) and the two equations

εIJKL BIJμνB

KLνρ = 0 (996a)

4εIJKL BIJμνB

KLρσ

εμνρσεIJKL BIJμνB

KLρσ

= εμνρσ (996b)

(no sum over repeated spacetime indices in the first equation) does Itfollows that if we consider BF theory plus the additional constraint (996)the resulting theory has precisely all the solutions of GR

Two remarks First GR has far more solutions than BF theory which is topologi-cal The fact that adding a constraint on B increases the number of solutions shouldnot surprise us In BF theory B plays the role of Lagrange multiplier enforcing thevanishing of the curvature By constraining B in the action we reduce the number of

93 Models 351

independent components of the Lagrange multiplier Hence we get less constraints onthe curvature Hence more solutions If you jail some guards more thieves go free

Second the system (996) has other classes of solutions beside (988) In particular

BIJ = minusεIJKL eK and eL and BIJ = plusmneI and eJ (997)

The first simply redefines orientation The others give a topological term in the actionthat has no effect in the classical equations of motion I refer the reader to the literaturefor a full discussion of this point

The constraint (996a) can be implemented in the quantum theory byidentifying the couple of variables BIJ

μνBKLνρ that share an index with gen-

erators of the representations associated with faces that share an edgeIn turn for the constraint (996b) we identify the two BKL

ρσ BKLνρ variables

without common indices with generators of the representations associatedto opposite faces of a tetrahedron

There are different ways of discretizing these constraints that have beenconsidered in the literature yielding different BC models There are alsodifferent possible choices of the face and edge amplitudes Af and Ae thathave been considered in the literature The situation is still unclear as towhich variants correspond to discretized GR I shall not enter into thedetails of motivations of the different choices here (see [19] for a detaileddiscussion) Rather I simply illustrate some of the models referring tothe literature for their motivation

BCA The simplest of the BC models denoted BCA model (BarrettndashCrane model version A) is simply obtained by choosing Ae = Af = 1 Itis given by

ZBCA =sum

simple jf

sum

ie

prod

v

131313

i1

i2

i3

i5

i4

j1

j2

j3

j4

j5

j6j8j9

j7

j10

(998)

BCB A second model denoted BCB (BarrettndashCrane model version B)assumes that the intertwiners are constrained by (996) to the form

i(aaprime)(bbprime)(ccprime)(ddprime)BC =

sum

j

(2j + 1) vabfvfcd vaprimebprimef prime

vfprimecprimedprime (999)

where indices in an SO(4) representation are given by couples of indices inan SU(2) representation and the indices f and f prime are in the representationj This is called the BC intertwiner See [269] for details on the relation


between this intertwiner and (996) The BC intertwiner has the propertyof being formed by a simple virtual link in any decomposition It is theunique state with this property Using the representation (686) it is givenby

ibc =sum

j (2j + 1)

j

j (9100)

Choosing Ae = 1 we obtain the sum

ZBCB =sum

simple jf

prod

f

dim(jf )prod

v

131313

iBC

iBC

iBC

iBC

iBC

j1

j2

j3

j4

j5

j6j8j9

j7

j10

(9101)

The vertex amplitude

A(j1 j10) =

1313

iBC

iBC

iBC

iBC

iBC

j1

j2

j3

j4

j5

j6j8j9 j7

j10

(9102)

=sum

i1i5

1313

i1

i2

i3

i5

i4

j1

j2

j3

j4

j5

j6j8j9 j7

j10

1313

i1

i2

i3

i5

i4

j1

j2

j3

j4

j5

j6j8j9 j7

j10

depends on ten spins and is called a 10j symbol Thus we write (9101)as

ZBCB =sum

simple jf

prod

f

dim(jf )prod

v

10j (9103)

Therefore the BCB model has the general form (921) with thefollowing choices

93 Models 353


bull The representations summed over are the simple unitary irreduciblerepresentations of SO(4) All intertwiners are fixed to be iBC

bull The vertex amplitude is Av = 10j

These choices define a tentative quantization of 4d riemannian GR on afixed two-complex namely with a cut-off at the high-frequency modesClearly a fixed two-complex can accommodate only a finite number ofdegrees of freedom and cannot capture all the degrees of freedom of thetheory which are infinite

BCC A variant of the model B is of particular interest since as we shallsee it is perturbatively finite This is obtained by adding an edge ampli-tude to the BCB model

ZBCC =sum

simple jf

prod

f

dim(jf )prod

e

Ae(je1 je4)prod

v

10j (9104)

where Ae is a function of the four representations je1 je4 associatedto the four faces bounded by the edge e and is defined as follows LetHj1j4 be the tensor product of the four representations j1 j4 andH0

j1j4its invariant subspace Then Ae is the ratio of the dimensions of

these spaces

Ae(j1 j4) =dimH0

j1j4

dimHj1j4

(9105)

As we shall see below this amplitude emerges naturally in the group fieldtheory (GFT) context

At present it is not clear which of these variants or others is the mostphysically interesting one Interpreting the choice as a choice of sum-over-paths measure and imposing diff invariance Bojowald and Perez haveobtained indications in favor of certain models [270] In [271] the differentstatistical properties of these models have been analyzed numericallyhowever it is not yet clear which are the ldquocorrectrdquo statistical propertiesto be expected

Geometrical interpretation of the Plebanski constraints The constraintson the representations and intertwiners that define the BC models canbe given a geometrical interpretation In fact they were first obtainedfrom an independent set of considerations based on this geometrical in-terpretation Consider a tetrahedron embedded in R4 Denote vectors in


R4 as v = (vI) I = 1 4 Label the four vertices of the tetrahedron asvi i = 1 4 A vector

v(ij) = vj minus vi (9106)

describes the edge (ij) of the tetrahedron The triangle (ijk) can be de-scribed by the ldquobivectorrdquo

vIJ(ijk) = vI(ij) vJ(jk) minus vJ(ij) v

I(jk) (9107)

often written asv(ijk) = v(ij) and v(jk) (9108)

For instance the triangle determined by the three vertices (123) is de-scribed by the bivector

vIJ(123) = vI1 vJ2 minus vI2 vJ1 + vI2 vJ3 minus vI3 vJ2 + vI3 vJ1 minus vI1 vJ3 =sum

i

εijkvIj vJk

(9109)

obtained by inserting (9106) in (9107) The area of the triangle (ijk) isthe norm of the bivector

AIJ(ijk) = vIJ(ijk)v(ijk)IJ (9110)

From the definition

εIJKL vIJ(ijk)vKL(ijk) = 0 (9111)

εIJKL vIJ(ijk)vKL(ijl) = 0 (9112)

Furthermore consider a four-simplex embedded in R4 with vertices viwhere i = 1 5 This defines the four tetrahedra (ijkl) and the tentriangles (ijk) Then for two triangles sharing a vertex i

εIJKL vIJ(ijk)vKL(ilm) =

sum

i

εijklmεIJKL vIj vJk v

Kl vLm (9113)

which is independent of i Hence we can write

εIJKL vIJ(ijk)vKL(ilm)

sumi ε

ijklm εIJKL vIj vJk v

Kl vLm

= εjklm (9114)

Notice the remarkable similarity of the bivector equations (9111)(9112) and (9114) with the Plebanski constraints (989)ndash(996) Thebivector equations can be interpreted as a discretization of the Plebanskiconstraints Expressed the other way around the Plebanski constraintscan be interpreted as the requirement that the B field of the BF theoryis an infinitesimal area element of elementary triangles in spacetime

93 Models 355

Quantum tetrahedron The historical path to the BC models has been the onedescribed above I remember long hours of discussion with Louis Crane searching un-successfully for a way to implement the Plebanski constraint (989) within his TOCYmodel But in the seminal work [269] which brilliantly solves the problem in terms ofsimple representations no reference is made to the Plebanski constraint and the rela-tion between BF and GR The paper is based on a description of the intrinsic metricdegrees of freedom of a single tetrahedron embedded in R4 and a ldquoquantizationrdquo ofthese degrees of freedom as I now describe

Consider a single tetrahedron embedded in R4 View it as a physical sys-tem whose dynamical variables are given by its geometry We can expectthat the quantum properties of this system are described by a quantumstate space H and dynamical variables be represented by operators in thisstate space The geometry of the tetrahedron can be described in terms ofthe bivectors vIJ(ijk) defined in the previous section Thus bivectors will berepresented by operators on H This construction defines a sort of ldquoquan-tum tetrahedronrdquo Since SO(4) acts naturally on the bivectors we expectthat H carries a representation of SO(4) Since classical bivectors trans-form in the adjoint representation we expect the quantum operators todo the same It follows that bivector operators vIJ(ijk) are the infinitesimalgenerators of a representation j of SO(4) in H The quadratic expression(9110) giving the area of the triangle is precisely one of the two Casimirsof j Hence H will carry a representation jijk such that its Casimir isthe area of the triangle (ijk) The other Casimir of SO(4) is given by(9111) constrained to vanish Hence H will contain only representationsin which this Casimir vanishes These are the simple representations Thesum (9101) can then be constructed as a sum-over-states in the Hilbertspace where the bivector operators live

This method of arriving at the BC models has the shortcoming of hidingits relation with GR as well as with conventional quantization proceduresOn the other hand it has the virtue of opening an entirely new interestingperspective on quantum spacetime The convergence of different ways ofthinking of a model which shed light on each other is always valuable

In fact the idea of the quantum tetrahedron as an elementary systemcan perhaps be taken seriously on physical grounds Ordinary QFT on abackground has two natural physical interpretations The two correspondto two different choices of families of observables For instance in free elec-tromagnetism we can measure the electric field or the magnetic field andinterpret the theory as a theory for a continuous field Alternatively wecan measure energies and momenta and interpret the theory as describ-ing particles moving in spacetime the photons There is no contradictionof course between the two descriptions for the same reason that there isno contradiction between the continuity of the elongation of a harmonicoscillator and the discreteness of its energy


Similarly we can construct the QFT starting from the quantization ofthe classical field theory and deriving the existence of the particles Or wecan construct the QFT starting from the particles define the quantumtheory of a single particle then the ldquomany-particlerdquo quantum theory ofan arbitrary number of particles and so on The two roads yield the sametheory as is well known In an interacting QFT nontrivial dynamics canbe expressed by simple interaction vertices between the particle states

In GR loop quantization shows that space has a granular structureat short scale Space can be thought of as made up of individual quantaof space (which can be connected to each other) These quanta of spaceare like the particles eigenstates of certain measurable quantities It isthen not unreasonable to think that we can reinterpret quantum GR asa ldquomany-particlerdquo theory built up from a quantum theory of a singlequantum of space The mathematics of the ldquoquantum tetrahedronrdquo canperhaps be seen as a first step in this direction If we take this pointof view then dynamics can be expressed by simple interaction verticesbetween these ldquoparticlerdquo states For instance an elementary vertex suchas the one in Figure 93 can be interpreted as one quantum of space(connected to three other quanta not represented) decaying into threequanta of space (connected to each other and to the three other quantanot represented) and so on Spacetime is then an history of interactionsof a variable number of quanta of space


The remaining step to arrive at a model with some chance of describingquantum GR is to implement a sum over two-complexes so that theinfinite number of degrees of freedom of the theory could be captured

Notice that in order to capture all degrees of freedom we do not have the option ofrefining the triangulation (or the two-complex) as one does in lattice QCD The reasonis that there is in fact nothing to refine no parameter such as the lattice spacing oflattice QCD which can be set to zero

In fact the cut-off introduced by the BarrettndashCrane models is not an ultravioletcut-off The theory does not have an ultraviolet sector because there are no degreesof freedom beyond Planck scale Rather it is a sort of infrared cut-off in the sensethat a fixed triangulation cannot capture configurations that can be written on a largertriangulation (a triangulation with more n-simplices) The sum includes arbitrary largegeometries because it includes arbitrary high jf (not in the quantum group case) Buton a fixed Δ a large geometry can be represented only by large jf and not by smalljf over a larger triangulation Clearly this restriction reduces dramatically the class ofcontinuous fields that the spinfoam can approximate

We could think of defining the model in the limit of large Δ This is certainly aninteresting direction to explore On the other hand in summing over colorings of a largeΔ we have to include configurations in which the representations are trivial except ona subset Δprime of Δ The amplitude of this configuration can be viewed as associated withΔprime rather than Δ Hence we naturally fall back to a sum-over-triangulations

93 Models 357

For fixed boundary conditions yielding a classical geometry of volume Nl4P it may bereasonable to assume that triangulations with a number of four-simplices much largerthan N would not contribute much Hence the expansion in the size of the triangulationmight be of physical interest

How do we sum over two-complexes The problem is to select a classof two-complexes to sum over and to fix the relative weight Now theduality that I have illustrated above in Section 933 provides precisely aprescription for summing over two-complexes It is therefore natural totake a dual formulation of the BC models as a natural ansatz for thecomplete sum over two-complexes But is there a dual formulation of theBC models or is duality a feature of the much simpler topological BFmodel

Remarkably a dual formulation of the BC models exists BC modelsare obtained from the TOCY models by restricting representations to thesimple ones This restriction implements the constraints that transformBF theory into GR In the dual picture the sum over representations isobtained as an expansion of the field over the group in modes A genericfield can be expanded in a sum over all unitary irreducible representationsHow can we pick a field whose expansion contains only simple represen-tations The answer turns out to be easy

Pick a fixed SO(3) subgroup H of SO(4) Then the following holds Afield φ(g) on SO(4) is invariant under the action of H namely satisfies

φ(g) = φ(gh) forallh isin H (9115)

if and only if its mode expansion contains only simple irreducible repre-sentations This is an elementary result described in Appendix A3

Consider the field theory defined in Section 933 It is useful to slightlysimplify this notation First write the action in the shorthand notation

S[φ] =12

intφ2 +

λ

5

intφ5 (9116)

Second instead of demanding the field to satisfy (962) we can take anarbitrary field φ not necessarily satisfying (963) and use the projectionoperator PG defined by

PGφ(g1 g2 g3 g4) =int

SO(4)dg φ(g1g g2g g3g g4g) (9117)

We can also define the projector GP acting on the left

GPφ(g1 g2 g3 g4) =int

SO(4)dg φ(gg1 gg2 gg3 gg4) (9118)

Define now the projector PH on the simple representations

PHφ(g1 g2 g3 g4) =int

H4dh1 dh4 φ(g1h1 g2h2 g3h3 g4h4) (9119)


GFTTOCY The action

S[φ] =12

int(PGφ)2 +

λ

5

int(PGφ)5 (9120)

for a generic field is equivalent to the action (963) and yields the TOCYmodel as discussed in Section 933 Now by simply inserting the projec-tor PH into this action we obtain the following surprising results

GFTA Consider the action

SA [φ] =12

int(GPPHφ)2 +

λ

5

int(GPPHφ)5 (9121)

The Feynman expansion of the partition function of this theory gives

ZA =int

Dφ eminusS[φ] =sum

Γ

λv[Γ]

sym[Γ]ZA [Γ] (9122)

The amplitude of a Feynman graph turns out to be precisely the partitionfunction (998) of the BCA model where the model is over the two-complex determined by the Feynman graph That is

ZA [Γ] =sum

simple jf

prod

v

15j = ZBCA (9123)

GFTB The action

SB [φ] =12

int(PGPHφ)2 +

λ

5

int(PGPHφ)5 (9124)

gives the partition function (9101) of the BCB model

ZB [Γ] =sum

simple jf

prod

f

dim(jf )prod

v

10j = ZBCB (9125)

GFTC The action

SC [φ] =12

int(PGφ)2 +

λ

5

int(PHPGφ)5 (9126)

yields the partition function (9104) of the BCC model

ZC [Γ] =sum

simple jf

prod

f

dim(jf )prod

e

Ae(je1 je4)prod

v

10j

= ZBCC (9127)

The derivation of these relations is a rather straightforward applicationof the mode expansion described in Section 933 I leave it as a good

93 Models 359

exercise for the reader It can be found in the original papers quoted atthe end of the chapter

Now in the TOCY case the sum over Feynman graphs is trivial di-vergent and without physical motivation It is trivial in the sense that allterms are equal due to triangulation invariance It is divergent becausewe sum an infinite number of equal terms It has no physical motiva-tion because all the degrees of freedom of the classical theory are alreadycaptured by a finite triangulation

In the case of the BC models on the other hand a choice of a fixedtwo-complex reduces the number of degrees of freedom of the theoryTherefore a sum over two-complexes is necessary if we hope to definequantum GR and the Feynman expansion provides precisely such a sumSince the BC models are not triangulation invariant the sum is not triv-ial each two-complex contributes in a different manner to the sum Thesum over two-complexes defined by the GFT defines in these cases a newspinfoam model where the number of degrees of freedom are not cut offWe denote these as GFT spinfoam models In particular we denote thesum defined in (9122) as the group field theory version A or GFTA andthe corresponding ones in cases B and C as the GFTB and GFTC

What about finiteness Remarkably the GFTC appears to be finiteat all orders in λ The proof has been completed up to certain degeneratetwo-complexes which however are likely not to spoil the result Thereforethe GFTC model or some variant of it can be taken as a tentative ansatzfor a covariant definition of transition amplitudes in euclidean quantumgravity

In particular we can consider expectation values of spin networks as in(985)

W (s) =int

Dφ fs[φ] eminus12

int(P

Gφ)2 + λ

5

int(P

GPHφ)5 (9128)

These quantities are well defined and are likely to be convergent at everyorder in λ We can tentatively interpret them as transition amplitudes ofeuclidean quantum gravity


The fact that expression (9128) provides a finite tentative definition ofquantum gravitational transition amplitudes is certainly exciting but theGFT models described above are all euclidean There are two possibledirections to recover the physical lorentzian theory from here

SO(3) and SO(21) lorentzian GFT One direction for defining lorentzianamplitudes is to study the lorentzian analogs of these models These canbe obtained by simply replacing SO(4) with the Lorentz group SO(3 1)Let φ(g1 g4) be a field on [SO(3 1)]4 Define the projectors PG and


PH as in (9117) and (9119) To define the projector H there are twonatural choices for the subgroup H sub SO(3 1) The first is to take it tobe a fixed H = SO(3) subgroup of SO(3 1) This is the subgroup thatkeeps a chosen timelike vector invariant The second is to take it as anH = SO(2 1) subgroup of SO(3 1) This is the subgroup that keeps achosen spacelike vector invariant Consider the action

SH [φ] =12

int(PGφ)2 +

λ

5

int(PGPHφ)5 (9129)

for the two cases The Feynman expansion of the partition function de-fines two lorentzian spinfoam sums denoted the SO(3) and the SO(2 1)lorentzian GFT respectively

The set of the representations that appear in these spinfoam sums isdetermined by the mode expansion of the field This implies that therepresentations summed over are the unitary irreducible representationsThese are infinite-dimensional for SO(3 1) which is noncompact and la-beled also by a continuous variable Therefore in the lorentzian spinfoammodels sums over internal indices are replaced by integrals and the spin-foam sum contains an integral over a continuous set of representationsStill the technology developed above extends to this case quite well Thereis an extensive literature on this to which I refer the interested readersee bibliographical notes at the end of the chapter Most of the featuresof the euclidean theory persist in the lorentzian case Most remarkablyfiniteness results have been extended to the lorentzian SO(3) GFT

Unitary representations of SO(3 1) fall naturally into two classesspacelike and timelike ones distinguished by the sign of the Casimir rep-resenting the square of the area The action S

SO(3)above gives rise to a

sum over just the timelike representations while the action SSO(21)

givesrise to a sum over both kinds of representations The kind of the represen-tation associated with a triangle f gives a spacelike or timelike characterto the triangles of the triangulation This is physically appealing but sev-eral aspects of this issue are unclear at this time For instance the sign ofthe square of the area appears to be opposite to what one would expecton the basis of the hamiltonian theory This is an intriguing open problemthat I signal to the reader

Analytic continuation The second possible direction for the constructionof the physical theory is to define the lorentzian transition amplitudesfrom the euclidean ones by analytic continuation as is done in conven-tional QFT Standard theorems relating euclidean transition amplitudesto a lorentzian QFT are grounded on Poincare invariance and do not ex-tend to gravity However this does not imply that the project of definingphysical transition amplitudes from the euclidean theory by some form of


analytic continuation must necessarily fail In particular analytic continu-ation in the time coordinate is likely to be completely nonappropriate in abackground-independent context but analytic continuation in a physicaltime might be viable I discuss this possibility below in Section 941


A spinfoam model can be used to compute an amplitude W (s) associatedto any boundary spin network s How can we relate these amplitudes tophysical measurements

Relation with hamiltonian theory The spinfoam formalism has formalsimilarities with lattice gauge theory The interpretation of the two for-malisms however is quite different In the case of lattice theories thediscretized action depends on a parameter the lattice spacing a Thephysical theory is recovered as a is taken to zero In this limit the dis-cretization introduced by the lattice is removed In particular we canapproximate continuous boundary fields in terms of sequences of latticediscretization In gravity on the other hand there is no lattice spacingparameter a in the discretized action Therefore there is no sense in thea rarr 0 limit The discrete structure of the spinfoams must reflect actualfeatures of the physical theory In particular boundary states are givenby spin networks not by continuous field configurations approximated bysequences of discretized fields

The hamiltonian theory of Chapter 6 provides a physical interpreta-tion for the boundary spin network s Spin networks are eigenstates ofarea and volume operators and can therefore be interpreted as describ-ing the result of measurements of the geometry of a 3d surface Suchmeasurements do not correspond to complete observables since they donot commute with the hamiltonian operator However they correspondto partial observables a notion explained in Chapter 3 Therefore theyare still described by operators in the kinematical quantum state spaceas argued in Chapter 5 In particular the discrete spectrum of these oper-ators can be interpreted as a physical prediction on the possible outcomeof a physical measurement of these observables As recalled at the be-ginning of this chapter transition amplitudes do not in general dependon classical configurations they depend on quantum eigenstates This iswhy we expect quantum gravity transition amplitudes to be functions ofspin networks and not of continuous three-geometries The fact that thespinfoam formalism yields precisely such functions of spinfoams is there-fore satisfying There is thus a strong and encouraging consistency be-tween the physical picture of nonperturbative quantum gravity providedby spinfoam theory and by hamiltonian LQG


It would be very good if we were able to translate directly between thetwo formalisms A sketch of a formal derivation of a spinfoam sum fromthe loop formalism was given at the beginning of this chapter Expressingthis the other way around it would be very interesting to reconstruct indetail the hamiltonian Hilbert space as well as kinematical and dynam-ical operators of the loop theory starting from the covariant spinfoamdefinition of the theory At present neither of these two paths is undercomplete control The first would amount to a derivation of a FeynmanndashKac formula (see for instance [272]) valid in the diffeomorphism-invariantcontext

The second would amount at an extension to the diffeomorphism-invariant context of the Wightman and OsterwalderndashSchrader reconstruc-tion theorems [273] There are two ways in which we can derive the Hilbertstate space from a spinfoam model One is to identify K with a linear clo-sure of the set of the boundary data This is the philosophy I have usedabove The other is to directly reconstruct H from the amplitudes W (s)using the GelfandndashNeimarkndashSiegal construction This approach has beendeveloped in [274] (see also [275])

At an even more naive level there are several gaps between the hamil-tonian loop theory and the spinfoam models that have been discussedso far These concern the role of the selfdual connection the role of theSO(3 1) rarr SO(3) gauge-fixing used in the hamiltonian framework thefact that in the spinfoam models so far considered there are only four-valent nodes the fact that in the GFTB and C models there is no freeboundary intertwiner variable associated with the node the eventual roleof the quantum group deformation in the hamiltonian theory [228] andothers All these aspects of the relation between the two formalisms needto be clarified before being able to cleanly translate between the two

On the one hand there is much latitude in the definition of the hamil-tonian operator as explained in Chapter 7 In general covariant methodsdeal more easily with symmetries and interaction vertices have a simpleform in the covariant picture For instance compare the complexity of thefull QED hamiltonian with the simplicity of the QED single vertex whichcompactly summarizes all hamiltonian interaction terms The spinfoamformalism could suggest the correct form of the hamiltonian operator

On the other hand hamiltonian methods are more precise and rigorousthan covariant sums-over-paths As we have seen the hamiltonian pic-ture provides a clean and well-motivated physical interpretation for theboundary spin network states as well as a general justification of the dis-creteness of spacetime Therefore the two formalisms shed light on eachother and their relation needs to be studied in detail

(Note added in the paperback edition The precise equivalence of thespinfoam and hamiltonian LQG formalisms has been proven rigorouslyby Karim Noui and Alejandro Perez See Bibliographical notes below)



Finally I sketch here a way to compute the Minkowski vacuum statefrom the spinfoam formalism following the general theory described inSection 54 This is the first step to define particle states Consider athree-sphere formed by two ldquopolarrdquo in and out regions and one ldquoequato-rialrdquo side region Let the matter + gravity field on the three-sphere besplit as ϕ = (ϕout ϕin ϕside) Fix the equatorial field ϕside to take thespecial value ϕRT defined as follows Consider a cylindrical surface ΣRT

of radius R and height T in R4 as defined above Let Σin (and Σout) be a(3d) disk located within the lower (and upper) basis of ΣRT and let Σside

be the part of ΣRT outside these disks so that ΣRT = ΣincupΣoutcupΣside LetgRT be the metric of Σside and let ϕRT = (gRT 0) be the boundary field onΣside determined by the metric being gRT and all other fields being zeroGiven arbitrary values ϕout and ϕin of all the fields including the metricin the two disks consider W [(ϕout ϕin ϕRT )] In writing the boundaryfield as composed of three parts ϕ = (ϕout ϕin ϕside) we are in fact split-ting K as K = Hout otimesHlowast

in otimesHside Fixing ϕside = ϕRT means contractingthe covariant vacuum state |0Σ〉 in K with the bra state 〈ϕRT | in HsideFor large enough R and T we expect the resulting state in Hout otimes Hlowast

in

to reduce to the Minkowski vacuum That is (again braket mismatch isapparent only)

limRTrarrinfin

〈ϕRT |0Σ〉 = |0M〉 otimes 〈0M| (9130)

For a generic in configuration and up to normalization

ΨM[ϕ] = limRTrarrinfin

W [(ϕϕin ϕRT )] (9131)

(Below I shall use a simpler geometry for the boundary) These formu-las allow us to extract the Minkowski vacuum state from a euclideanspinfoam formalism n-particle scattering states can then be obtained bygeneralizations of the flat-space formalism and if this is well defined byanalytic continuation in the single variable T

Consider a spin network that we denote as sprime = ssT composed oftwo parts s and sT connected to each other where s is arbitrary and sTis a weave state (6143) of the three-metric gT defined as follows Take a3-sphere of radius T in R4 Remove a spherical 3-ball of unit radius gT isthe three-metric of the three-dimensional surface (with boundary) formedby the sphere with removed ball The quantity

ψM[s] = limTrarrinfin

intDΦ fssT [Φ] eminus

12

int(P

Gφ)2 + λ

5

int(P

GPHφ)5 (9132)

represents an ansatz for the Minkowski vacuum state in a ball of unitradius


(Note added in the paperback edition A technique for computing n-point functions from background independent quantum gravity has beenproposed and developed This has allowed the derivation the gravitonpropagator which is so to say the derivation of ldquoNewton lawrdquo from thetheory without space and time See Bibliographical notes below)

mdashmdash


John Baez gives a nice and readable introduction to BF theory and spin-foams in [17] which contains also an invaluable carefully annotated bib-liography A good general review is Daniele Oritirsquos [18] In [19] AlejandroPerez describes the group field theory in detail and gives an extensiveoverview on different models On the derivation from hamiltonian looptheory see [11]

The idea of describing generally covariant QFT in terms of a ldquosum-over-surfacesrdquo was initially discussed in [276ndash278] the formal derivationfrom LQG in [279ndash281] The relation spinfoamstriangulated spacetimewas clarified by Fotini Markopoulou [282]

The PonzanondashRegge model was introduced in [283] On the precise re-lation between 6j symbols and Einstein action see [284] Regge calculusis introduced in [285] the TuraevndashViro model in [286] The duality be-tween spinfoam models and QFT on groups was pointed out by Boulatovin [287] as a duality between a QFT on [SU(2)]4 and the 3d PonzanondashRegge model Boulatovrsquos aim was to extend from 2d to 3d the dualitybetween the ldquomatrix modelsrdquo and 2d quantum gravity [21] or ldquozero di-mensional string theoryrdquo [288] (For a while it was hoped that matrixmodels would provide a background-independent definition of string the-ory More recently they have been extensively developed and used in arange of applications) The result was extended by Ooguri to 4d in [289]yielding the TOCY model described in this chapter see also [290] (BFtheories were discussed in [291]) The precise construction of the quantumdeformed version of this model and the proof of its triangulation indepen-dence were given in [292] The relation between PonzanondashRegge modelLQG and length quantization was pointed out in [293]

For the construction of the BC models I have followed Roberto DePietriand Laurent Freidel [294] see also [295 296] The idea of the quantumtetrahedron was discussed by Andrea Barbieri in [297] and by John Baezand John Barrett in [298] and used to construct the spinfoam model forGR in [269] and [299] where the cute term ldquospinfoamrdquo was introducedThe models BCA and BCB were defined in [300] The model BCC wasdefined in [301] and [302] The statistical behavior of different modelsis tentatively explored in [271] Arguments for preferring some of these


models on the basis of diff invariance are considered in [270] The factthat the duality between spinfoam models and QFT on groups extendsto BC models was noticed by DePietri and is presented in [300] Theremarkable fact that it can be extended to arbitrary spinfoam modelswas noticed by Michael Reisenberger and presented in [303] see also [304]The GFT finiteness proof appeared in [305] For recent (2007) discussionson the group field theory approach and updated references see [306] and[307] An intriguing recent result is the observation by Laurent Freideland Etera Livine that a quantum theory of gravity plus matter in 3d isequivalent to a matter theory over a noncommutative spacetime [308]

The convergence of the full series in 3d is discussed in [309] LorentzianPonzanondashRegge models are discussed in [310] lorentzian BC models in[311] Quantum group versions of lorentzian models have been studiedin [231] a paper I recommend for a detailed introduction to the subjectand its extensive references As for the euclidean case the quantum defor-mation of the group controls the divergences and can be related to a cos-mological term in the classical action The lorentzian GFT model SSO(3)

is defined in [312] The lorentzian GFT finiteness proof for this modelappeared in [313] The model SSO(21) is defined in [314] On the problemof the timelikespacelike character of the representations see [315] andreferences therein Functional integral methods to define spinfoam modelsare discussed in [316]

A different approach to the lorentzian sum-over-histories is developedin [317] An interesting variant of the spinfoam formalism in which prop-agation forward and backward in time can be distinguished has beenintroduced in [318]

On the reconstruction of the Hilbert space from the amplitudes of aspinfoam model see [274] and [275] On the relation between spinfoamand canonical LQG see also [319 320] A complete proof of the equiva-lence in 3d has been obtained by Karim Noui and Alejandro Perez in [321]On the role of the Immirzi parameter in BF theory see [322] The ansatzfor the derivation of the Minkowski vacuum from spinfoam amplitudesappeared in [145] A hamiltonian approach to the construction of co-herent states representing classical solutions of the Einstein equations isbeing developed by Thiemann and Winkler [323] A causal version ofthe spinfoam formalism has been studied by Fotini Markopoulou and LeeSmolin [324] The extension of the spinfoam models to matter coupling isstill embryonic see [325326] On the PeterndashWeyl theorem and harmonicanalysis on groups in general see for instance [327328]

A background independent difinition of n-point functions has beengiven in [329] The graviton propagator has been derived in [330] Seealso [331]

10Conclusion

In this book I have tried to present a compact and unified perspective on quantumgravity its technical aspects and its conceptual problems the way I understand themThere are a great number of other aspects of quantum gravity that I would have wishedto cover but my energies are limited Just to mention a few of the major topics I haveleft out 2 + 1 gravity the Kodama state and related results quantum gravity phe-nomenology supergravity in LQG coherent states Here I briefly summarize the phys-ical picture that emerges from LQG and the solution it proposes to the characteristicconceptual issues of quantum gravity I conclude with a short summary of the mainopen problems and the main results of the theory


The effort to develop a quantum theory of gravity forces us to revisesome conventional physical ideas This was expected given the conceptualnovelty of the two ingredients GR and QM and the tension between thetwo The physical picture presented in this book has emerged from severaldecades of research and is a tentative solution of the puzzle Here is abrief summary of its conceptual consequences

1011 GR and QM

The first conclusion of loop gravity is that GR and QM do not contradicteach other A quantum theory that has GR as its classical limit appearsto exist In order to merge both QM and classical GR have to be suit-ably formulated and interpreted More precisely they both modify someaspects of classical prerelativistic physics and therefore some aspects ofeach other

GR changes the way we understand dynamics It changes the structureof mechanical systems classical and quantum Classical and quantummechanics admit formulations consistent with this conceptual noveltyThese formulations have been studied in Chapters 3 and 5 respectively

366


Both classical and quantum mechanics are well defined consistent andpredictive in this generalized form although the relativistic formulationof QM has aspects that need to be investigated further

The main novelty is that dynamics treats all physical variables (partialobservables) on equal footing and predicts their correlations It does notsingle out a special variable called ldquotimerdquo to describe evolution withrespect to it Dynamics is not about time evolution it is about relationsbetween partial observables

QM modifies the picture of the world of classical GR as well In quan-tum theory a physical system does not follow a trajectory A classicaltrajectory of the gravitational field is a spacetime Therefore QM impliesthat continuous spacetime is ultimately unphysical in the same sensein which the notion of the trajectory of a Schrodinger particle is mean-ingless GR remains a meaningful theory even giving up the notion ofclassical spacetime as Maxwell theory remains a physically meaningfultheory in the quantum regime even if the notion of classical Maxwellfield is lost

To make sense of GR in the absence of spacetime we must readEinsteinrsquos major discovery in a light which is different from the conven-tional ones Einsteinrsquos major discovery is that spacetime and gravitationalfield are the same object A common reading of this discovery is that thereis no gravitational field there is just a dynamical spacetime In the viewof quantum theory it is more illuminating and more useful to say thatthere is no spacetime there is just the gravitational field From this pointof view the gravitational field is very much a field like any other fieldEinsteinrsquos discovery is that the fictitious background spacetime introducedby Newton does not exist Physical fields and their relations are the onlycomponents of reality


What does a physical theory predict in the absence of time evolutionand spacetime Any physical measurement that we perform is ultimatelya measurement of some local property of a quantum field These measure-ments are represented by an operator in a suitable kinematical quantumspace The theory then gives two kinds of predictions

bull First the spectral properties of the operator predict the quantiza-tion properties of the corresponding physical quantity They deter-mine the list of the values that the quantity may take

bull Second quantum dynamics predicts correlation probabilities be-tween observations That is it associates a correlation probabilityamplitude to ensembles of measurement outcomes

368 Conclusion

This conceptual structure is sufficient to formulate a meaningful and pre-dictive theory of the physical world even in the absence of a backgroundspacetime and in particular in the absence of a background time

There are no specific ldquoquantum gravity observablesrdquo Any measurementinvolving the gravitational field is also a quantum gravitational measure-ment Any measurement with which we test classical GR and whose out-comes we predict using classical GR is in principle a ldquoquantum gravitymeasurementrdquo as well The distinction is one of experimental accuracy

A measurement of a quantity that depends on the gravitational field isrepresented by an operator in the quantum theory of gravity In particulargeometric measurements such as measurements of volumes and areas areof this kind The theory illustrated in Chapter 6 constructs well-definedoperators corresponding to these measurements Their spectra are knownand provide quantitative quantum gravitational predictions

Some experiments can be viewed as ldquoscatteringrdquo experiments happen-ing in a finite region surrounded by detectors The traditional descriptionof this setting is based on two distinct kinds of measurements

(i) clocks and meters which measure the relative position of the detec-tors

(ii) particle detectors or other instruments which measure field proper-ties

In prerelativistic physics the measurements in class (i) refer to the loca-tion on background spacetime while the measurements of class (ii) referto the dynamical variables of the field theory The distinction between (i)and (ii) disappears in gravity This is because distances and time intervalsare nothing other than properties of the gravitational field and there-fore the measurements of class (i) fall into class (ii) both refer to thevalue of the field on the boundary of the experimental region Given anensemble of detectors having measured a certain ensemble of field prop-erties and their relative distances the theory should yield an associatedcorrelation probability amplitude that allows us to compute the relativefrequency of a given outcome with respect to a different outcome of thesame measurement


Space The disappearance of conventional physical space is a character-istic feature of the LQG picture There are quantum excitations of thegravitational field that have given probability amplitudes of transform-ing into each other These ldquoquanta of gravityrdquo do not live immersed in a


spacetime They are space The idea of space as the inert ldquocontainerrdquo ofthe physical world has disappeared

Instead the physical space that surrounds us is an aggregate of indi-vidual quanta of the gravitational field represented by the nodes of aspin network More precisely it is a quantum superposition of such aggre-gates

As observed in Chapter 2 the disappearance of the space-containeris not very revolutionary after all it amounts to a return to the viewof space as a relation between things which was the dominant tra-ditional way of understanding space in the Western culture beforeNewton

Perhaps I could add that in the pre-copernican world the cosmic organization wasquite hierarchical and structured Hence objects were located only with respect to oneanother but this was sufficient to grant every object a rather precise position in thegrand scheme of things This position marked the ldquostatusrdquo of each object vile objectsdown here noble objects above in the heavens With the copernican revolution thisoverall grand structure was lost Objects no longer knew ldquowhererdquo they were Newtonoffered reality a global frame It is a frame that for Newton was grounded in Godspace was the ldquosensoriumrdquo of God the World as perceived by God With or withoutsuch an explicit reference to God space has been held for three centuries as a preferredentity with respect to which all other entities are located Perhaps with the twentiethcentury and with GR we are learning that we do not need this frame to hold realityReality holds itself Objects interact with other objects and this is reality Reality isthe network of these interactions We do not need an external entity to hold the net

Time The disappearance of conventional physical time is the secondcharacteristic feature of nonperturbative quantum gravity This is per-haps a more radical step than the disappearance of space This book isas much about time as about quantum gravity A central idea defendedin this book is that in order to formulate the quantum theory of gravitywe must abandon the idea that the flow of time is an ultimate aspect ofreality We must not describe the physical world in terms of time evolu-tion of states and observables Instead we must describe it in terms ofcorrelations between observables

This shift of point of view is already forced upon us by classical GRbut in classical GR each solution of the Einstein equations still provides anotion of continuous spacetime It is only in the quantum theory of gravitywhere classical solutions disappear that we truly confront the absence oftime at the fundamental level Basic physics without time is viable Theformalism and its interpretation remain consistent In fact as soon as wegive up the idea that the ldquotimerdquo partial observable is special mechanicstakes a far more compact and elegant form as shown in Chapter 3

370 Conclusion

Unitarity In conventional QM and QFT unitarity is a consequence ofthe time translation symmetry of the dynamics In GR there isnrsquot ingeneral an analogous notion of time translation symmetry Thereforethere is no sense in which conventional unitarity is necessary in the theoryOne often hears that without unitarity a theory is inconsistent This isa misunderstanding that follows from the erroneous assumption that allphysical theories are symmetric under time translations

Some people find the absence of time difficult to accept I believe this is just a sort ofnostalgia for the old newtonian notion of the absolute ldquoTimerdquo along which everythingflows But this notion has already been shown to be inappropriate for understanding thereal world by special relativity Holding on to the idea of the necessity of unitary timeevolution or to Poincare invariance is an anchorage to a notion that is inappropriateto describe general-relativistic quantum physics


All sorts of open problems in theoretical physics (and outside it) havebeen related to quantum gravity For many of these I see no connectionwith quantum gravity In particular

bull Interpretation of quantum mechanics I see no reason why a quan-tum theory of gravity should not be sought within a standard in-terpretation of quantum mechanics (whatever one prefers) Severalarguments have been proposed to connect these two problems Acommon one is that in the Copenhagen interpretation the observermust be external but it is not possible to be external from the grav-itational field I think that this argument is wrong if it was correctit would apply to the Maxwell field as well We can consistently usethe Copenhagen interpretation to describe the interaction betweena macroscopic classical apparatus and a quantum gravitational phe-nomenon happening say in a small region of (macroscopic) space-time The fact that the notion of spacetime breaks down at shortscale within this region does not prevent us from having the regioninteracting with an external Copenhagen observer1

bull Quantum mechanical collapse Roger Penrose has proposed a subtleargument to relate GR and the QM collapse issue The argument isbased on the fact that in the Schrodinger equation there is a timevariable but the flow of physical time is affected by the gravitationalfield I think that this argument is correct but it only shows thatthe Schrodinger picture with an external time variable is not viablein quantum gravity

1However see Section 564


bull Unification of all interactions To quantize the electromagnetic fieldwe did not have to unify it with other fields And to find the quantumtheory of the strong interactions we do not have to unify them withother interactions The only vague hint that the problem of quantumgravity and the problem of the unification of all interactions mightbe related is the fact that the scale at which the running couplingconstants of the standard model meet is not very far from the Planckscale But it is not very close either

bull Particle masses cosmological constant standard modelrsquos families consciousness There are many aspects of the Universe whichwe do not understand There is no reason why all of these have tobe related to the problem of quantizing gravity or the problem ofunderstanding background-independent QFT We are far from theldquoend of physicsrdquo and there is much we do not yet understand

On the other hand there are two important open problems to whichquantum gravity is strongly connected

bull Ultraviolet divergences The disappearance of the ultraviolet diver-gences is one of the major successes of loop gravity This is achievedin a physically clear and compelling way via the short-scale quan-tization of space

bull Spacetime singularities There is no general result so far But loopquantum cosmology mentioned in Chapter 8 shows that the clas-sical initial singularity can be controlled by the theory


The formalism presented in Chapters 6 and 7 provides a well-definedbackground-independent quantum theory of gravity and matter The the-ory exists in euclidean and lorentzian versions In more detail

bull Background independence The main ambition of LQG was to com-bine GR and QM into a theory capable of merging the insightson Nature gathered by the two theories The problem was to un-derstand what is a general-relativistic QFT or a QFT constructedwithout using a fictitious background spacetime LQG achieves thisgoal Whether or not it is physically correct it proves that a QFTcan be general-relativistic and background independent It providesa nontrivial example of a background-independent QFT

bull A physical picture LQG offers a novel tentative unitary picture ofthe world that incorporates GR and QM In the book I have tried tospell out in detail this picture its assumptions and implications The

372 Conclusion

picture of the background-independent structure of the quantumgravitational field and matter is simple and compelling Spin net-work states describe Planck-scale quantum excitations which them-selves define localization and spatial relations as the solutions ofthe Einstein equations do Physical space is a quantum superpo-sition of spin networks Spin networks are not primary concreteldquoobjectsrdquo like particles in classical mechanics rather they describethe way the gravitational field interacts like the energy quanta ofan oscillator do The elements of the theory with a direct physicalinterpretation are elements of the algebra of the partial observablesof which spin networks characterize the spectrum

bull Quantitative physical predictions The spectra of area and volumedescribed in Chapter 6 provide a large body of precise quantitativephysical predictions These are unambiguous up to a single over-all multiplicative factor the Immirzi parameter or equivalentlythe bare value of the Newton constant Todayrsquos technology is notcapable of directly testing these spectra Indirect testing is not nec-essarily ruled out What is interesting about these predictions onthe other hand is the fact that they exist at all A theory is not ascientific theory unless it can provide a large body of precise quan-titative predictions capable at least in principle of being verifiedor falsified As far as I know no other current tentative quantumtheory of gravity provides a similar large set of predicted numbers

bull Ultraviolet divergences LQG appears to be free from ultravioletdivergences even when coupled with the standard model

bull Black-hole thermodynamics Although some aspects of the pictureare still unclear (in particular the determination of the Immirziparameter) LQG provides a compelling explanation of black-holeentropy as described in Section 82

bull Big-Bang singularity The classical initial cosmological singularityis controlled in the application of LQG to cosmology described inSection 81

The main aspects of LQG that are still missing or not sufficiently de-veloped are the following

bull Scattering amplitudes Having a well-defined physical theory is dif-ferent from knowing how to extract physics from it We can becapable of writing the full Schrodinger equation for the iron atomand be confident that this equation could predict the iron spectrumBut computing this spectrum is a different matter In a sense we arein a similar situation in LQG We have a well-defined theory but so


far we do not have a great capacity of systematic calculations of ob-servable amplitudes from the basic formalism of the theory What ismissing is a systematic formalism for doing so in some appropriateform of perturbation expansion

The difficulty in developing this formalism is of course due to thefact that a perturbative expansion around a classical solution of thegravitational field does not work The reason why this happens isclear nonperturbative effects dominate at the Planck scale yield-ing the discrete quantized structure of space We have to find analternative way for performing perturbative calculations One di-rection of research on this issue utilizes the covariant spinfoam for-malism described in Chapter 9 But this formalism is not yet at thepoint of providing a systematic technique for computing transitionamplitudes (Note added in the paperback edition Research is devel-oping rapidly in this direction General covariant n-point functionshave been defined and computed See the Bibliographical notes atthe end of the previous chapter)

bull Semiclassical limit The description of a macroscopic configurationof the electromagnetic field within quantum electrodynamics is nottrivial but it can be achieved using for instance coherent-state tech-niques Describing a macroscopic solution of the Einstein equationswith LQG is a similar problem Can we find a state in Kdiff that ap-proximates a given macroscopic solution Research programs in thisdirection are being pursued in particular by the groups of ThomasThiemann and Abhay Ashtekar I refer the reader to their worksfor a description of the state of the art in this rapidly developingdirection of research

LQG is in a peculiar specular position with respect to many tra-ditional approaches to quantum gravity The most common diffi-culty is to arrive at a description of Planck-scale physics manyformalisms tend to diverge or break down in some other way at thebackground-independent Planck-scale level LQG on the contraryprovides a formalism that gives a simple and compact descriptionof the background-independent Planck-scale physics but the recov-ery of low-energy physics appears more difficult (Note added in thepaperback edition The recent computation of the n-point functionsmentioned above provide a way of testing the large distance limit ofthe theory In particular the correct large-distance behavior of thepropagator obtained in [330] can be interpreted as the recovery ofthe Newton law from the background independent theory See theBibliographical notes at the end of the previous chapter)

374 Conclusion

(Note added in the paperback edition An important recent resultin this direction is the proof of the precise equivalence of the twoformalishs in 3d [321])

bull The Minkowski vacuum The most important state that we need isthe coherent state |0M〉 corresponding to Minkowski space This isessential to connect the theory to the usual formalism of QFT andto define particle-scattering amplitudes A direction for computingthis state was suggested at the end of Chapter 9 but it is too earlyto see if this will work

bull The form of the hamiltonian As discussed in Section 713 theprecise form of the quantum hamiltonian is not yet settled Thereare a number of quantization ambiguities and a number of possiblevariants that have been proposed The difficulty of selecting thecorrect form of the hamiltonian is not only due to the lack of directempirical guidance on the Planck scale but also to the little controlthat we have in extracting physical predictions from the theory asdiscussed above

bull Relation between the spinfoam and the hamiltonian formalism Fi-nally the relation between the lagrangian approach of Chapter 9 andthe hamiltonian approach of Chapters 6 and 7 is not yet sufficientlyclear

There are many problems that we have to solve before we can say wehave a credible and complete quantum theory of spacetime I hope thatamong the readers that have followed this book until this point there arethose that will be able to complete the journey

I close borrowing Galileorsquos marvelous prose

Ora perche e tempo di por fine ai nostri discorsi mi resta a pregarviche se nel riandar piu posatamente le cose da me arrecate incontrastedelle difficolta o dubbi non ben resoluti scusiate il mio difetto si per lanovita del pensiero si per la debolezza del mio ingegno si per la grandezzadel suggetto e si finalmente perche io non pretendo ne ho preteso da altriquellrsquoassenso chrsquoio medesimo non presto a questa fantasia2

2ldquoNow since it is time to end our discussion it remains for me to pray of you that ifin reconsidering more carefully what I have presented you find difficulties or doubtsthat were not well resolved you excuse my deficiency either because of the noveltyof the ideas the weakness of my understanding the magnitude of the subject orfinally because I neither ask nor have I ever asked from others that they attach tothis imagination that certainty which I myself do not haverdquo [332]

Part III

Appendices

Appendix A

Groups and recoupling theory


SU(2) is the group of the unitary 2times2 complex matrices with determinant1 We write these matrices as UA

B where the indices A and B take thevalues AB = 0 1 The fundamental representation of the group is definedby the natural action of these matrices on C2 The representation spaceis therefore the space of complex vectors with two components These arecalled spinors and denoted

ψA =(ψ0

ψ1

) (A1)

Consider the space formed by completely symmetric spinors with n indicesψA1An This space transforms into itself under the action of SU(2) onall the indices Therefore it defines a representation of SU(2)

ψA1An rarr UA1Aprime

1 UAn

Aprimen

ψAprime1A

primen (A2)

This representation is irreducible has dimension 2j + 1 and is called thespin-j representation of SU(2) where j = 1

2n All unitary irreduciblerepresentations have this form

The antisymmetric tensor εAB (defined with ε01 = 1) is invariant underthe action of SU(2)

UAC UB

D εCD = εAB (A3)

Contracting this equation with εAB (defined with ε01 = 1) we obtain thecondition that the determinant of U is 1

detU =12εAC εBD UA

B UCD = 1 (A4)

sinceεAB εAB = 2 (A5)

377

378 Appendix A

The inverse of an SU(2) matrix can be written simply as

(Uminus1)AB = minusεBD UDC εCA (A6)

Most of SU(2) representation theory follows directly from the invari-ance of εAB For instance consider the tensor product of the fundamentalrepresentation j = 12 with itself This defines a reducible representationon the space of the two-index spinors ψAB

(ψ otimes φ)AB = ψAφB (A7)

We can decompose any two-index spinor ψAB into its symmetric and itsantisymmetric part

ψAB = ψ0εAB + ψAB

1 (A8)

whereψ0 =

12εABψ

AB (A9)

and ψAB1 is symmetric Because of the invariance of εAB this decomposi-

tion is SU(2) invariant The one-dimensional invariant subspace formedby the scalars ψ0 defines the trivial representation j = 0 The three-dimensional invariant subspace formed by the symmetric spinors ψAB

1

defines the adjoint representation j = 1 Hence the tensor product of twospin-12 representations is the sum of a spin-0 and a spin-1 representa-tion 12 otimes 12 = 0 oplus 1

In general if we tensor a representation of spin j1 with a representationof j2 we obtain the space of spinors with 2j1 + 2j2 indices symmetric inthe first 2j1 and in the last 2j2 indices By symmetrizing all the indiceswe obtain an invariant subspace transforming in the representation j1+j2Alternatively we can contract k indices of the first group with k indicesof the second using k times the tensor εAB and then symmetrize theremaining 2(j1 + j2 minus k) indices This defines an invariant subspace ofdimension 2(j1 + j2 minus k) The maximum value of k is clearly the smallestbetween 2j1 and 2j2 Hence the tensor product of the representations j1and j2 gives the sum of the representations |j1minusj2| |j1minusj2|+2 (j1 +j2)

Thus each irreducible j3 appears in the product of two representationsat most once and if and only if

j1 + j2 + j3 = N (A10)

is integer and|j1 minus j2| le j3 le (j1 + j2) (A11)

These two conditions are called the ClebshndashGordon conditions They areequivalent to the requirement that there exist three nonnegative integers


a = 3 b = 2

c = 2

j2 = 52

2j2 = 5

j1 = 52

2 j1 = 5 2 j3 = 4

j3 = 2

=

Fig A1 ClebshndashGordon condition

a b and c such that

2j1 = a + c 2j2 = a + b 2j3 = b + c (A12)

If we have three representations j1 j2 j3 the tensor product of the threecontains the trivial representation if and only if one is in the product ofthe other two namely only if the ClebshndashGordon conditions are satisfiedThe invariant subspace in the product of the three is formed by invarianttensors with 2(j1 + j2 + j3) indices symmetric in the first 2j1 in thesecond 2j2 and in the last 2j3 indices There is only one such tensor up toscaling because it must be formed by combinations of the sole invarianttensor εAB It is given by simply taking a tensors εAB b tensors εBC andc tensors εCA that is

vA1A2j1 B1B2j2

C1C2j3

= (εA1B1 εAaBa) (εBa+1C1 εBa+bCb) (εCb+1Aa+1 εCb+cAa+c)(A13)

This is an intertwiner between the representations j1 j2 j3 We can choosea preferred intertwiner by demanding that the intertwiner is normalizednamely multiplying vA1A2j1

B1B2j2 C1C2j3 by a normalization fac-

tor K (which I give below) The normalized intertwiner is called theWigner 3j symbol

There is a simple graphical interpretation to the tensor algebra of theSU(2) irreducibles suggested by the existence of the three integers a b csee Figure A1 A representation of spin j is the symmetrized product of 2jfundamentals When three representations come together all fundamen-tals must be contracted among themselves There will be a fundamentalscontracted between j1 and j2 and so on Let us represent each irreducibleof spin j as a line formed by 2j strands An invariant tensor is a trivalent

380 Appendix A

node where three such lines meet and all strands are connected across thenode a strands flow from j1 to j2 and so on The meaning of the ClebshndashGordon conditions is then readily apparent (A10) simply demands thatthe total number of strands is even so they can pair (A11) demands thatj3 is neither larger than j1 + j2 because then some strands of j3 wouldremain unmatched nor smaller than |j1 minus j2| because then the largestamong j1 and j2 would remain unmatched Indeed this relation betweenthe lines and the strands reproduces precisely the relation between spinnetworks and loops Below this graphical representation is developed indetail

Orthonormal basis The space of the symmetric spinors with n indiceshas (complex) dimension 2j + 1 It is often convenient to choose a basisformed by 2j+1 orthonormal vectors eα1αn

α in this space For instance ifj = 1 the basis eAB

i = 1radic2σABi defined using the Pauli matrices transforms

under SU(2) in the fundamental representation of SO(3) The matricesσABi are obtained from the Pauli matrices

σAiB =

(0 11 0

)

(0 minus ii 0

)

(1 00 minus 1

)(A14)

by raising an index with εCB

σABi = σA

iCεCB =

(minus1 00 1

)

(i 00 i

)

(0 11 0

) (A15)

In general if the spin j is integer then the real section of the representationdefines a real irreducible representation

Wigner 3j symbols In an arbitrary orthonormal basis we write the nor-malized invariant tensors (A13) as

K vα1α2α3 =(j1 j2 j3α1 α2 α3

) (A16)

If we chose the basis that diagonalizes the third component of the angularmomentum (α equiv m) which we do below then these are proportional tothe Wigner 3j symbols The normalization K is fixed by

K vα1α2α3 K vα1α2α3 = 1 (A17)

For instance we have easily(

12 12 1A B i

)=

1radic6σiAB (A18)

and (1 1 1i j k

)=

1radic6εijk (A19)


Wigner 6j symbols Contracting four 3j symbols with the invariant ten-sor (minus1)jminusα defines a 6j symbol(j1 j4 j6j3 j2 j5

)=

sum

α1α6

(minus1)sum

A(j1minusα1)

(j3 j6 j2α3 α6 α2

) (j2 j1 j5α2 α1 α5

)

times(j6 j4 j1α6 α4 α1

) (j4 j3 j5α4 α3 α5

) (A20)

Since the indices are all contracted this quantity does not depend on thebasis chosen in the representation space The pattern of contraction isdictated by the geometry of a tetrahedron There is one 3j symbol foreach vertex of the tetrahedron and one representation for each edge

j1

j2

j3

j4 j6

j5 (A21)

Consider a tetrahedron with six spins j1 j6 associated with itsedges as above Denote Hj the representation space of the SU(2) irre-ducible representation of spin j Given a reducible representation H ofSU(2) denote by [H]j its component of spin j The Wigner 6j symbol isthe dimension of the intersection between the subspaces

[[Hj1 otimesHj2 ]j6 otimesHj3 ]j4 and [Hj1 [otimesHj2 otimesHj3 ]j5 ]j4 (A22)

of the space Hj1 otimesHj2 otimesHj3

Intertwiners All intertwiners can be built starting from the three-valentones For instance a four-valent intertwiner between representations j1j2 j3 and j4 can be defined (up to the normalization) by contracting twothree-valent intertwiners

vα1α2α3α4i = iα1α2α iα

α3α4 (A23)

where α is an index in a representation i The space of the intertwinersis then spanned by the tensors vi as i ranges over all representations thatsatisfy the two relevant ClebshndashGordon conditions namely such that thethree-valent intertwiners exist The representation i is said to be associ-ated with a ldquovirtual linkrdquo joining the two three-valent nodes into whichthe four-valent node has been decomposed

382 Appendix A

A different basis on this same intertwiner space is obtained by couplingthe first and the third leg instead of the first and the second That is

wα1α2α3α4i = iα1α3α iα

α2α4 (A24)

The change of basis between the vi and the wi is given by the Wigner 6jsymbols as we will show below (equation (A65))

vi =sum

j

(2j + 1)(j1 j2 jj3 j4 i

)wj (A25)

Graphically

j1

j2

j4

j3

i =sum

j

(2j + 1)(j1 j2 jj3 j4 i

)

j1

j2

j4

j3

j (A26)

Five-valent intertwiners can be constructed contracting a three-valentand a four-valent intertwiner and can thus be labeled with two irre-ducibles and so on In general we can decompose an n-valent node intonminus 2 three-valent nodes connected by nminus 2 virtual links and constructintertwiners accordingly

vα1αni1inminus3

= iα1α2β1 iβ1α3β2 iβ2

α4β3 iβnminus3αnminus1αn (A27)

where the indices βn belong to the representation in

Pauli matrices identities Define τi = minus i2 σi where σi are the Pauli ma-

trices (A14) We have the following identities

tr[τiτj ] = minus12δij (A28)

tr[τiτjτk] = minus14εijk (A29)

δijτiABτj

CD = minus1

4

(δADδ

BC minus εACεBD

) (A30)

δijtr[Aτi]tr[Bτj ] = minus14

tr[AB] minus tr[ABminus1]

(A31)

Aminus1AB = εACεBDA

DC (A32)

δABδDC = δACδ

DB + εADεBC (A33)

tr[A]tr[B] = tr[AB] + tr[ABminus1] (A34)

where A and B are SL(2 C) matrices




In his doctoral thesis Roger Penrose introduced the idea of writing ten-sor expressions in which there are sums of indices in a graphical way abeautiful idea that is at the root of spin networks1 Consider in particu-lar the calculus of spinors Penrose represents the basic element of spinorcalculus as

ψA = ψA

(A35)

ψA = ψA (A36)

δ AC =

C

A(A37)

εAC = A C (A38)

εAC = A C(A39)

and generally any tensor object as

XCAB = X

A BC (A40)

The idea is then to represent index contraction by simply joining the openends of the lines and dropping the index This convention provides thepossibility of writing the product of any two tensors in a graphical wayFor example

εABψAψB = ψ

ψ (A41)

However notice that the meaning of a diagram is not invariant if wesmoothly deform the lines For instance

εABηAηB = η

AηB

(A42)

= minusεADεBCεCDηAηB = minus η

AηB

C D (A43)

= minusεCDδDA δCBη

AηB = minusη ηA BC D

(A44)

Penrose introduced a modification of this graphical spinor calculus whichhe denoted as binor calculus that makes it invariant under deformations

1I have recently learned that Penrose called this notation ldquoloop notationrdquo

384 Appendix A

of the lines The binor calculus is obtained by adding two conventionsto the calculus above In translating a diagram into tensor notation wemust also ensure the following

(i) We assign a minus sign to each minimum and

(ii) assign a minus sign to each crossing

(iii) Maxima and minima are taken with respect to a fixed direction inthe plane (This direction is conventionally taken to be the verticaldirection on the written page)

(iv) A vertical segment represents a Kronecker delta

The advantage of these additional rules is that they make the calculustopologically invariant namely one can arbitrarily smoothly deform agraphical expression without changing its meaning

Expressed the other way around any curve can now be decomposedinto a product of δs and εs and any two curves that are ambient isotopicie that can be transformed one into the other by a sequence of Reide-meister moves represent the tensorial expression as products of epsilonsand deltas

A closed loop with this convention has value minus2 because = minusεAB εAB = minus2 (A45)

and we have the basic binor identity which reads

+ + = (minus1) δCB δDA + δCA δDB + (minus1) εAB εCD = 0 (A46)

Remarkably the two graphical identities = minus2 (A47)

= minus minus (A48)

are sufficient to generate a very rich graphical calculus

Kauffman brackets Equations (A47)ndash(A48) can be seen as a particular case of a richerstructure In the context of knot theory Lou Kauffman has defined a function of tanglesnamely planar graphical representations of knots which is now denoted the Kauffmanbrackets A planar tangle is a set of lines on a plane that overcross or undercross atintersections It represents the 2d projection of a 3d node The Kauffman brackets ofa tangle K are indicated as 〈K〉 and are completely determined by the two relations

lang

rang= A

lang rang+ Aminus1

lang rang(A49)

and lang cup Krang

= dlangK

rang (A50)


where d = minusA2 minus Aminus2 and K is any diagram that does not intersect the added loopBy applying equation (A49) to all crossings the Kauffman brackets of the tangle canbe reduced to a linear combination of Kauffman brackets of nonintersecting tangles Byrepeated application of (A50) we can then associate a number to the tangle Penrosebinor calculus and (A47)ndash(A48) are recovered for A = minus1 In this case undercrossingand overcrossing are not distinct


Following Kauffman and Linsrsquo book [174] one can pose the followingdefinitions

The antisymmetrizer Write n parallel lines as a single line labeled withn

n equiv (A51)

(The precise relation between the graphical calculus defined by this andthe following equations and the graphical calculus used in Chapter 6defined by equation (686) is discussed below in Section A23) Definethe antisymmetrizer as

n=

1n

sum

p

(minus1)|p| P (p)n (A52)

where P(p)n p = 1 n represents all the possible ways of connecting n

incoming lines with n outgoing lines obtained as n permutations and|p| is the sign of the permutation

The 3-vertex A special sum of tangles is indicated by a 3-vertex Eachline of the vertex is labeled with a positive integer n m or p

n

m

p

(A53)

and it is assumed that n = a+ b m = a+ c and p = b+ c where a b c arepositive integers This last condition is called the admissibility conditionfor the 3-vertex (mn p) The 3-vertex is then defined as

n

m

p

equiva

b c

p

n m

(A54)

386 Appendix A

Compare this definition with the discussion of the ClebshndashGordon coef-ficients and Wigner 3j symbols given above in A1 It is clear that the3-vertex in Penrose binor notation represents precisely the nonnormal-ized intertwiner (A13) In turn the Wigner 3j symbol can be obtainedby normalizing this intertwiner (Take care The KL 3-vertex representsthe nonnormalized intertwiner (A13) while the spin network vertex de-fined in (686) corresponds to the Wigner 3j symbol Therefore the twovertices differ by a normalization (See A23 below))

Chromatic evaluation If we join several trivalent vertices by their edgeswe obtain a trivalent spin network Thus in the present context a trivalentnetwork is defined as a trivalent graph with links labeled by an admissiblecoloring Notice that in this context networks are not embedded in athree-dimensional space A link of color n represents n parallel lines andan antisymmetrizer Thus a trivalent spin network determines a closedtangle The Penrose evaluation (or the Kauffman bracket with A = minus1)of this tangle is called the chromatic evaluation or network evaluation

Contractions of intertwiners and Wigner 3j symbols can therefore becomputed as chromatic evaluations of colored diagrams using only therelations (A47)ndash(A48)

As an example consider the spin network formed by two trivalent ver-tices joined to each other This is called the θ network Consider the casewith edges of color 2 1 1 Applying the definitions given we have

121

=1

1

2 13

=12

13

minus 12

=

=12(minus2)2 minus 1

2(minus2) = 3 (A55)

Therefore121

= 3 (A56)

The general formula of the chromatic evaluation of a generic θ network isgiven below in (A59)

Formulas from KL recoupling theory Direct computation using the defi-nitions above give the following formulas (See also the appendix of [195])(1) The dimension

Δn =n

= (minus1)n(n + 1) (A57)


Notice that if we write n = 2j then (n+ 1) = (2j + 1) is the dimension ofthe SU(2) spin-j representation(2) The exchange of lines in a 3-vertex

a

b

c

= λabc

ab

c

(A58)

where λabc = (minus1)(a+bminusc)2 (minus1)(a

prime+bprimeminuscprime)2 and xprime = x(x + 2)(3) The θ evaluation

θ(a b c) =

a

b

c

13

=(minus1)m+n+p(m + n + p + 1) m n p

a b c (A59)

where m = (a + bminus c)2 n = (b + cminus a)2 p = (c + aminus b)2(4) The tetrahedral net

Tet[A B EC D F

]=

B

A

C

D

F

E (A60)

=IE

sum

mleSleM

(minus1)S(S + 1)prod

i (S minus ai)prod

j (bj minus S) (A61)

where

a1 =A + D + E

2 b1 =

B + D + E + F

2

a2 =B + C + E

2 b2 =

A + C + E + F

2

a3 =A + B + F

2 b3 =

A + B + C + D

2

a4 =C + D + F

2

m = maxai M = minbj

E = ABCDEF I =prod

ij(bj minus ai)

388 Appendix A

(5) The reduction formulas

b

a

a

13c

=

abc

a

a (A62)

b

c

d

ef

a

a

=

cb

d

e

f

a

a a (A63)

Strictly speaking the identity factors on the right-hand side (the nonin-tersecting a-tangles) of both of these equations should include the anti-symmetrizer When embedded in spin networks this gets absorbed intothe nearest vertex

Also

q

p

rp

q

2 =

Tet[p p rq q 2

]

θ(p q r) q

pr middot (A64)

(6) The recoupling theorem

a

b

d

c

j =sum

i

a b ic d j

a

b

d

c

i (A65)

a b ic d j

=

Δi Tet[a b ic d j

]

θ(a d i)θ(b c i) (A66)

These formulas are sufficient for most computations performed in loopquantum gravity

A23 Normalizations

Finally it is time to relate the KauffmanndashLins recoupling theory diagramsgiven in this appendix with the spin networks recoupling diagrams thatwe have used in Chapter 6 and which are defined by equation (686)There are two main differences The first is trivial lines are labeled bythe spin j in spin network recoupling diagrams while they are labeled by


the color n = 2j in the KauffmanndashLins diagrams Thus⎛

⎝ j

⎞

⎠

spin network

=

⎛

⎝ n = 2j

⎞

⎠

KauffmanminusLins

middot (A67)

The second and more important difference is that the trivalent nodes ofthe spin network diagrams represent normalized intertwiners Thereforethey are proportional to the recoupling theory trivalent nodes and theproportionality factor is easily obtained from (A59) Thus up to possiblephase factors

⎛

⎝

j

jprime

jprimeprime

⎞

⎠

spin network

=

⎛

⎜⎜⎜⎝

1radicθ(a b c)

a=2j

b=2jprime

c=2jprimeprime

⎞

⎟⎟⎟⎠

KauffmanminusLins

middot

(A68)

These two equations provide the complete translation rules It follows forinstance that the Wigner 6j-symbol is given by

(j1 j2 j5j3 j4 j6

)=

⎛

⎜⎝

j2

j1

j3

j4

j6

j5

⎞

⎟⎠

spin network

=

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

2j2

2j1

2j3

2j4

2j6

2j5

radic2j12j2

2j6

2j32j4

2j6

2j12j4

2j5

2j22j3

2j5

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

KauffmanminusLins

=Tet

[2j1 2j2 2j52j3 2j4 2j6

]

radicθ(2j1 2j2 2j6)θ(2j3 2j4 2j6)θ(2j1 2j4 2j5)θ(2j2 2j3 2j5)

(A69)

And the recoupling theorem (A26) follows from (A65)The reason I have described two distinct normalization conventions for

the diagrams is that they are both utilized in the loop quantum gravityliterature and both turn out to be useful The diagrammatic notationdeveloped in this appendix is the one used by Kauffman and Lins in

390 Appendix A

[174] This notation has often been used for computing matrix elementsof loop operators

On the other hand one can find many results on the Wigner 3nj sym-bols and a well-developed graphical calculus for the representation theoryof SU(2) in the general literature These results are routinely used for in-stance in atomic and nuclear physics A standard reference is for instanceBrink and Satcheler [333] The BrinkndashSatcheler diagrams are written withthe convention I have used in Chapter 6 for the spin networks2


I collect here some facts on the representation theory of SO(n) Labelfinite-dimensional irreducible representations of SO(n) by their highestweight Λ Here Λ is a vector of length n = [d2] ([middot] is the integral part)Λ = (N1 middot middot middot Nn) where Ni are integers and N1 ge ge Nn If we areinterested in representations of Spin(n) we let the Ni be half-integers Therepresentation labeled by the highest weight Λ = (N 0 0) are calledsimple or spherical Let Xij 1 le i j le n be a basis of the Lie algebra ofSO(n) The simple representations are those for which the ldquosimplicityrdquorelations

X[ijXij] middot VN = 0 (A70)

are satisfied The representation space VN of a simple representation canbe realized as a space of spherical harmonics that is harmonic homoge-neous polynomials on Rn Any L2 function on the sphere can be uniquelydecomposed in terms of these spherical harmonics

L2(Snminus1) = oplusinfinN=0VN (A71)

In the case of SO(4) since Spin(4) = SU(2)timesSU(2) there is an alter-nate description of the representation as products of two representationsjprime and jprimeprime of SU(2) The relation with the highest weight presentation isgiven by

N1 = jprime + jprimeprime N2 = jprime minus jprimeprime (A72)

The simple representations are therefore the representation in whichjprime = jprimeprime equiv j Thus we can label simple representations with a half-integerspin j Notice that the integer ldquocolorrdquo N = 2j is also the (nonvanishingcomponent of the) highest weight of the representation

2With a minor difference in the angular momentum literature Wignerrsquos 6j symbolsare generally indicated with curl brackets while following Kauffman I have denoted6j symbols by round brackets reserving the curl brackets in (A65) to denote therecoupling matrix of a four-valent node which differs from the 6j symbol by thenormalization factor Δi


An elementary illustration of simple representations can be given asfollows The vectors vα of the representation Λ = (jprime jprimeprime) can be writ-ten as spinors ψA1Ajprime B1Bjprimeprime with jprime ldquoundottedrdquo symmetrized indicesAi = 1 2 transforming under one of the SU(2) and jprimeprime ldquodottedrdquo anti-symmetrized indices Bi=1 2 transforming under the other Consider theparticular vector wα which (in the spinor notation and in a given basis)has components ψA1Ajprime B1Bjprimeprime = ε(A1B1 εAj)Bj where εAB is the unitantisymmetric tensor and the symmetrization is over the Ai indices onlyThe subgroup of SO(4) that leaves this vector invariant is an SO(3) sub-group of SO(4) (which depends on the basis chosen) Clearly since εAB isthe only object invariant under this SU (2) a normalized SO(3)-invariantvector (and only one) exists in these simple representations only

Equivalently the simple representations of SO(4) are those defined bythe completely symmetric traceless 4d tensors of rank N The invariantvector w is then the traceless part of the tensor with (in the chosen basis)all components vanishing except w444 The SO(3) subgroup is given bythe rotations around the fourth coordinate axis The relation between thevector and spinor representation is obtained contracting the spinor indiceswith the (four) Pauli matrices vμ1μj = ψA1Ajprime B1Bjprimeprimeσμ1

A1B1 σ

μj

AjBj

Let VΛ be a representation of SO(n) we say that ω isin VΛ is a sphericalvector if it is invariant under the action of SO(nminus1) Such a vector existsif and only if the representation is simple In that case this vector is uniqueup to normalization

Let ω be a vector of VΛ and consider an orthonormal basis vi of VΛWe can construct the following functions on G

Θi(g) = 〈ω|Rminus1(g)|vi〉 (A73)

These functions span a subspace of L2(G) The group acts on this sub-space by the right regular representation and the corresponding rep-resentation is equivalent to the representation VΛ If ω is sphericalthen these functions are in fact L2 functions on the quotient spaceSO(n)SO(n minus 1) = Snminus1 and VΛ is therefore a spherical representa-tion On the other hand if the representation is spherical then we canconstruct a spherical vector Θω(g) =

sumi Θi(g)Θi(1) When ω is spheri-

cal the spherical function Θω is a function on the double-quotient spaceSO(nminus1)SO(n)SO(nminus1) = U(1) It is now a standard exercise to showthat there is a unique harmonic polynomial on Rn invariant by SO(nminus1)of a given degree hence there is a unique spherical function

The space of the intertwiners of three representations of SO(4) isat most one-dimensional The dimension nΛ1Λ2Λ3 of the space of theintertwiners between three representations Λ1Λ2Λ3 is given by the

392 Appendix A

integral

nΛ1Λ2Λ3 =int

dg χΛ1(g)χΛ2(g)χΛ3(g) (A74)

where χΛ are the characters of the representation Λ Since any repre-sentation of SO(4) is the product of two representations of SU(2) theintertwining number of three SO(4) representations is the product ofSU(2) intertwining numbers These numbers can take the values 0 or1 for SU(2)

Representations of SO(n) are real This means that it is always possibleto choose a basis of VΛ such that the representation matrices are real Forthe half-integer spin representations of Spin(n) it is still true that Λis equivalent to its complex conjugate or dual but the isomorphism isnontrivial

mdashmdash


On Penrose calculus see [181] and [334] The basic reference on recouplingtheory that I have used here is [174] where the formulas given here arederived in the general case with arbitrary A see also the appendix of[195] for more details On the relation between recoupling theory andspin networks see [335 336] Chromatic evaluation is used to computeSU(2) ClebshndashGordon coefficients in for instance [337] A widely usedgraphical method for angular momentum theory is the one of Levinson[338] developed by Yutsin Levinson and Vanagas [339] and the slightlymodified version of Brink and Satcheler [333]

Appendix B

History

In this appendix I sketch the main lines of development of the research in quantumgravity from the first explorations in the early 1930s (the thirties) to nowadays

I have no ambition of presenting complete references to all the important workson the subject some of the references are to original works others to reviews wherereferences can be found Errors and omissions are unfortunately unavoidable and Iapologize for these I have made my best efforts to be balanced but in a field that hasnot yet succeeded in finding consensus my perspective is obviously subjective Tryingto write history in the middle of the developments is hard Time will pass dust willsettle and it will slowly become clear if we are right if some of us are right or ndash apossibility never to disregard ndash if we all are wrong

I am very much indebted to the many friends that have contributed to this histor-ical perspective I am particularly grateful to John Stachel Augusto Sagnotti GaryHorowitz Ludwig Faddeev Alejandro Corichi Jorge Pullin Lee Smolin Joy ChristianBryce DeWitt Cecile DeWitt Giovanni Amelino-Camelia Daniel Grumiller NikolaosMavromatos Stanley Deser Ted Newman and Gennady Gorelik


The quest for quantum gravity can be separated into three main lines ofresearch The relative weight of these lines has changed there have beenimportant intersections and connections between the three and there hasbeen research that does not fit into any of the three lines Neverthelessthe three lines have maintained a distinct individuality across 70 yearsof research They are often denoted ldquocovariantrdquo ldquocanonicalrdquo and ldquosum-over-historiesrdquo even if these names can be misleading and are often usedinterchangeably They cannot be characterized by a precise definitionbut within each line there is a certain methodological unity and a certainconsistency in the logic of the development of the research

bull The covariant line of research is the attempt to build the theory asa quantum field theory of the fluctuations of the metric over a flat

393

394 Appendix B

Minkowski space or some other background metric space The pro-gram was started by Rosenfeld Fierz and Pauli in the thirties TheFeynman rules of GR were laboriously found by DeWitt Feynmanand Faddeev in the sixties trsquoHooft and Veltman Deser and VanNieuwenhuizen and others found increasing evidence of nonrenor-malizability at the beginning of the seventies Then a search foran extension of GR giving a renormalizable or finite perturbationexpansion started Through high-derivative theory and supergrav-ity the search converged successfully to string theory in the lateeighties

bull The canonical line of research is the attempt to construct a quan-tum theory in which the Hilbert space carries a representation of theoperators corresponding to the full metric or some functions ofthe metric without any background metric to be fixed The pro-gram was initiated by Bergmann and Dirac in the fifties Unravel-ing the canonical structure of GR turned out to be laborious DiracBergmann and his group and Peres completed the task in the fiftiesTheir cumbersome formalism was drastically simplified by the intro-duction of new variables first by Arnowit Deser and Misner in thesixties and then by Ashtekar in the eighties The formal equationsof the quantum theory were written down by Wheeler and DeWittin the middle sixties but turned out to be too ill defined A well-defined version of the same equations was successfully found onlyin the late eighties with the formulation of LQG

bull The sum-over-histories line of research is the attempt to use someversion of Feynmanrsquos functional integral quantization to define thetheory The idea was introduced by Misner in the fifties followinga suggestion by Wheeler and developed by Hawking in the formof euclidean quantum gravity in the seventies Most of the discrete(lattice-like posets ) approaches and the spinfoam formalismintroduced more recently belong to this line as well

bull Others There are of course other ideas that have been explored

ndash Noncommutative geometry has been proposed as a key math-ematical tool for describing Planck-scale geometry and hasrecently obtained very surprising results particularly with thework of Connes and collaborators

ndash Twistor theory has been more fruitful on the mathematicalside than on the strictly physical side but it is still developing

ndash Finkelstein Sorkin and others pursue courageous and intrigu-ing independent paths

History 395

ndash Penrosersquos idea of a gravity-induced quantum state reductionhas recently found new life with the perspective of a possibleexperimental test

ndash

So far however none of these alternatives has been developed intoa detailed quantum theory of gravity

B2 Five periods

Historically the evolution of the research in quantum gravity can roughlybe divided into five periods summarized in Table B1

bull The Prehistory 1930ndash1957 The basic ideas of all three lines ofresearch appear as early as the thirties By the end of the fifties thethree research programs are clearly formulated

bull The Classical Age 1958ndash1969 The sixties see the strong develop-ment of two of the three programs the covariant and the canonicalAt the end of the decade the two programs have both achieved thebasic construction of their theory the Feynman rules for the gravi-tational field on one side and the WheelerndashDeWitt equation on theother To get to these beautiful results an impressive amount oftechnical labour and ingenuity proves necessary The sixties close ndashas they did in many regards ndash with the promise of a shining newworld

bull The Middle Ages 1970ndash1983 The seventies disappoint the hopes ofthe sixties It becomes increasingly clear that the WheelerndashDeWittequation is too ill defined for genuine field theoretical calculationsAnd evidence for the nonrenormalizability of GR piles up Both linesof attack have found their stumbling block

In 1974 Stephen Hawking derives black-hole radiation Trying todeal with the WheelerndashDeWitt equation he develops a version ofthe sum-over-histories as a sum over ldquoeuclideanrdquo (riemannian)geometries There is excitement with the idea of the wave func-tion of the Universe and the approach opens the way for thinking ofand computing topology change But for field theoretical quantitiesthe euclidean functional integral will prove as weak a calculationtool as the WheelerndashDeWitt equation

On the covariant side the main reaction to nonrenormalizabil-ity of GR is to modify the theory Strong hopes then disap-pointments motivate extensive investigations of supergravity and

Table B1 The search for a quantum theory of the gravitational field

Prehistory

1920 The gravitational field needs to be quantized

1930 ldquoFlat-space quantizationrdquo

1950 ldquoPhase spacequantizationrdquo

1957Constraint theory

ldquoFeynmanquantizationrdquo

Classical Age

1961ADM Tree-amplitudes

1962 Background fieldmethod

1963 Wave function of the 3-geometry spacetime foam

1967 Ghosts

1968 MinisuperspaceFeynman rules

completed

Middle Ages

1971 YM renormalization

1972 Twistors

1973Nonrenormalizability

1974 Black-holeradiation

1976 Asymptotic safety

1976 Supergravity

1977 High-derivative theories

1978Euclidean QG

1981

1983 Wave functionof the Universe

Renaissance

1984 String renaissance

1986 Connectionformulation of GR

TQFT

1987Superstring theory

1988Loop quantum gravity

2+1

1989 2d QG

1992 Weaves State sum models

1994 Noncommutative

geometry

Nowadays

1995Eigenvalues of area and volume

Null surfaceformulation Nonperturbative strings

1996 BH radiation from loops Spin foams BH radiation from strings

1997 ldquoQuantum gravity phenomenologyrdquo

Stringsndashnoncommutativegeometry

WheelermdashDeWitt equation

396

History 397

higher-derivative actions for GR The landscape of quantum gravityis gloomy

bull The Renaissance 1984ndash1994 Light comes back in the middle ofthe eighties In the covariant camp the various attempts to modifyGR to get rid of the infinities merge into string theory Perturbativestring theory finally delivers on the long search for a computable per-turbative theory for quantum gravitational scattering amplitudesTo be sure there are prices to pay such as the wrong dimensional-ity of spacetime and the introduction of supersymmetric particleswhich year after year are expected to be discovered but so far arenot But the result of a finite perturbation expansion long soughtafter is too good to be discarded just because the world insists inlooking different from our theoriesLight returns to shine on the canonical side as well Twenty yearsafter the WheelerndashDeWitt equation LQG finally provides a versionof the theory sufficiently well defined for performing explicit com-putations Here as well we are far from a complete and realistictheory and scattering amplitudes for the moment canrsquot be com-puted at all but the excitement for having a rigorously definednonperturbative generally covariant and background-independentquantum field theory in which physical expectation values can becomputed is strong

bull Nowadays 1995c Both string theory and LQG grow strongly fora decade until in the middle of the nineties they begin to deliverphysical results The BekensteinndashHawking black-hole entropy for-mula is derived within both approaches virtually simultaneouslyLQG leads to the computation of the first Planck-scale quantita-tive physical predictions the spectra of the eigenvalues of area andvolumeThe sum-over-histories tradition in the meantime is not dead Inspite of the difficulties of the euclidean integral it remains as a ref-erence idea and guides the development of several lines of researchfrom the discrete lattice-like approaches to the ldquostate sumrdquo for-mulation of topological theories Eventually the last motivate thespinfoam formulation a translation of LQG into a Feynman sum-over-histories formOther ideas develop in the meanwhile most notably noncommuta-tive geometry which finds intriguing points of contact with stringtheory towards the end of the decadeThe century closes with two well-developed contenders for a quan-tum theory of gravity string theory and LQG as well as a set of

398 Appendix B

intriguing novel new ideas that go from noncommutative geometryto the null surfaces formulation of GR to the attempt to mergestrings and loops And even on a very optimistic note the birth of anew line of research the self-styled ldquoquantum gravity phenomenol-ogyrdquo which investigates the possibility that Planck-scale type mea-surements might be within reach And thus that perhaps we couldfinally know which of the theoretical hypotheses if any make sense

I now describe the various periods and their main steps in more detail


General relativity is discovered in 1915 quantum mechanics in 1926 Afew years later around 1930 Born Jordan and Dirac are already capableof formalizing the quantum properties of the electromagnetic field Howlong did it take to realize that the gravitational field should presumablybehave quantum mechanically as well Almost no time already in 1916Einstein points out that quantum effects must lead to modifications inthe theory of general relativity [340] In 1927 Oskar Klein suggests thatquantum gravity should ultimately modify the concepts of space and time[341] In the early thirties Rosenfeld [342] writes the first technical pa-pers on quantum gravity applying Paulirsquos method for the quantizationof fields with gauge groups to the linearized Einstein field equations Therelation with a linear spin-2 quantum field is soon unraveled in the worksof Fierz and Pauli [343] and the spin-2 quantum of the gravitational fieldis already a familiar notion in the thirties Its name ldquogravitonrdquo is al-ready in use in 1934 when it appears in a paper by Blokhintsev andGalrsquoperin [344] (published in the ideological magazine Under the Bannerof Marxism) Bohr considers the idea of identifying the neutrino and thegraviton In 1938 Heisenberg [345] points out that the fact that the grav-itational coupling constant is dimensional is likely to cause problems withthe quantum theory of the gravitational field

The history of these early explorations of the quantum properties ofspacetime has recently been reconstructed by John Stachel [346] In par-ticular John describes in his paper the extensive but largely neglectedwork conducted in the middle thirties by a Russian physicist MatveiPetrovich Bronstein Persistent rumors claim that Bronstein was a nephewof Leon Trotsky and that he hid this relation that became dangerous butGennady Gorelik (of the Center for Philosophy and History of Science atBoston University and Institute for the History of Science and Technol-ogy of the Russian Academy of Sciences) assures me that this rumor isfalse Bronstein re-derives the RosenfeldndashPauli quantization of the lin-ear theory but realizes that the unique features of gravitation require aspecial treatment when the full nonlinear theory is taken into account

History 399

He realizes that field quantization techniques must be generalized in sucha way as to be applicable in the absence of a background geometry Inparticular he realizes that the limitation posed by general relativity onthe mass density radically distinguishes the theory from quantum elec-trodynamics and would ultimately lead to the need to ldquoreject riemanniangeometryrdquo and perhaps also to ldquoreject our ordinary concepts of space andtimerdquo [347] The reason Bronstein has remained unknown for so long haspartly to do with the fact that he was executed by the Soviet State Secu-rity Agency (the NKVD) at the age of 32 I am told that in Russia somestill remember Bronstein as ldquosmarter than Landaurdquo (but Gorelik doubtsthis opinion could be shared by a serious physicist) For a discussion ofBronsteinrsquos early work in quantum gravity see [348]

References and many details on these pioneering times are in the fas-cinating paper by John Stachel mentioned above Here I pick up thehistorical evolution after World War II In particular I start from 1949a key year for the history of quantum gravity

1949ndash Peter Bergmann starts his program of phase space quantization of

nonlinear field theories [349] He soon realizes that physical quantumobservables must correspond to coordinate-independent quantities only[350] The search for these gauge-independent observables is started in thegroup that forms around Bergmann at Brooklyn Polytechnic and then inSyracuse For instance Ted Newman develops a perturbation approachfor finding gauge-invariant observables order by order [351] The groupstudies the problems raised by systems with constraints and reaches aremarkable clarity unfortunately often forgotten later on on the problemof what are the observables in general relativity The canonical approachto quantum gravity is born

ndash Bryce DeWitt completes his thesis He applies Schwingerrsquos covariantquantization to the gravitational field

ndash Dirac presents his method for treating constrained hamiltonian sys-tems [113]

1952ndash Following the pioneering works of Rosenfeld Fierz and Pauli Gupta

[352] develops systematically the ldquoflat-space quantizationrdquo of the grav-itational field The idea is simply to introduce a fictitious ldquoflat spacerdquothat is Minkowski metric ημν and quantize the small fluctuations of themetric around Minkowski hμν = gμν minus ημν The covariant approach isfully born The first difficulty appears immediately when searching forthe propagator because of gauge invariance the quadratic term of thelagrangian is singular as for the electromagnetic field Guptarsquos treatmentuses an indefinite norm state space as for the electromagnetic field

400 Appendix B

1957ndash Charles Misner introduces the ldquoFeynman quantization of general rel-

ativityrdquo [353] He quotes John Wheeler for suggesting the expressionint

exp(ih)(Einstein action) d(field histories) (B1)

and studies how to have a well-defined version of this idea Misnerrsquos paper[353] is very remarkable in many respects It explains with complete claritynotions such as why the quantum hamiltonian must be zero why theindividual spacetime points are not defined in the quantum theory andthe need of dealing with gauge invariance in the integral Even moreremarkably the paper opens with a discussion of the possible directionsfor quantizing gravity and lists the three lines of directions ndash covariantcanonical and sum-over-histories ndash describing them almost precisely withthe same words we would today1

At the end of the fifties all the basic ideas and the research programsare clear It is only a matter of implementing them and seeing if theywork The implementation however turns out to be a rather herculeantask that requires the ingenuity of people of the caliber of Feynman andDeWitt on the covariant side and of Dirac and DeWitt on the canonicalside


1958ndash The Bergmann group [129] and Dirac [113 114] work out the gen-

eral hamiltonian theory of a constrained system For a historical recon-struction of this achievement see [354] At the beginning Dirac and theBergmann group work independently The present double classificationinto primary and secondary constraints and into first- and second-classconstraints still reflects this original separation

1959ndash By 1959 Dirac has completely unraveled the canonical structure of

GR [130]

1961ndash Arnowitt Deser and Misner complete what we now call the ADM for-

mulation of GR namely its hamiltonian version in appropriate variables

1To be sure Misner lists a fourth approach as well based on the Schwinger equationsfor the variation of the propagator but notices that ldquothis method has not been appliedindependently to general relativityrdquo a situation that might have changed only veryrecently [145 146]

History 401

which greatly simplify the hamiltonian formulation and make its geomet-ric reading transparent [131]

In relation to the quantization Arnowitt Deser and Misner present aninfluential argument for the finiteness of the self-energy of a point-particlein classical GR and use it to argue that nonperturbative quantum gravityshould be finite

ndash Tullio Regge defines the Regge calculus [285]

1962ndash Feynman attacks the task of computing transition amplitudes in

quantum gravity He shows that tree-amplitudes lead to the physics oneexpects from the classical theory [355]

ndash DeWitt starts developing his background field methods for the com-putation of perturbative transition amplitudes [356]

ndash Bergmann and Komar clarify what one should expect from a Hilbertspace formulation of GR [357]

ndash Following the ADM methods Peres writes the HamiltonndashJacobi for-mulation of GR [358]

G2(qabqcd minus12qacqbd)

δS(q)δqac

δS(q)δqbd

+ det q R[q] = 0 (B2)

which is our fundamental equation (49) written in metric variables andwill soon lead to the WheelerndashDeWitt equation here qab is the ADM3-metric

1963ndash John Wheeler realizes that the quantum fluctuations of the gravita-

tional field must be short-scale fluctuations of the geometry and intro-duces the physical idea of spacetime foam [359] Wheelerrsquos Les Houcheslecture notes are remarkable in many respects and are the source of manyof the ideas still current in the field To mention two others ldquoProblem56rdquo suggests that gravity in 2 + 1 dimensions may not be so trivial afterall and indicates it may be an interesting model to explore ldquoProblem57rdquo suggests studying quantum gravity by means of a Feynman integralover a spacetime lattice

Julian Schwinger introduces the tetrad spin-connection formulation inquantum gravity [80] On the strict relation between this formalism andYangndashMills theories he writes

Weyl the originator of the electromagnetic gauge invariance princi-ple also recognized that the gravitational field can be characterizedby a kind of gauge transformation [79] This is the possibility of alter-ing freely at each point the orientation of a local Lorentz coordinate

402 Appendix B

frame while suitably transforming certain gravitational potentials Ina subsequent development Yang and Mills introduced an arbitrarilyoriented 3d isotopic space at each spacetime point The occasional re-mark that the gravitational field can be viewed as a YangndashMills fieldis thus rather anachronistic

1964ndash Penrose introduces the idea of spin networks and of a discrete struc-

ture of space controlled by SU(2) representation theory The constructionexists only in the form of a handwritten manuscript It gets published onlyin 1971 [181] The idea will surprisingly re-emerge 25 years later whenspin networks will be found to label the states of LQG

ndash Beginning to study loop corrections to GR amplitudes Feynman ob-serves that unitarity is lost for naive diagrammatic rules DeWitt [360]develops the combinatorial means to correct the quantization (requir-ing independence of diagrams from the longitudinal parts of propaga-tors) These correction terms can be put in the form of loops of fictitiousfermionic particles the FaddeevndashPopov ghosts [361] The key role ofDeWitt in this context was emphasized by Veltman in 1974 [362]

Essentially due to this and some deficiencies in his combinatorialmethods Feynman was not able to go beyond one closed loop DeWittin his 1964 Letter and in his subsequent monumental work derivedmost of the things that we know of now That is he considered thequestion of a choice of gauge and the associated ghost particle Indeedhe writes the ghost contribution in the form of a local Lagrangiancontaining a complex scalar field obeying Fermi statistics Somewhatillogically this ghost is now called the FaddeevndashPopov ghost

The designation ldquoFaddeevndashPopov ghostrdquo is far from illogical in compar-ison with the complicated combinatorics of DeWitt the FaddeevndashPopovapproach has the merit of a far greater technical simplicity and a trans-parent geometric interpretation which justifies its popularity It is onlyin the work of Faddeev that the key role played by the gauge orbits asthe true dynamical variables is elucidated [363]

1967ndash Bryce DeWitt publishes the ldquoEinsteinndashSchrodinger equationrdquo [214]

((hG)2(qabqcd minus

12qacqbd)

δ

δqac

δ

δqbdminus det q R[q]

)Ψ(q) = 0 (B3)

which is the main quantum gravity equation (61) in metric variablesBryce will long denote this equation as the ldquoEinsteinndashSchrodinger equa-tionrdquo attributing it to Wheeler ndash while John Wheeler called it the DeWittequation ndash until finally in 1988 at an Osgood Hill conference DeWitt

History 403

gives up and calls it what everybody else had been calling it since thebeginning the ldquoWheelerndashDeWitt equationrdquo

The story of the birth of the WheelerndashDeWitt equation is worth tellingIn 1965 during an air trip John had to stop for a short time at theRaleighndashDurham airport in North Carolina Bryce lived nearby Johnphoned Bryce and proposed to meet at the airport during the wait be-tween two flights Bryce showed up with the HamiltonndashJacobi equationfor GR published by Peres in 1962 and mumbled the idea of doing pre-cisely what Schrodinger did for the hydrogen atom replace the squareof the derivative with a second derivative Surprising Bryce John wasenthusiastic (John is often enthusiastic of course) and declared immedi-ately that the equation of quantum gravity had been found The paperwith the equation the first of Brycersquos celebrated 1967 quantum gravitytrilogy [214 364] was submitted in the spring of 1966 but its publicationwas delayed until 1967 Among the reasons for the delay apparently weredifficulties with publication charges

ndash John Wheeler discusses the idea of the wave function Ψ(q) on thespace of the ldquo3-geometryrdquo q and the notion of superspace the space ofthe 3-geometries in [38]

ndash Roger Penrose starts twistor theory [365]ndash The project of DeWitt and Feynman is concluded A complete and

consistent set of Feynman rules for GR are written down [361 364]

1968ndash Ponzano and Regge define a quantization of 3d euclidean GR [283]

The model will lead to major developments

1969ndash Developing an idea in Brycersquos paper on canonical quantum gravity

Charles Misner starts quantum cosmology the game of truncating theWheelerndashDeWitt equation to a finite number of degrees of freedom [366]The idea is beautiful but it will develop into a long-lasting industry fromwhich after a while little new will be understood

The decade closes with the main lines of the covariant and the canon-ical theory clearly defined It will soon become clear that neither theoryworks


1970ndash The decade of the seventies opens with a word of caution Reviving a

point made by Pauli a paper by Zumino [367] suggests that the quanti-zation of GR may be problematic and might make sense only by viewing

404 Appendix B

GR as the low-energy limit of a more general theory More than thirtyyears later opinions still diverge on whether this is true

1971ndash Using the technology developed by DeWitt and Feynman for gravity

trsquoHooft and Veltman decide to study the renormalizability of GR Almostas a warm-up exercise they consider the renormalization of YangndashMillstheory and find that the theory is renormalizable ndash a result that wonthem the Nobel prize [368] In a sense one can say that the first physicalresult of the research in quantum gravity is the proof that YangndashMillstheory is renormalizable

ndash David Finkelstein writes his inspiring ldquospacetime coderdquo series of pa-pers [369] (which among other ideas discuss quantum groups)

1973ndash Following the program initiated with Veltman in 1971 trsquoHooft finds

evidence of nonrenormalizable divergences in GR with matter fieldsShortly after trsquoHooft and Veltman as well as Deser and Van Nieuwen-huizen confirm the evidence [370]

1974ndash Hawking announces the derivation of black-hole radiation [248] A

(macroscopic) Schwarzschild black hole of mass M emits thermal radia-tion at the temperature (825) The result comes as a surprise anticipatedonly by the observation by Bekenstein a year earlier that entropy is nat-urally associated with black holes and thus they could be thought of insome obscure sense as ldquohotrdquo [247] and by the BardeenndashCarterndashHawkinganalysis of the analogy between laws of thermodynamics and dynamicalbehavior of black holes [246] Hawkingrsquos result is not directly connectedto quantum gravity ndash it is a skillful application of quantum field theoryin curved spacetime ndash but has a very strong impact on the field It fostersan intense activity in quantum field theory in curved spacetime it opensa new field of research in ldquoblack-hole thermodynamicsrdquo and it opens thequantum-gravitational problems of understanding the statistical origin ofthe black-hole (the BekensteinndashHawking) entropy (827)

An influential clarifying and at the same time intriguing paper is writ-ten two years later by Bill Unruh The paper points out the existence ofa general relation between accelerated observers quantum theory grav-ity and thermodynamics [371] Something deep about Nature should behidden in this tangle of problems but we do not yet know what

1975ndash It becomes generally accepted that GR coupled to matter is not

renormalizable The research program started with Rosenfeld Fierz andPauli is dead

History 405

1976ndash A first attempt to save the covariant program is made by Steven Wein-

berg who explores the idea of asymptotic safety [39] developing earlierideas from Giorgio Parisi [372] Kenneth Wilson and others suggestingthat nonrenormalizable theories could nevertheless be meaningful

ndash To resuscitate the covariant theory even if in modified form thepath has already been indicated find a high-energy modification of GRPreserving general covariance there is not much one can do to modify GRAn idea that attracts much enthusiasm is supergravity [373] it seems thatby simply coupling a spin-32 particle to GR namely with the action (infirst-order form)

S[gΓ ψ] =int

d4xradicminusg

(1

2GRminus i

2εμνρσ ψμγ5γνDρψσ

) (B4)

one can get a theory finite even at two loopsndash Supersymmetric string theory is born [349]

1977ndash Another independent idea is to keep the same kinematics and change

the action The obvious thing to do is to add terms proportional to thedivergences Stelle proves that an action with terms quadratic in the cur-vature

S =int

d4xradicminusg

(αR + βR2 + γRμνRμν

) (B5)

is renormalizable for appropriate values of the coupling constants [375]Unfortunately precisely for these values of the constants the theory is badIt has negative energy modes that make it unstable around the Minkowskivacuum and not unitary in the quantum regime The problem becomesto find a theory renormalizable and unitary at the same time or to cir-cumvent nonunitarity

1978ndash The Hawking radiation is soon re-derived in a number of ways

strongly reinforcing its credibility Several of these derivations point tothermal techniques [376] thus motivating Hawking [40] to revive theWheelerndashMisner ldquoFeynman quantization of general relativityrdquo [353] in theform of a ldquoeuclideanrdquo integral over riemannian 4-geometries g

Z =int

Dg eminusint radic

gR (B6)

Time-ordering and the concept of positive frequency are incorporated intothe ldquoanalytic continuationrdquo to the euclidean sector The hope is double todeal with topology change and that the euclidean functional integral willprove to be a better calculation tool than the WheelerndashDeWitt equation

406 Appendix B

1980ndash Within the canonical approach the discussion focuses on understand-

ing the disappearance of the time coordinate from the WheelerndashDeWitttheory The problem has actually nothing to do with quantum gravitysince the time coordinate disappears in the classical HamiltonndashJacobiform of GR as well and in any case physical observables are coordinateindependent and thus in particular independent from the time coordi-nate in whatever correct formulation of GR But in the quantum contextthere is no single spacetime as there is no trajectory for a quantum par-ticle and the very concepts of space and time become fuzzy This factraises much confusion and a vast interesting discussion (whose many con-tributions I can not possibly summarize here) on the possibility of doingmeaningful fundamental physics in the absence of a fundamental notionof time For early references on the subject see for instance [42 44]

1981ndash Polyakov [377] shows that the cancellation of the conformal anomaly

in the quantization of the string action

S =1

4παprime

intd2σ

radicg gμνpartμX

apartνXbηab (B7)

leads to the critical dimension A new problem is created how to recoverour 4-dimensional world from a string theory which is defined in the crit-ical dimension

1983ndash The hope is still high for supergravity now existing in various ver-

sions as well as for higher-derivative theories whose rescue from nonuni-tarity is explored using a number of ingenious ideas (large-N expansionslarge-d expansions LeendashWick mechanisms ) At the tenth GRG con-ference in Padova in 1983 two physicists of indisputable seriousness GaryHorowitz and Andy Strominger summarize their contributed paper [378]with the words

In sum higher-derivative gravity theories are a viable option forresolving the problem of quantum gravity

At the same conference supergravity is vigorously advertised as the fi-nal solution of the quantum gravity puzzle But very soon it becomesclear that supergravity is nonrenormalizable at higher loops and thathigher-derivatives theories do not lead to viable perturbative expansionsExcitement hope and hype fade away

In its version in 11 dimensions supergravity will find new importancein the late 1990s in connection with string theory High-derivative cor-rections will also re-appear in the low-energy limit of string theory

History 407

ndash Hartle and Hawking [157] introduce the notion of the ldquowave func-tion of the Universerdquo and the ldquono-boundaryrdquo boundary condition for theHawking integral opening up a new intuition on quantum gravity andquantum cosmology But the euclidean integral does not provide a wayof computing genuine field theoretical quantities in quantum gravity anybetter than the WheelerndashDeWitt equation and the atmosphere at themiddle of the eighties is again rather gloomy On the other hand JimHartle [26] develops the idea of a sum-over-histories formulation of GRinto a fully fledged extension of quantum mechanics to the generally co-variant setting The idea will later be developed and formalized by ChrisIsham [379]

ndash Sorkin introduces his poset approach to quantum gravity [380]


1984ndash Green and Schwarz realize that strings might describe ldquoour Universerdquo

[381] Excitement starts to build up around string theory in connectionwith the unexpected anomaly cancellation and the discovery of the het-erotic string [382]

ndash The relation between the ten-dimensional superstrings theory andfour-dimensional low-energy physics is studied in terms of compactifica-tion on CalabyndashYau manifolds [383] and orbifolds The dynamics of thechoice of the vacuum remains unclear but the compactification leads to4d chiral models resembling low-energy physics

ndash Belavin Polyakov and Zamolodchikov publish their analysis of con-formal field theory [384]

1986ndash Goroff and Sagnotti [29] finally compute the two-loop divergences of

pure GR definitely nailing the corpse of pure GR perturbative quantumfield theory into its coffin the divergent term in the effective action is

ΔS =209

737 280π4

1ε

intd4x

radicminusg RμνρσR

ρσεθR

εθμν (B8)

ndash Penrose suggests that the wave function collapse in quantum mechan-ics might be of quantum-gravitational origin [385] The idea is radical andimplies a re-thinking of the basis of mechanics Remarkably the idea maybe testable work is today in progress to study the feasibility of an exper-imental test

ndash String field theory represents a genuine attempt to address the mainproblem of string theory finding a fundamental background-independentdefinition of the theory [386] The string field path however turns out tobe hard

408 Appendix B

ndash The connection formulation of GR is developed by Abhay Ashtekar[132] on the basis of some results by Amitaba Sen [82] At the time thisis denoted the ldquonew variablesrdquo formulation It is a development in clas-sical general relativity but it has long-ranging consequences on quantumgravity as the basis of LQG

1987ndash Fredenhagen and Haag explore the general constraint that general

covariance puts on quantum field theory [387]ndash Green Schwarz and Witten publish their book on superstring theory

In the gauge in which the metric has no superpartner the superstringaction is

S =1

4παprime

intd2σ

radicg

(gμνpartμX

apartνXb minus iψaγμpartμψ

b)ηab (B9)

Interest in the theory grows very rapidly To be sure string theory stillobtains a very small place at the 1991 Marcel Grossmann meeting [388]But the research in supergravity and higher-derivative theories has mergedinto strings and string theory is increasingly viewed as a strong compet-ing candidate for the quantum theory of the gravitational field As aside product many particle physicists begin to study general relativityor at least some bits of it Strings provide a consistent perturbative the-ory The covariant program is fully re-born The problem becomes under-standing why the world described by the theory appears so different fromours

1988ndash Ted Jacobson and Lee Smolin find loop-like solutions to the Wheelerndash

DeWitt equation formulated in the connection formulation [178] openingthe way to LQG

ndash The ldquoloop representation of quantum general relativityrdquo is introducedin [176 177] It is based on the new connection formulation of GR [132]on the JacobsonndashSmolin solutions [178] and on Chris Ishamrsquos ideas onthe need of nongaussian or nonFock representations in quantum gravity[43] Loop quantization had been previously and independently developedby Rodolfo Gambini and his collaborators for YangndashMills theories [179]In the gravitational context the loop representation leads immediately totwo surprising results an infinite family of exact solutions of the WheelerndashDeWitt equation is found and knot theory controls the physical quantumstates of the gravitational field Classical knot theory with its extensionsbecomes a branch of mathematics relevant to describe the diff-invariantstates of quantum spacetime [215] The theory transforms the oldWheelerndashDeWitt theory into a formalism that can be concretely used tocompute physical quantities in quantum gravity The canonical programis fully re-born Nowadays the theory is called ldquoloop quantum gravityrdquo

History 409

ndash Ed Witten introduces the notion of topological quantum field theory(TQFT) [389] In a celebrated paper [390] he uses a TQFT to give afield theoretical representation of the Jones polynomial a knot theoryinvariant The expression used by Witten has an interpretation in LQGit can be seen as the ldquoloop transformrdquo of a quantum state given by theexponential of the ChernndashSimon functional [215]

Formalized by Atiyah [391] the idea of TQFT will have beautiful de-velopments and will strongly influence later developments in quantumgravity General topological theories in any dimensions and in particularBF theory are introduced by Gary Horowitz shortly afterwards [392]

ndash Witten finds an ingenious way of quantizing GR in 2 + 1 spacetimedimensions [393] (thus solving ldquoProblem 56rdquo of the 1963 Wheelerrsquos LesHouches lectures) opening up a big industry of analysis of the theory(for a review see [394]) The quantization method is partially a sumover histories and partially canonical Covariant perturbative quantizationseemed to fail for this theory The theory had been studied a few yearsearlier by Deser Jackiw trsquoHooft Achucarro Townsend and others [395]

1989ndash Amati Ciafaloni and Veneziano find evidence that string theory

implies that distances smaller than the Planck scale cannot be probed[396]

ndash In the string world there is excitement for some nonperturbativemodels of strings ldquoin 0 dimensionrdquo equivalent to 2d quantum gravity[397] The excitement dies fast as is often the case but the models will re-emerge in the nineties [398] and will also inspire the spinfoam formulationof quantum gravity [278]

1992ndash Turaev and Viro [286] define a state sum that on the one hand is a

rigorously defined TQFT and on the other hand can be seen as a reg-ulated and well-defined version of the PonzanondashRegge [283] quantizationof 2+1 gravity Turaev and Ooguri [289] soon find a 4d extension whichwill have a remarkable impact on later developments

ndash The notion of weave is introduced in LQG [189] It is evidence of adiscrete structure of spacetime emerging from LQG The first exampleof a weave which is considered is a 3d mesh of intertwined rings Notsurprisingly the intuition was already in Wheeler (See Figure B1 takenfrom Misner Thorne and Wheeler [399])

1993ndash Gerard rsquot Hooft introduces the idea of holography developed by Lenny

Susskind [375] According to the ldquoholographic principlerdquo the informationon the physical state in the interior of a region can be represented onthe regionrsquos boundary and is limited by the area of this boundary This

410 Appendix B

Fig B1 The weave in Wheelerrsquos vision From Ref [374]

principle can be also interpreted as referring to the information on thesystem accessible from the outside of the region in which case it makesmuch more sense to me

1994ndash Noncommutative geometry often indicated as a tool for describing

certain aspects of Planck-scale geometry finds a strict connection to GR inthe work of Alain Connes Remarkably the ConnesndashChamseddine ldquospec-tral actionrdquo just the trace of a simple function of a suitably defined Dirac-like operator D

S = tr[f(D2(hG))] (B10)

where f is the characteristic function of the [0 1] interval turns out toinclude the standard model action as well as the EinsteinndashHilbert action[401]


1995ndash Nonperturbative aspects of string theory begin to appear branes

[402] dualities [403] the matrix model formulation of M theory[404] (for a review see for instance [405]) The interest in stringsbooms At the plenary conference of a meeting of the American Mathe-matical Society in Baltimore Ed Witten claims that

History 411

The mathematics of the next millennium will be dominated by stringtheory

causing a few eyebrows to raiseThe various dualities appear to relate the different versions of the the-

ory pointing to the existence of a unique fundamental theory The actualconstruction of the fundamental background-independent theory how-ever is still missing and string theory exists so far only in the form of anumber of (related) expansions over assigned backgrounds

ndash Two results in loop gravity appear (i) the overcompleteness of theloop basis is resolved by the discovery of the spin network basis [171] (ii)eigenvalues of area and volume are computed [191] The latter result israpidly extended and derived in a number of alternative ways

The rigorous mathematical framework for LQG starts to be developed[200 201]

ndash Ted Newman and his collaborators introduce the Null Surface For-mulation of GR [406]

1996ndash The BekensteinndashHawking black-hole entropy (822) is computed

within LQG as well as within string theory almost at the same timeThe loop result is obtained by computing the number of (spin network)

states which endow a 2-sphere with a given area [239 249] as well asby loop quantizing the classical theory of the field outside the hole andstudying the boundary states [238] These gravitational surface states[254] can be identified with the states of a ChernndashSimon theory on asurface with punctures [253] The computation is valid for various realisticblack holes The 14 factor in (827) is obtained by fixing the Immirziparameter

In string theory the computation exploits a strong couplingweak cou-pling duality which in certain supersymmetric configurations preservesthe number of states the physical black hole is in a strong coupling situ-ation but the number of its microstates can be computed in a weak-fieldconfiguration that has the same charges at infinity One obtains preciselythe 14 factor of (827) as well as other aspects of the Hawking radiationphenomenology [407] However the calculation method is indirect andworks only for extremal or near-extremal black holes

ndash A rigorously defined finite and anomaly-free hamiltonian constraintoperator is constructed by Thomas Thiemann in LQG [133] Some doubtsare raised on whether the classical limit of this theory is in fact GR (theissue is still open) but the construction defines a consistent generallycovariant quantum field theory in 4d

ndash Intriguing state sum models obtained modifying a TQFT are pro-posed by Barrett and Crane Reisenberger Iwasaki and others as a

412 Appendix B

tentative model for quantum GR All these models appear as sums ofldquospinfoamsrdquo branched surfaces carrying spins

ndash The loop representation is ldquoexponentiatedrdquo a la Feynman givingrise again to a spinfoam model corresponding to canonical LQG Thesedevelopments revive the sum-over-histories approach

1997ndash Intriguing connections between noncommutative geometry and string

theory appear [408]ndash There is a lively discussion on the difficulties of the lattice approaches

in finding a second-order phase transition [409]

1998ndash Juan Maldacena shows [410] that the large-N limit of certain confor-

mal field theories includes a sector describing supergravity on the productof anti-deSitter spacetimes and spheres He conjectures that the com-pactifications of Mstring theory on an anti-deSitter spacetime is dualto a conformal field theory on the spacetime boundary This leads to anew proposal for defining M theory itself in terms of the boundary the-ory an effort to reach background independence (for M theory) usingbackground-dependent methods (for the boundary theory)

A consequence of this ldquoMaldacena conjecturerdquo is an explosion of interestfor rsquot Hooft and Susskind holographic principle (see year 1993)

ndash Two papers in the influential journal Nature [265] raise the hopethat seeing spacetime-foam effects and testing quantum gravity theoriesmight not be as forbidding as usually assumed The idea is that thereare a number of different instances (the neutral kaon system gamma-rayburst phenomenology interferometers ) in which presently operatingmeasurement or observation devices or instruments that are going to besoon constructed involve sensitivity scales comparable to ndash or not toofar from ndash the Planck scale [266] If this direction fails testing quantumgravity might require the investigation of very early cosmology [411]

1999 I stop here because too-recent history is not yet history

B3 The divide

The lines of research that I have summarized in Appendix B2 have foundmany points of contact in the course of their development and have oftenintersected For instance there is a formal way of deriving a sum-over-histories formulation from a canonical theory and vice versa the perturba-tive expansion can also be obtained by expanding the sum-over-historiesstring theory today faces the problem of finding its nonperturbative

History 413

formulation and thus the typical problems of a canonical theory andLQG has mutated into the spinfoam models a sum-over-histories formu-lation using techniques that can be traced to a development of stringtheory of the early nineties However in spite of this continuous cross-fertilization the three main lines of development have kept their essentialseparateness

The three directions of investigation were already clearly identified byCharles Misner in 1959 [353] In the concluding remark of the ConferenceInternationale sur les Theories Relativistes de la Gravitation in 1963Peter Bergmann notes [412]

In view of the great difficulties of this program I consider it a verypositive thing that so many different approaches are being broughtto bear on the problem To be sure the approaches we hope willconverge to one goal

This was 40 years ago The divide is particularly strong between the covariant line of research

more connected to the particle-physics tradition and the canonicalsum-over-histories one more connected to the relativity tradition This dividehas remained through over 70 years of research in quantum gravity Hereis a typical comparison arbitrarily chosen among many On the particle-physics side at the First Marcel Grossmann Meeting Peter van Neuwen-huizen writes [413]

gravitons are treated on exactly the same basis as other parti-cles such as photons and electrons In particular particles (includinggravitons) are always in flat Minkowski space and move as if theyfollowed their geodesics in curved spacetime because of the dynamicsof multiple graviton exchange Pure relativists often become some-what uneasy at this point because of the following two aspects entirelypeculiar to gravitation (1) One must decide before quantizationwhich points are spacelike separated but it is only after quantizationthat the fully quantized metric field can tell us this spacetime struc-ture (2) In a classical curved background one needs positiveand negative frequency solutions but in non-stationary spacetimes itis not clear whether one can define such solutions The strategy of par-ticle physicists has been to ignore these problems for the time beingin the hope that they will ultimately be resolved in the final theoryConsequently we will not discuss them any further

On the relativity side Peter Bergmann comments [414]

The world-point by itself possesses no physical reality It acquires real-ity only to the extent that it becomes the bearer of specific propertiesof the physical fields imposed on the spacetime manifold

The conceptual divide is huge Partially it reflects the different under-standing of the world held by the particle-physics community on the one

414 Appendix B

hand and the relativity community on the other The two communitieshave made repeated and sincere efforts to talk to each other and under-stand each other But the divide remains Both sides have the feeling thatthe other side is incapable of appreciating something basic and essentialthe structure of QFT as it has been understood over half a century of in-vestigation on the particle-physics side the novel physical understandingof space and time that has appeared with GR on the relativity side Bothsides expect that the point of view of the other will turn out at the endof the day to be not very relevant One side because the experience withQFT is on a fixed metric spacetime and thus is irrelevant in a genuinelybackground-independent context The other because GR is only a low-energy limit of a much more complex theory and thus cannot be takentoo seriously as an indication about the deep structure of Nature Hope-fully the recent successes of both lines will force the two sides finallyto face the problems that the other side considers prioritary backgroundindependence on the one hand control of a perturbation expansion onthe other

After 70 years of research there is no consensus no established theoryand no theory that has yet received any direct or indirect experimentalsupport In the course of 70 years many ideas have been explored fashionshave come and gone the discovery of the Holy Grail has been several timesannounced with much later scorn Ars longa vita brevis

However in spite of its age the research in quantum gravity does notseem to have been meandering meaninglessly when seen in its entiretyOn the contrary one sees a logic that has guided the development of theresearch from the early formulation of the problem and the research di-rections in the fifties to nowadays The implementation of the programshas been extremely laborious but has been achieved Difficulties haveappeared and solutions have been proposed which after much difficultyhave led to the realization at least partial of the initial hopes It wassuggested in the early seventies that GR could perhaps be seen as thelow-energy theory of a theory without uncontrollable divergences today30 years later such a theory ndash string theory ndash is known In 1957 CharlesMisner indicated that in the canonical framework one should be able tocompute eigenvalues and in 1995 37 years later eigenvalues were com-puted ndash within loop quantum gravity The road is not yet at the endmuch remains to be understood some of the current developments mightlead nowhere But looking at the entire development of the subject it isdifficult to deny that there has been progress

Appendix C

On method and truth

I collect in this appendix some simple reflections on scientific methodology and onthe content of scientific theories relevant for quantum gravity In particular I try tomake more explicit the methodological assumptions at the root of some of the researchdescribed in this book and to give it some justification

I am no professional philosopher and what follows has no ambition in that senseI am convinced however of the utility of a dialog between physics and philosophyThis dialog has played a major role during the other periods in which science has facedfundamental problems I think that most physicists underestimate the effect of theirown epistemological prejudices on their research And many philosophers underestimatethe effect ndash positive or negative ndash they have on fundamental research On the one handa more acute philosophical awareness would greatly help the physicists engaged infundamental research As I have argued in Chapter 1 during the second half of thetwentieth century fundamentals were clear in theoretical physics and the problems weretechnical but today foundational problems are back on the table as they were at thetime of Newton Faraday Heisenberg and Einstein These physicists couldnrsquot certainlyhave done what they have done if they werenrsquot nurtured by (good or bad) philosophyOn the other hand I wish contemporary philosophers concerned with science would bemore interested in the ardent lava of the fundamental problems science is facing todayIt is here I believe that stimulating and vital issues lie


Part of the reflection about the science of the last decades has emphasizedthe ldquononcumulativerdquo aspect in the development of scientific knowledgethe evolution of scientific theories is marked by large or small break-ing points in which to put it crudely empirical facts are reorganizedwithin new theories which are to some extent ldquoincommensurablerdquo withrespect to their antecedent These ideas ndash correctly understood or misun-derstood ndash have had a strong influence on the physicists

The approach to quantum gravity described in this book assumes a dif-ferent reading of the evolution of scientific knowledge Indeed I have basedthe discussion on quantum gravity on the expectation that the central

415

416 Appendix C

physical tenets of QM and GR represent our best guide for accessingeven the extreme and unexplored territories of the quantum-gravitationalregime In my opinion the emphasis on the incommensurability betweentheories has clarified an important aspect of science but risks to obscuresomething of the internal logic according to which historically physicshas extended knowledge There is a subtle but very definite cumulativeaspect in the progress of physics which goes far beyond the growth in thevalidity and precision of the empirical content of the theories In movingfrom a theory to the theory that supersedes it we do not save only theverified empirical content of the old theory but more This ldquomorerdquo is acentral concern for good physics It is the source I think of the spectac-ular and undeniable predictive power of theoretical physics I think thatby playing it down one risks misleading theoretical research into a lesseffective methodology

Let me illustrate this point with a historical case There was a problembetween Maxwell equations and Galileo transformations There were twoobvious ways out To consider Maxwell theory as a theory of limitedvalidity a phenomenological theory of some yet-to-be-discovered aetherrsquosdynamics Or to consider that galilean equivalence of inertial systemshad limited validity thus accepting the idea that inertial systems arenot equivalent in electromagnetic phenomena Both directions are logicaland were pursued at the end of the nineteenth century Both are soundapplications of the idea that a scientific revolution changes in depth whatold theories teach us about the world Which of the two ways did Einsteinsuccessfully take

Neither For Einstein Maxwell theory was a source of awe He rhap-sodizes about his admiration for the theory For him Maxwell had openeda new window on the world Given the astonishing success of Maxwelltheory empirical (electromagnetic waves) technological (radio) as wellas conceptual (understanding the nature of light) Einsteinrsquos admirationis comprehensible But Einstein had a tremendous respect for Galileorsquosinsight as well Young Einstein was amazed by a book with Huygensrsquoderivation of collision theory virtually out of galilean invariance aloneEinstein understood that Galileorsquos great intuition ndash that the notion ofvelocity is only relative ndash could not be wrong I am convinced that in thisfaith of Einstein in the core of the great galilean discovery there is verymuch to learn for the philosophers of science as well as for the con-temporary theoretical physicists So Einstein believed the two theoriesMaxwell and Galileo and assumed that their tenets would hold far beyondthe regime in which they had been tested He assumed that Galileo hadgrasped something about the physical world which was simply correctAnd so had Maxwell Of course details had to be adjusted The coreof Galileorsquos insight was that all inertial systems are equivalent and that


velocity is relative not the details of the galilean transformations Einsteinknew the Lorentz transformations (found by Poincare) and was able tosee that they do not contradict Galileorsquos insight If there was contradictionin putting the two together the problem was ours we were surreptitiouslysneaking some incorrect assumption into our deductions He found the in-correct assumption which of course was that simultaneity could be welldefined It was Einsteinrsquos faith in the essential physical correctness of theold theories that guided him to his spectacular discovery

There are very many similar examples in the history of physics thatcould equally well illustrate this point Einstein found GR ldquoout of purethoughtrdquo having Newton theory on the one hand and special relativ-ity ndash the understanding that any interaction is mediated by a field ndash onthe other Dirac found quantum field theory from Maxwell equations andquantum mechanics Newton combined Galileorsquos insight that accelerationgoverns dynamics with Keplerrsquos insight that the source of the force thatgoverns the motion of the planets is the Sun The list could be long Inall these cases confidence in the insight that came with some theory orldquotaking a theory seriouslyrdquo led to major advances that greatly extendedthe original theory itself Far be it from me to suggest that there is any-thing simple or automatic in figuring out where the true insights areand in finding the way of making them work together But what I amsaying is that figuring out where the true insights are and finding theway of making them work together is the work of fundamental physicsThis work is grounded on confidence in the old theories not on a randomsearch for new ones

One of the central concerns of the modern philosophy of science is toface the apparent paradox that scientific theories change but are never-theless credible Modern philosophy of science is to some extent an af-tershock reaction to the fall of newtonian mechanics A tormented recog-nition that an extremely successful scientific theory can nevertheless beuntrue But I think that a notion of truth which is challenged by the eventof a successful physical theory being superseded by a more successful oneis a narrow-minded notion of truth

A physical theory is a conceptual structure that we develop and use inorder to organize read and understand the world and make predictionsabout it A successful physical theory is a theory that does so effectivelyand consistently In the light of our experience there is no reason not toexpect that a more effective conceptual structure might always exist Aneffective theory may always show its limits and be replaced by a betterone However a novel conceptualization cannot but rely on what theprevious one has already understood Thought is in constant evolutionand in constant reorganization It is not a static entity Science is itselfthe process of the evolution of thinking

418 Appendix C

When we move to a new city we are at first confused about its geog-raphy Then we find some reference points and we make a first roughmental map of the city in terms of these points Perhaps there is partof the city on the hills and part on the plain As time goes on the mapimproves There are moments in which we suddenly realize that we hadit wrong Perhaps there were two areas with hills and we were previ-ously confusing the two Or we had mistaken a large square called Earthsquare for the downtown while downtown was farther away around asquare called Sun square So we update the mental map Sometime laterwe have learned names and features of neighborhoods and streets andthe hills as references fade away The neighborhood structure of knowl-edge is more effective than the hillplain one The structure changesbut the knowledge increases And Earth square now we know it is notdowntown and we know it forever

There are discoveries that are forever That the Earth is not the cen-ter of the Universe that simultaneity is relative that absolute velocity ismeaningless That we do not get rain by dancing These are steps human-ity takes and does not take back Some of these discoveries amount simplyto clearing from our thinking wrong encrusted or provisional credencesBut also discovering classical mechanics or discovering electromagnetismor quantum mechanics are discoveries forever Not because the details ofthese theories cannot change but because we have discovered that a largeportion of the world admits to being understood in certain terms andthis is a fact that we will have to keep facing forever

One of the main theses of this book is that general relativity is theexpression of one of these insights which will stay with us ldquoforeverrdquo Theinsight is that the physical world does not have a stage that localiza-tion and motion are relational only that this background independenceis required for any fundamental description of our world

How can a theory be effective even outside the domain for which it wasfound How could Maxwell predict radio waves Dirac predict antimatterand GR predict black holes How can theoretical thinking be so magicallypowerful

It has been suggested that these successes are due to chance and seemgrand only because of the historically deformed perspective A sort ofdarwinian natural selection for theories has been suggested there arehundreds of theories proposed most of them die the ones that surviveare the ones remembered There is alway somebody who wins the lotterybut this is not a sign that humans can magically predict the outcome ofthe lottery My opinion is that such an interpretation of the developmentof science is unjust and worse misleading It may explain something butthere is more in science There are tens of thousands of persons playingthe lottery there were only two relativistic theories of gravity in 1916


when Einstein predicted that light would be deflected by the Sun preciselyby an angle of 175 seconds of arc Familiarity with the history of physicsI feel confident to claim rules out the lottery picture

I think that the answer is simpler Say somebody predicts that theSun will rise tomorrow and the Sun rises This successful prediction isnot a matter of chance there arenrsquot hundreds of people making randompredictions about all sorts of strange objects appearing at the horizonThe prediction that tomorrow the Sun will rise is sound However itcannot be taken for granted A neutron star could rush in close to thespeed of light and sweep the Sun away Who or what grants the rightof induction Why should I be confident that the Sun will rise just be-cause it has been rising so many times in the past I do not know theanswer to this question But what I know is that the predictive power ofa theory beyond its own domain is precisely of the same sort Simply welearn something about Nature and what we learn is effective in guidingus to predict Naturersquos behavior Thus the spectacular predictive powerof theoretical physics is nothing less and nothing more than commoninduction it follows from the successful assumption that there are reg-ularities in Nature at all levels The spectacular success of science inmaking predictions about territories not yet explored is as comprehensi-ble (or as incomprehensible) as my ability to predict that the Sun willrise tomorrow Simply Nature around us happens to be full of regular-ities that we recognize whether or not we understand why regularitiesexist at all These regularities give us strong confidence ndash although notcertainty ndash that the Sun will rise tomorrow as well as that the basic factsabout the world found with QM and GR will be confirmed not violatedin the quantum-gravitational regimes that we have not yet empiricallyprobed

This view is not dominant in theoretical physics nowadays Other at-titudes dominate The ldquopessimisticrdquo scientist has little faith in the pos-sibilities of theoretical physics because he worries that all possibilitiesare open and anything might happen between here and the Planck scaleThe ldquowildrdquo scientist observes that great scientists had the courage tobreak with ldquoold and respected assumptionsrdquo and to explore some novelldquostrangerdquo hypotheses From this observation the ldquowildrdquo scientist con-cludes that to do great science one has to explore all sorts of strangehypotheses and to violate respected ideas The wilder the hypothesis thebetter I think wildness in physics is sterile The greatest revolutionaries inscience were extremely almost obsessively conservative So was certainlythe greatest revolutionary Copernicus and so was Planck Copernicuswas pushed to the great jump from his pedantic labor on the minute tech-nicalities of the ptolemaic system (fixing the equant) Kepler was forcedto abandon the circles by his extremely technical work on the details

420 Appendix C

of Marsrsquo orbit He was using ellipses as approximations to the epicycle-deferent system when he began to realize that the approximation wasfitting the data better than the supposedly exact curve And Einsteinand Dirac were also extremely conservative Their vertigo-inducing leapsahead were not pulled out of the blue sky They did not come from thethrill of violating respected ideas or trying a new pretty idea They wereforced out of respect towards previous physical insights Today insteadwe have plenty of seminars on ldquoa new pretty ideardquo regularly soon forgot-ten and superseded by a new fad In physics novelty has always emergedfrom new data or from a humble devoted interrogation of the old theo-ries From turning these theories over and over in onersquos mind immersingoneself in them making them clash merge talk until through them themissing gear can be seen

Finally the ldquopragmaticrdquo scientist ignores conceptual questions andphysical insights and only cares about developing a theory This is anattitude successful in the sixties in arriving at the standard model But inthe sixties empirical data were flowing in daily to keep research on trackToday theoreticians have no new data The ldquopragmaticrdquo theoretician doesnot care He does not trust the insight of the old theories He focuses onlyon the development of the novel theory and cannot care less if the worldpredicted by the theory resembles less and less the world we see He iseven excited that the theory looks so different from the world thinkingthat this is evidence of how much ahead he has advanced in knowledgewhich is a complete nonsense Theoretical physics becomes a mental gameclosed in on itself and the connection with reality is lost

In my opinion precious research energies are today wasted in these at-titudes A philosophy of science that downplays the component of factualknowledge in physical theories might have part of the responsibility

C2 On realism

A scientific theory is a conceptual structure that we use to read organizeand understand the world at some level of our knowledge It is one stepalong a process since knowledge increases In my view scientific thinkingis not much different from common-sense thinking In fact it is only a bet-ter instance of the same activity thinking about the world and updatingour mental schemes Science is the organized enterprise of continuouslyexploring the possible ways of thinking about the world and constantlyselecting the ones that work best

If this is correct there cannot be any qualitative difference between thetheoretical notions introduced in science and the terms in our everydaylanguage A fundamental intuition of classical empiricism is that nothinggrants us that the concepts that we use to organize our perceptions refer


to ldquorealrdquo entities Some modern philosophy of science has emphasized theapplication of this intuition to the concepts introduced by science Thuswe are warned to doubt the ldquorealityrdquo of the theoretical objects (electronsfields black holes ) I find these warnings incomprehensible Not be-cause they are ill founded but because they are not applied consistentlyThe fathers of empiricism consistently applied this intuition to any phys-ical object Who grants me the reality of a chair Why should a chairbe more than a theoretical concept organizing certain regularities in myperceptions I will not venture here in disputing nor in agreeing with thisdoctrine What I find incomprehensible is the position of those who grantthe solid status of reality to a chair but not to an electron The argu-ments against the reality of the electron apply to the chair as well Thearguments in favor of the reality of the chair apply to the electron as wellA chair as well as an electron is a concept that we use to read organizeand understand the world They are equally real They are equally volatileand uncertain

Perhaps this curious schizophrenic attitude of being antirealist withelectrons and iron-realist with chairs is the result of a tortuous historicalevolution initiated by the rebellion against ldquometaphysicsrdquo and with itthe granting of confidence to science alone From this point of view meta-physical questioning on the reality of chairs is sterile ndash true knowledge is inscience Thus it is to scientific knowledge that we apply empiricist rigorBut understanding science in empiricistsrsquo terms required making sense ofthe raw empirical data on which science is based With time the idea ofraw empirical data showed more and more its limits The common-senseview of the world was reconsidered as a player in our picture of knowl-edge This common-sense view should give us a language and a foundationfrom which to start ndash the old antimetaphysical prejudice still preventingus however from applying empiricist rigor to this common-sense view ofthe world But if one is not interested in questioning the reality of chairsfor the very same reason why should one be interested in questioning theldquoreality of the electronsrdquo

Again I think this point is important for science itself The factual con-tent of a theory is our best tool The faith in this factual content does notprevent us from being ready to question the theory itself if sufficientlycompelled to do so by novel empirical evidence or by putting the theoryin relation to other things we know about the world or we learn about itScientific antirealism in my opinion is not only a shortsighted applicationof a deep classical empiricist insight it is also a negative influence overthe development of science H Stein (private communication) has recentlybeautifully illustrated a case in which a great scientist Poincare wasblocked from getting to a major discovery (special relativity) by a philos-ophy that restrained him from ldquotaking seriouslyrdquo his own findings

422 Appendix C

Science teaches us that our naive view of the world is imprecise inap-propriate biased It constructs better views of the world (Better for someuse worse for others of course which is why it is silly to think of ourgirlfriend as a collection of electrons) Electrons if anything are ldquomorerealrdquo than chairs not ldquoless realrdquo in the sense that they underpin a way ofconceptualizing the world which is in many respects more powerful Onthe other hand the process of scientific discovery and the experience ofthe twentieth century in particular has made us painfully aware of theprovisional character of any form of knowledge Our mental and mathe-matical pictures of the world are only mental and mathematical picturesBetween our images of reality and our experience of reality there is alwaysan hiatus This is true for abstract scientific theories as well as for theimage we have of our dining room (not to even mention the image we haveof our girlfriend) Nevertheless the pictures are effective and we canrsquot doany better

C3 On truth

So is there anything we can say with confidence about the ldquoreal physicalworldrdquo A large part of the recent reflection on science has taught us thatraw data do not exist that any information about the world is alreadydeeply filtered and interpreted by the theory and that theories are alllikely to be superseded It has been useful and refreshing to learn thisFar more radically the European reflection and part of the American aswell has emphasized the fact that truth is always internal to the theorythat we can never leave language that we can never exit the circle ofdiscourse within which we are speaking As a scientist I appreciate andshare these ideas

But the fact that the only notion of truth is internal to our discoursedoes not imply that we should lose confidence in it If truth is internal toour discourse then this internal truth is what we mean by truth Indeedthere may be no valid notion of truth outside our own discourse but it isprecisely ldquofrom withinrdquo this discourse not from without it that we canand do assert the truth of the reality of the world and the truth of whatwe have learned about it More significantly still it is structural to ourlanguage to be a language about the world and to our thinking to be athinking of the world1

Therefore there is no sense in denying the truth of what we have learnedabout the world precisely because there is no notion of truth except the

1The rational investigation of the world started with the pre-Socratic λoγoς (logos)which is the principle (that we seek) governing the cosmos as well as human reasoningand speaking about the cosmos it is at the same time the truth and our reasoningabout it


one within our own discourse If there is no place we can go which isoutside our language in which place are they standing those who questionthe truth we find It can only be a pleasant short dreamy place wherewe are happy to stay for a short while smiling as if we were wise andthen come back to reality The world is real solid and understandableprecisely because the language our only home states so The best we cansay about the physical world and about what is physically real out thereis what good physics says about it2

At the same time there is no reason that our perceiving understand-ing and conceptualizing the world should not be in continuous evolutionScience is the form of this evolution At every stage the best we can sayabout the reality of the world is precisely what we are saying The factwe will understand it better later on does not make our present under-standing less valuable or less credible When we walk in the mountainswe do not dismiss our map just because there may exist a better mapwhich we donrsquot have Searching for a fixed point on which to rest ourrestlessness is in my opinion naive useless and counterproductive forthe development of knowledge It is only by believing our insights andat the same time questioning our mental habits that we can go ahead Ibelieve that this process of cautious faith and self-confident doubt is thecore of scientific thinking Science is the human adventure that consists inexploring possible ways of thinking of the world Being ready to subvertif required anything we have been thinking so far

I think this is among the best of human adventures Research in quan-tum gravity in its effort to conceptualize quantum spacetime and thusmodify in depth the notions of space and time is a step in this adventure

2I certainly do not wish to suggest that the physical description of the world exhaustsit It would be like saying that if I understand the physics of a brick I immediatelyknew why a cathedral stands or why it is splendid

References

Preface and terminology and notation

[1] C Rovelli Loop space representation In New Perspectives in CanonicalGravity ed A Ashtekar et al (Napoli Bibliopolis 1988)

[2] C Rovelli Ashtekar formulation of general relativity and loop space non-perturbative Quantum Gravity a report Class and Quantum Grav 8(1991) 1613ndash1675

[3] A Ashtekar Non-perturbative Canonical Gravity (Singapore World Sci-entific 1991)

[4] L Smolin Time measurement and information loss in quantum cos-mology In Brill Feschrift Proceedings ed B Hu and T Jacobson(Cambridge Cambridge University Press 1993) Recent developments innonperturbative quantum gravity In Quantum Gravity and Cosmology edJ Perez-Mercader J Sola and E Verdaguer (Singapore World Scientific1993)

[5] J Baez Knots and Quantum Gravity (Oxford Oxford University Press1994)

[6] B Brugmann Loop representations In Canonical Gravity from Classicalto Quantum ed J Ehlers and H Friedrich (Berlin Springer-Verlag 1994)

[7] R Gambini and J Pullin Loops Knots Gauge Theories and QuantumGravity (Cambridge Cambridge University Press 1996)

[8] J Kowalski-Glikman Towards quantum gravity Lecture Notes in Physics541 (2000) (Berlin Springer)

[9] A Ashtekar Background independent quantum gravity A Status reportClass Quant Grav 21 (2004) R53

[10] L Smolin An invitation to loop quantum gravity hep-th0408048[11] T Thiemann Lectures on loop quantum gravity Lecture Notes in Physics

631 (2003) 41ndash135 gr-qc0210094[12] C Rovelli Loop quantum gravity Living Reviews in Relativity electronic

journal http wwwlivingreviewsorgArticlesVolume11998-1rovelli

424

References 425

[13] A Ashtekar Quantum geometry and gravity recent advances to appear inthe Proc 16th Int Conf on General Relativity and Gravitation DurbanS Africa July 2001 gr-qc9901023

[14] M Gaul and C Rovelli Loop quantum gravity and the meaning of diffeo-morphism invariance Lecture Notes in Physics 541 (2000) 277ndash324 (BerlinSpringer) gr-qc9910079

[15] C Rovelli and P Upadhya Loop quantum gravity and quanta of space aprimer gr-qc9806079

[16] J Baez and J Muniain Gauge Fields Knots and Gravity (SingaporeWorld Scientific 1994)

[17] JC Baez An introduction to spin foam models of BF theory and quan-tum gravity In Geometry and Quantum Physics ed H Gausterer andH Grosse Lecture Notes in Physics 543 (1999) 25ndash94 (Berlin Springer-Verlag) gr-qc9905087

[18] D Oriti Spacetime geometry from algebra spin foam models fornon-perturbative quantum gravity Rept Prog Phys 64 (2001) 1489gr-qc0106091

[19] A Perez Spin foam models for quantum gravity Class and QuantumGrav 20 (2002) gr-qc0301113

[20] T Thiemann Modern Canonical Quantum General Relativity (CambridgeCambridge University Press 2004 in press) a preliminary version is in gr-qc0110034

[21] J Ambjorn B Durhuus and T Jonsson Quantum Geometry (CambridgeCambridge University Press 1997)

Chapter 1 General ideas and heuristic picture

[22] M Gell-Mann Strange Beauty (London Vintage 2000) pp 303ndash304[23] L Smolin Towards a background independent approach to M theory

hep-th9808192 The cubic matrix model and duality between strings andloops hep-th0006137 A candidate for a background independent formu-lation of M theory Phys Rev D62 (2000) 086001 hep-th9903166

[24] L Smolin Strings as perturbations of evolving spin networks Nucl PhysProc Suppl 88 (2000) 103ndash113 hep-th9801022


[26] J Hartle Spacetime quantum mechanics and the quantum mechanicsof spacetime In Proceedings 1992 Les Houches School Gravitation andQuantisation ed B Julia and J Zinn-Justin (Paris Elsevier Science1995) p 285

[27] SA Fulling Aspects of Quantum Field Theory in Curved Spacetime(Cambridge Cambridge University Press 1989)

[28] RM Wald Quantum Field Theory on Curved Spacetime and Black HoleThermodynamics (Chicago University of Chicago Press 1994)

426 References

[29] MH Goroff and A Sagnotti Quantum gravity at two loops Phys LettB160 (1985) 81 The ultraviolet behaviour of Einstein gravity Nucl PhysB266 (1986) 709

[30] G Horowitz Quantum gravity at the turn of the millenium plenary talkat the Marcell Grossmann Conf Rome 2000 gr-qc0011089

[31] S Carlip Quantum gravity a progress report Rept Prog Phys 64 (2001)885 gr-qc0108040

[32] CJ Isham Conceptual and geometrical problems in quantum gravity InRecent Aspects of Quantum Fields ed H Mitter and H Gausterer (BerlinSpringer-Verlag 1991) p 123

[33] C Rovelli Strings loops and the others a critical survey on the present ap-proaches to quantum gravity In Gravitation and Relativity At the turn ofthe Millenium ed N Dadhich and J Narlikar (Pune Inter-University Cen-tre for Astronomy and Astrophysics 1998) pp 281ndash331 gr-qc9803024

[34] C Callender and N Hugget eds Physics Meets Philosophy at the PlanckScale (Cambridge Cambridge University Press 2001)

[35] C Rovelli Halfway through the woods In The Cosmos of Scienceed J Earman and JD Norton (University of Pittsburgh Press andUniversitats Verlag-Konstanz 1997)

[36] C Rovelli Quantum spacetime what do we know In Physics MeetsPhilosophy at the Planck Length ed C Callender and N Hugget(Cambridge Cambridge University Press 1999) gr-qc9903045

[37] C Rovelli The century of the incomplete revolution searching for generalrelativistic quantum field theory J Math Phys Special Issue 2000 41(2000) 3776 hep-th9910131

[38] JA Wheeler Superspace and the nature of quantum geometrodynamicsIn Batelle Rencontres 1967 ed C DeWitt and JW Wheeler Lectures inMathematics and Physics 242 (New York Benjamin 1968)

[39] S Weinberg Ultraviolet divergences in quantum theories of gravitationIn General Relativity An Einstein Centenary Survey ed SW Hawkingand W Israel (Cambridge Cambridge University Press 1979)

[40] SW Hawking The path-integral approach to quantum gravity In GeneralRelativity An Einstein Centenary Survey ed SW Hawking and W Israel(Cambridge Cambridge University Press 1979)

[41] SW Hawking Quantum cosmology In Relativity Groups and TopologyLes Houches Session XL ed B DeWitt and R Stora (Amsterdam NorthHolland 1984)

[42] K Kuchar Canonical methods of quantization In Oxford 1980 Proceed-ings Quantum Gravity 2 (Oxford Oxford University Press 1984)

[43] CJ Isham Topological and global aspects of quantum theory In RelativityGroups and Topology Les Houches 1983 ed BS DeWitt and R Stora(Amsterdam North Holland 1984)

[44] CJ Isham Quantum gravity an overview In Oxford 1980 ProceedingsQuantum Gravity 2 (Oxford Oxford University Press 1984)

References 427

[45] C J Isham Structural problems facing quantum gravity theory In Proc14th Int Conf on General Relativity and Gravitation ed M FrancavigliaG Longhi L Lusanna and E Sorace (Singapore World Scientific 1997)pp 167ndash209

[46] MB Green J Schwarz and E Witten Superstring Theory (CambridgeCambridge University Press 1987) J Polchinski String Theory(Cambridge Cambridge University Press 1998)

[47] C Rovelli A dialog on quantum gravity Int J Mod Phys 12 (2003) 1hep-th0310077

[48] L Smolin How far are we from the quantum theory of gravity hep-th0303185

[49] A Connes Non Commutative Geometry (New York Academic Press1994)

[50] L Smolin Three Roads to Quantum Gravity (Oxford Oxford UniversityPress 2000)

[51] KS Robinson Blue Mars (New York Bantam 1996)[52] G Egan Schild Ladder (London Gollancz 2001)[53] E Palandri Anna prende il volo (Milano Feltrinelli 2000)

Chapter 2 General relativity

[54] S Holst Barberorsquos Hamiltonian derived from a generalized Hilbert-Palatini action Phys Rev D53 (1996) 5966ndash5969

[55] L Russo La rivoluzione dimenticata (Milano Feltrinelli 1997)[56] JP Bourguignon and P Gauduchon Spineurs operateurs de Dirac et

variations de metriques Comm Math Phys 144 (1992) 581[57] T Schucker Forces from Connesrsquo geometry hep-th0111236 Lectures at

the Autumn School Topology and Geometry in Physics Rot an der Rot2001 ed E Bick and F Steffen (Lecture Notes in Physics Springer 2004)

[58] L Russo Flussi e riflussi (Feltrinelli Milano 2003)[59] M Faraday Experimental Researches in Electricity (London Bernard

Quaritch 1855) pp 436ndash437[60] R Descartes Principia Philosophiae (1644) Translated by VR Miller and

RP Miller (Dordrecht Reidel 1983)[61] I Newton De Gravitatione et Aequipondio Fluidorum translation in

Unpublished Papers of Isaac Newton ed AR Hall and MB Hall(Cambridge Cambridge University Press 1962)

[62] I Newton Principia Mathematica Philosophia Naturalis 1687 Englishtranslation The Principia Mathematical Principles of Natural Philosophy(City University of California Press 1999)

[63] A Einstein and M Grossmann Entwurf einer verallgemeinerten Rela-tivitatstheorie und einer Theorie der Gravitation Z fur Mathematik undPhysik 62 (1914) 225

428 References

[64] A Einstein Grundlage der allgemeinen Relativitatstheorie Ann der Phys49 (1916) 769ndash822

[65] M Pauri and M Vallisneri Ephemeral point-events is there a last rem-nant of physical objectivity DIALOGOS 79 (2002) 263ndash303 L Lusannaand M Pauri General covariance and the objectivity of space-time point-events the physical role of gravitational and gauge degrees of freedomhttpphilsci-archivepitteduarchive00000959 (2002)

[66] P Hajicek Lecture Notes in Quantum Cosmology (Bern University ofBern 1990)

[67] AS Eddington The Nature of the Physical World (New York MacMillan1930) pp 99ndash102

[68] SJ Earman A Primer on Determinism (Dordrecht D Reidel 1986)[69] B DeWitt in Gravitation An Introduction to Current Research ed

L Witten (New York Wiley 1962)[70] JD Brown and D Marolf On relativistic material reference systems Phys

Rev D53 (1996) 1835[71] C Rovelli What is observable in classical and quantum gravity Class

and Quantum Grav 8 (1991) 297 Quantum reference systems Class andQuantum Grav 8 (1991) 317

[72] PG Bergmann Phys Rev 112 (1958) 287 Observables in general covari-ant theories Rev Mod Phys 33 (1961) 510

[73] BW Parkinson and JJ Spilker eds Global Positioning System The-ory and Applications Prog in Astronautics and Aeronautics Nos 163ndash164 (Amer Inst Aero Astro Washington 1996) ED Kaplan Under-standing GPS Principles and Applications Mobile Communications Se-ries (Boston Artech House 1996) B Hofmann-Wellenhof H Lichteneg-ger and J Collins Global Positioning System Theory and Practice (NewYork Springer-Verlag 1993)

[74] B Guinot Application of general relativity to metrology Metrologia 34(1997) 261 F de Felice MG Lattanzi A Vecchiato and PL BernaccaGeneral relativistic satellite astrometry I A non-perturbative approachto data reduction Astron Astrophy 332 (1998) 1133 TB BahderFermi Coordinates of an Observer Moving in a Circle in MinkowskiSpace Apparent Behavior of Clocks Army Research Laboratory AdelphiMaryland USA Technical Report ARL-TR-2211 May 2000 ARThompson JM Moran and GW Swenson Interferometry and Synthe-sis in Radio Astronomy (Malabar Florida Krieger Pub Co 1994) pp138ndash139 PNAM Visser Gravity field determination with GOCE andGRACE Adv Space Res 23 (1999) 771

[75] S Weinberg Gravitation and Cosmology (New York Wiley 1972)[76] RM Wald General Relativity (Chicago The University of Chicago Press

1989)[77] Y Choquet-Bruhat C DeWitt-Morette and M Dillard-Bleick Analysis

Manifolds and Physics (Amsterdam North Holland 1982)

References 429

[78] I Ciufolini and J Wheeler Gravitation and Inertia (Princeton PrincetonUniversity Press 1996)

[79] H Weyl Electron and gravitation Z Physik 56 (1929) 330[80] J Schwinger Quantized gravitational field Phys Rev 130 (1963) 1253[81] JF Plebanski On the separation of Einsteinian substructures J Math

Phys 18 (1977) 2511[82] A Sen Gravity as a spin system Phys Lett 119B (1982) 89[83] J Samuel A lagrangian basis for Ashtekarrsquos reformulation of canonical

gravity Pramana J Phys 28 (1987) L429 T Jacobson and L SmolinCovariant action for Ashtekarrsquos form of canonical gravity Class andQuantum Grav 5 (1988) 583

[84] R Capovilla J Dell and T Jacobson General relativity without the met-ric Phys Rev Lett 63 (1991) 2325 R Capovilla J Dell T Jacobsonand L Mason Selfndashdual 2ndashforms and gravity Class and Quantum Grav8 (1991) 41

[85] JD Norton How Einstein found his field equations 1912ndash1915 HistoricalStudies in the Physical Sciences 14 (1984) 253ndash315 Reprinted in Einsteinand the History of General Relativity Einstein Studies ed D Howard andJ Stachel Vol I (Boston Birkhauser 1989) pp 101ndash159

[86] J Stachel Einsteinrsquos search for general covariance 1912ndash1915 In Einsteinand the History of General Relativity Einstein Studies ed D Howard andJ Stachel Vol 1 (Boston Birkhauser 1989) pp 63ndash100

[87] E Kretschmann Uber den physikalischen Sinn der RelativitatpostulateAnn Phys Leipzig 53 (1917) 575

[88] JL Anderson Principles of Relativity Physics (New York AcademicPress 1967)

[89] J Barbour Absolute or Relative Motion (Cambridge Cambridge Univer-sity Press 1989)

[90] J Earman and J Norton What price spacetime substantivalism The holestory Brit J Phil Sci 38 (1987) 515ndash525

[91] J Earman World Enough and Space-time Absolute Versus RelationalTheories of Spacetime (Cambridge MIT Press 1989)

[92] G Belot Why general relativity does need an interpretation Phil Sci 63(1998) S80ndashS88

[93] J Earman and G Belot Pre-Socratic quantum gravity In Physics MeetsPhilosphy at the Planck Scale ed C Callander (Cambridge CambridgeUniversity Press 2001)

[94] C Rovelli Analysis of the different meaning of the concept of time indifferent physical theories Il Nuovo Cimento 110B (1995) 81

[95] JT Fraser Of Time Passion and Knowledge (Princeton PrincetonUniversity Press 1990)

[96] H Reichenbach The Direction of Time (Berkeley University of CaliforniaPress 1956) PCW Davies The Physics of Time Asymmetry (England

430 References

Surrey University Press 1974) R Penrose in General Relativity AnEinstein Centenary Survey ed SW Hawking and W Israel (CambridgeCambridge University Press 1979) HD Zee The Physical Basis ofthe Direction of Time (Berlin Springer 1989) J Halliwel and JAPerez-Mercader eds Proceedings of the International Workshop PhysicalOrigins of Time Asymmetry Huelva Spain September 1991 (CambridgeCambridge University Press 1992)

[97] CJ Isham Canonical quantum gravity and the problem of timeLectures presented at the NATO Advanced Institute Recent Problems inMathematical Physics Salamanca June 15 1992 K Kuchar Time andinterpretations of quantum gravity In Proc 4th Canadian Conference onGeneral Relativity and Relativistic Astrophysics ed G Kunstatter D Vin-cent and J Williams (Singapore World Scientific 1992) A Ashtekar andJ Stachel eds Proc Osgood Hill Conference Conceptual Problems inQuantum Gravity Boston 1988 (Boston Birkhauser 1993)

[98] C Rovelli Time in quantum gravity an hypothesis Phys Rev D43 (1991)442

[99] J Hartle Classical physics and hamiltonian quantum mechanics as relicsof the big bang Physica Scripta T36 (1991) 228

[100] A Grunbaum Philosophical Problems of Space and Time (New YorkKnopf 1963) T Gold and DL Shumacher eds The Nature of Time(Ithaca Cornell University Press 1967) P Kroes Time its Structure andRole in Physical Theories (Dordrecht D Reidel 1985)

[101] C Rovelli GPS observables in general relativity Phys Rev D65 (2002)044017 gr-qc0110003

[102] TB Bahder Navigation in curved space-time Amer J Phys 69 (2001)315ndash321

[103] M Blagojevic J Garecki FW Hehl and Yu N Obukhov Real nullcoframes in general relativity and GPS type coordinates gr-qc0110078

Chapter 3 Mechanics

[104] VI Arnold Matematiceskie Metody Klassiceskoj Mechaniki (MoskowMir 1979) See in particular Chapter IX Section C

[105] JM Souriau Structure des Systemes Dynamiques (Paris Dunod 1969)[106] JL Lagrange Memoires de la Premiere Classe des Sciences Mathema-

tiques et Physiques (Paris Institute de France 1808)[107] WR Hamilton On the application to dynamics of a general mathematical

method previously applied to optics British Association Report (1834)513ndash518

[108] C Crnkovic and E Witten Covariant description of canonical formal-ism in geometrical theories In Newtonrsquos Tercentenary Volume ed SWHawking and W Israel (Cambridge Cambridge University Press 1987)

References 431

A Ashtekar L Bombelli and O Reula In Mechanics Analysis and Geom-etry 200 Years after Lagrange ed M Francaviglia (Amsterdam Elsevier1991)

[109] MJ Gotay J Isenberg and JE Marsden (with the collaboration of RMontgomery J Sniatycki and PB Yasskin) Momentum maps and classi-cal relativistic fields Part 1 covariant field theory physics9801019

[110] T DeDonder Theorie Invariantive du Calcul des Variationes (ParisGauthier-Villars 1935)

[111] Hesiod Theogony translated by HG Evelyn-White (London HarvardUniversity Press 1914) pp 125ndash130 [Instigated by mother Γαια Kρoνoςthen slaughters and castrates father O

vρανoς]

[112] C Rovelli The statistical state of the universe Class and Quantum Grav10 (1993) 1567

[113] PAM Dirac Generalized Hamiltonian dynamics Can J Math Phys 2(1950) 129ndash148

[114] PAM Dirac Lectures on Quantum Mechanics (New York Belfer Gradu-ate School of Science Yeshiva University 1964)

[115] A Hanson T Regge and C Teitelboim Constrained Hamiltonian Systems(Roma Accademia nazionale dei Lincei 1976) M Henneaux and C Teit-elboim Quantization of Gauge Systems (Princeton Princeton UniversityPress 1972)

[116] C Rovelli Partial observables Phy Rev D65 (2002) 124013 gr-qc0110035

[117] C Rovelli A note on the foundation of relativistic mechanics I Relativis-tic observables and relativistic states In Proc 15th SIGRAV Conferenceon General Relativity and Gravitational Physics 2002 (Bristol IOP Pub-lishing 2004) in press gr-qc0111037

[118] C Rovelli Covariant hamiltonian formalism for field theory symplecticstructure and HamiltonndashJacobi equation on the space G In Decoherenceand Entropy in Complex Systems Selected Lectures from DICE 2002Lecture Notes in Physics 633 ed HT Elze (Berlin SpringerndashVerlag2003) gr-qc0207043

[119] M Montesinos C Rovelli and T Thiemann SL(2 R) model with twoHamiltonian constraints Phys Rev D60 (1999) 044009

[120] H Weil Geodesic fields in the calculus of variations Ann Math 36 (1935)607ndash629

[121] J Kijowski A finite dimensional canonical formalism in the classical fieldtheory Comm Math Phys 30 (1973) 99ndash128 M Ferraris and M Fran-caviglia The Lagrangian approach to conserved quantities in general rel-ativity In Mechanics Analysis and Geometry 200 Years after Lagrangeed M Francaviglia (Amsterdam Elsevier Sci Publ 1991) pp 451ndash488IV Kanatchikov Canonical structure of classical field theory in the poly-momentum phase space Rep Math Phys 41 (1998) 49 F Helein andJ Kouneiher Finite dimensional Hamiltonian formalism for gauge and

432 References

field theories math-ph0010036 H Rund The HamiltonndashJacobi Theory inthe Calculus of Variations (New York Krieger 1973) H Kastrup Canon-ical theories of Lagrangian dynamical systems in physics Phys Rep 101(1983) 1

[122] C Rovelli Statistical mechanics of gravity and thermodynamical origin oftime Class and Quantum Grav 10 (1993) 1549

[123] A Connes and C Rovelli Von Neumann algebra automorphisms and timeversus thermodynamics relation in general covariant quantum theoriesClass and Quantum Grav 11 (1994) 2899

[124] P Martinetti and C Rovelli Diamondsrsquo temperature Unruh effect forbounded trajectories and thermal time hypothesis Class and QuantumGrav 20 (2003) 4919ndash4932 gr-qc0212074

[125] M Montesinos and C Rovelli Statistical mechanics of generally covariantquantum theories a Boltzmann-like approach Class and Quantum Grav18 (2001) 555ndash569

Chapter 4 Hamiltonian general relativity

[126] D Giulini Ashtekar variables in Classical General Relativity In CanonicalGravity From Classical to Quantum ed J Ehlers and H Friedrich (BerlinSpringer-Verlag 1994) p 81

[127] S Alexandrov E Buffenoir P Roche Plebanski theory and covariantcanonical formulation gr-qc0612071

[128] A Perez C Rovelli Physical effects of the Immirzi parameter Phys RevD73 (2006) 044013

[129] P Bergmann Phys Rev 112 (1958) 287 Rev Mod Phys 33 (1961)P Bergmann and A Komar The phase space formulation of general rel-ativity and approaches towards quantization Gen Rel Grav 1 (1981)pp 227ndash254 In General Relativity and Gravitation ed A Held (1981) pp227ndash254 A Komar General relativistic observables via HamiltonndashJacobifunctionals Phys Rev D4 (1971) 923ndash927

[130] PAM Dirac The theory of gravitation in Hamiltonian form Proc RoyalSoc London A246 (1958) 333 Phys Rev 114 (1959) 924

[131] R Arnowitt S Deser and CW Misner The dynamics of general relativityIn Gravitation An Introduction to Current Research ed L Witten (NewYork Wiley 1962) p 227

[132] A Ashtekar New variables for classical and quantum gravity Phys RevLett 57 (1986) 2244 New Hamiltonian formulation of general relativityPhys Rev D36 (1987) 1587

[133] T Thiemann Anomaly-free formulation of nonperturbative 4-dimensionalLorentzian quantum gravity Phys Lett B380 (1996) 257

[134] F Barbero Real Ashtekar variables for Lorentzian signature spacetimesPhys Rev D51 (1995) 5507 gr-qc9410014 Phys Rev D51 (1995) 5498

References 433

[135] G Immirzi Quantum gravity and Regge calculus Nucl Phys Proc Suppl57 (1997) 65 Real and complex connections for canonical gravity Classand Quantum Grav 14 (1997) L177ndashL181

[136] L Fatibene M Francaviglia C Rovelli On a Covariant Formulation ofthe Barbero-Immirzi Connection gr-qc0702134

[137] C Rovelli and T Thiemann The Immirzi parameter in quantum generalrelativity Phys Rev D57 (1998) 1009ndash1014 gr-qc9705059

[138] G Esposito G Gionti and C Stornaiolo Space-time covariant form ofAshtekarrsquos constraints Nuovo Cimento 110B (1995) 1137ndash1152

[139] C Rovelli A note on the foundation of relativistic mechanics II Covarianthamiltonian general relativity gr-qc0202079

[140] M Ferraris and M Francaviglia The Lagrangian approach to conservedquantities in General Relativity In Mechanics Analysis and Geometry200 Years after Lagrange ed M Francaviglia (Amsterdam Elsevier SciPubl 1991) pp 451ndash488 W Szczyrba A symplectic structure of the setof Einstein metrics a canonical formalism for general relativity CommMath Phys 51 (1976) 163ndash182 J Sniatcki On the canonical formulationof general relativity In Proc Journees Relativistes (Caen Faculte des Sci-ences 1970) J Novotny On the geometric foundations of the Lagrangeformulation of general relativity In Differential Geometry ed G Soos andJ Szenthe (Amsterdam North-Holland 1982)

[141] A Peres Nuovo Cimento 26 (1962) 53 U Gerlach Phys Rev 177 (1969)1929 K Kuchar J Math Phys 13 (1972) 758 P Horava On a covariantHamiltonndashJacobi framework for the EinsteinndashMaxwell theory Class andQuantum Grav 8 (1991) 2069 ET Newman and C Rovelli Generalizedlines of force as the gauge invariant degrees of freedom for general relativityand YangndashMills theory Phys Rev Lett 69 (1992) 1300 J Kijowski and GMagli Unconstrained Hamiltonian formulation of General Relativity withthermo-elastic sources Class and Quantum Grav 15 (1998) 3891ndash3916

Chapter 5 Quantum mechanics

[142] E Schrodinger Quantisierung als Eigenwertproblem Ann der Phys 79(1926) 489 Part 2 English translation in Collected Papers on QuantumMechanics (Chelsea Publications 1982)

[143] E Schrodinger Quantisierung als Eigenwertproblem Ann der Phys 79(1926) 361 Part 1 English translation op cit

[144] M Reisenberger and C Rovelli Spacetime states and covariant quan-tum theory Phys Rev D65 (2002) 124013 gr-qc0111016 D Marolf andC Rovelli Relativistic quantum measurement Phys Rev D66 (2002)023510 gr-qc0203056

[145] F Conrady L Doplicher R Oeckl C Rovelli and M Testa Minkowskivacuum from background independent quantum gravity Phys Rev D164(2004) 064019 gr-qc0307118

434 References

[146] F Conrady and C Rovelli Generalized Schrodinger equation in Euclideanquantum field theory Int J Mod Phys in press hep-th0310246

[147] M Montesinos The double role of Einsteinrsquos equations as equations ofmotion and as vanishing energy-momentum tensor gr-qc0311001

[148] PAM Dirac Principles of Quantum Mechanics 1st edition (OxfordOxford University Press 1930)

[149] C Rovelli Is there incompatibility between the ways time is treated ingeneral relativity and in standard quantum mechanics In ConceptualProblems of Quantum Gravity ed A Ashtekar and J Stachel (New YorkBirkhauser 1991)

[150] J Halliwell The WheelerndashdeWitt equation and the path integral in mini-superspace quantum cosmology In Conceptual Problems of QuantumGravity A Ashtekar and J Stachel (New York Birkhauser 1991)

[151] C Rovelli Quantum mechanics without time a model Phys Rev D42(1991) 2638

[152] C Rovelli Quantum evolving constants Phys Rev D44 (1991) 1339[153] R Oeckl A lsquogeneral boundaryrsquo formulation for quantum mechanics and

quantum gravity hep-th0306025 Schroedingerrsquos cat and the clock lessonsfor quantum gravity gr-qc0306007

[154] L Doplicher Generalized TomonagandashSchrodinger equation from theHadamard formula gr-qc0405006

[155] S Tomonaga Prog Theor Phys 1 (1946) 27 J Schwinger Quantumelectrodynamics I A covariant formulation Phys Rev 74 (1948) 1439

[156] NC Tsamis and RP Woodard Physical Greenrsquos functions in quantumgravity Annals of Phys 215 (1992) 96

[157] JB Hartle and SW Hawking Wave function of the Universe Phys RevD28 (1983) 2960

[158] D Marolf Group averaging and refined algebraic quantization where arewe In Proceedings of the IXth Marcel Grossmann Conference Rome ItalyJuly 2ndash9 2000 ed RT Jantzen GM Keiser and R Ruffini (World Sci-entific 1996) gr-qc0011112

[159] C Rovelli Relational quantum mechanics Int J Theor Phys 35 (1996)1637ndash1678

[160] C Rovelli Incerto tempore incertisque loci Can we compute the exacttime at which a quantum measurement happens Foundations of Physics28 (1998) 1031ndash1043

[161] F Laudisa The EPR argument in a relational interpretation of quantummechanics Foundations of Physics Letters 14 (2) (2001) 119ndash132

[162] A Grinbaum Elements of information theoretic derivation of the formal-ism of quantum theory Int J Quant Information 1 (2003) 1

[163] F Laudisa and C Rovelli Relational quantum mechanics In The StanfordEncyclopedia of Philosophy (Spring 2002 Edition) ed Edward N ZaltaURL httpplatostanfordeduarchivesspr2002entriesqm-relational

References 435

[164] M Bitbol Relations et correlations en Physique Quantique In Un Sieclede Quanta ed M Crozon and Y Sacquin (Paris EDP Sciences 2000)

[165] J Wheeler Information physics quantum the search for the links Proc3rd Int Symp Foundations of Quantum Mechanics Tokyo 1989 p 354

[166] J Wheeler It from Bit In Sakharov Memorial Lectures on Physics Vol 2ed L Keldysh and V Feinberg (New York Nova Science 1992)

[167] CU Fuchs Quantum foundations in the light of quantum information InProc NATO Advanced Research Workshop on Decoherence and its Impli-cations in Quantum Computation and Information Transfer ed A Gonis(New York Plenum 2001) quant-ph0106166quant-ph0205039

[168] D Finkelstein Quantum Relativity (Berlin Springer 1996)

Chapter 6 Quantum space

[169] L Freidel and ER Livine Spin networks for non-compact groups J MathPhys 44 (2003) 1322ndash1356

[170] C Rovelli Loop representation in quantum gravity In Conceptual Prob-lems of Quantum Gravity ed A Ashtekar and J Stachel (New YorkBirkhauser 1991)

[171] C Rovelli and L Smolin Spin networks and quantum gravity Phys RevD52 (1995) 5743ndash5759 gr-qc9505006

[172] J Lewandowski ET Newman and C Rovelli Variations of the parallelpropagator and holonomy operator and the Gauss law constraint J MathPhys 34 (1993) 4646

[173] A Ashtekar and J Lewandowski Quantum theory of geometry I Areaoperators Class and Quantum Grav 14 (1997) A55 II Volume operatorsAdv Theor Math Phys 1 (1997) 388ndash429

[174] LH Kauffman and SL Lins Temperley-Lieb Recoupling Theory and In-variant of 3-Manifolds (Princeton Princeton University Press 1994)

[175] R De Pietri and C Rovelli Geometry eigenvalues and scalar product fromrecoupling theory in loop quantum gravity Phys Rev D54 (1996) 2664gr-qc9602023 T Thiemann Closed formula for the matrix elements of thevolume operator in canonical quantum gravity J Math Phys 39 (1998)3347ndash3371 gr-qc9606091

[176] C Rovelli and L Smolin Knot theory and quantum gravity Phys RevLett 61 (1988) 1155

[177] C Rovelli and L Smolin Loop space representation for quantum generalrelativity Nucl Phys B331 (1990) 80

[178] T Jacobson and L Smolin Nonperturbative quantum geometries NuclPhys B299 (1988) 295

[179] R Gambini and A Trias Phys Rev D22 (1980) 1380 On the geometricalorigin of gauge theories Phys Rev D23 (1981) 553 Nucl Phys B278(1986) 436 C di Bartolo F Nori R Gambini and A Trias Loop spaceformulation of free electromagnetism Nuovo Cimento Lett 38 (1983) 497

436 References

[180] B Brugmann R Gambini and J Pullin Knot invariants as nondegeneratequantum geometries Phys Rev Lett 68 (1992) 431 Jones polynomials forintersecting knots as physical states of quantum gravity Nucl Phys B385(1992) 587 Gen Rel Grav 25 (1993) 1 J Pullin in Proc 5th MexicanSchool of Particles and Fields ed J Lucio (Singapore World Scientific1993)

[181] R Penrose Theory of quantized directions unpublished manuscript An-gular momentum an approach to combinatorial spacetime In QuantumTheory and Beyond ed T Bastin (Cambridge Cambridge UniversityPress 1971) pp 151ndash180

[182] L Smolin The future of spin networks gr-qc9702030[183] JC Baez Spin networks in gauge theory Adv Math 117 (1996) 253 JC

Baez Spin networks in nonperturbative quantum gravity In Interface ofKnot Theory and Physics ed L Kauffman (Providence Rhode IslandAmerican Mathematical Society 1996) gr-qc9504036

[184] N Grott and C Rovelli Moduli spaces structure of knots with intersec-tions J Math Phys 37 (1996) 3014

[185] W Fairbairn and C Rovelli Separable Hilbert space in loop quantumgravity J Math Phys to appear gr-qc0403047

[186] J Zapata A combinatorial approach to diffeomorphism invariant quantumgauge theories J Math Phys 38 (1997) 5663ndash5681 A combinatorial spacefor loop quantum gravity Gen Rel Grav 30 (1998) 1229

[187] J Lewandowski A Okolow H Sahlmann T Thiemann Comm MathPhys 267 (2006) 703ndash733

[188] C Fleischhack Representations of the Weyl Algebra in Quantum Geome-try math-ph0407006

[189] A Ashtekar C Rovelli and L Smolin Weaving a classical metric withquantum threads Phys Rev Lett 69 (1992) 237 hep-th9203079

[190] C Rovelli A generally covariant quantum field theory and a prediction onquantum measurements of geometry Nucl Phys B405 (1993) 797

[191] C Rovelli and L Smolin Discreteness of area and volume in quantumgravity Nucl Phys B442 (1995) 593 Erratum Nucl Phys B456 (1995)734

[192] S Frittelli L Lehner and C Rovelli The complete spectrum of the areafrom recoupling theory in loop quantum gravity Class and Quantum Grav13 (1996) 2921

[193] R Loll The volume operator in discretized quantum gravity Phys RevLett 75 (1995) 3048 Spectrum of the volume operator in quantum gravityNucl Phys B460 (1996) 143ndash154 gr-qc9511030

[194] J Lewandowski Volume and quantizations Class and Quantum Grav 14(1997) 71ndash76

[195] T Tsushima The expectation value of the Gaussian weave state in loopquantum gravity gr-qc0212117

References 437

[196] T Thiemann A length operator for canonical quantum gravity J MathPhys 39 (1998) 3372ndash3392 gr-qc9606092

[197] A Ashtekar A Corichi and J Zapata Quantum theory of geometry IIINoncommutativity of Riemannian structures Class and Quantum Grav15 (1998) 2955

[198] S Major Operators for quantized directions Class Quant Grav 16(1999) 3859ndash3877

[199] S Major A Spin Network Primer Am J Phys 67 (1999) 972ndash980[200] A Ashtekar and CJ Isham Representations of the holonomy algebra

of gravity and non-abelian gauge theories Class and Quantum Grav 9(1992) 1433 hep-th9202053

[201] A Ashtekar and J Lewandowski Representation theory of analytic holon-omy Clowast-algebras In Knots and Quantum Gravity ed J Baez (OxfordOxford University Press 1994) Differential geometry on the space of con-nections via graphs and projective limits J Geom and Phys 17 (1995)191

[202] A Ashtekar Mathematical problems of non-perturbative quantum generalrelativity In Les Houches Summer School on Gravitation and Quantiza-tions Les Houches France Jul 5ndashAug 1 1992 ed J Zinn-Justin andB Julia (Amsterdam North-Holland 1995) gr-qc9302024

[203] J Baez Generalized measures in gauge theory Lett Math Phys 31 (1994)213ndash223

[204] A Ashtekar J Lewandowski D Marolf J Mourao and T ThiemannQuantization of diffeomorphism invariant theories of connections with localdegrees of freedom J Math Phys 36 (1995) 6456 gr-qc9504018

[205] R De Pietri On the relation between the connection and the loop rep-resentation of quantum gravity Class and Quantum Grav 14 (1997) 53gr-qc9605064

[206] A Ashtekar C Rovelli and L Smolin Gravitons and loops Phys RevD44 (1991) 1740ndash1755 hep-th9202054 J Iwasaki and C Rovelli Gravi-tons as embroidery on the weave Int J Mod Phys D1 (1993) 533 Gravi-tons from loops non-perturbative loop-space quantum gravity containsthe graviton-physics approximation Class and Quantum Grav 11 (1994)1653 M Varadarajan Gravitons from a loop representation of linearizedgravity Phys Rev D66 (2002) 024017 gr-qc0204067

[207] A Corichi and JM Reyes A Gaussian weave for kinematical loop quan-tum gravity Int J Mod Phys D10 (2001) 325 gr-qc0006067

Chapter 7 Dynamics and matter

[208] R Borissov R De Pietri and C Rovelli Matrix elements of Thiemannrsquoshamiltonian constraint in loop quantum gravity Class and QuantumGrav 14 (1997) 2793 gr-qc9703090

438 References

[209] T Thiemann The phoenix project master constraint programme for loopquantum gravity gr-qc0305080

[210] M Gaul and C Rovelli A generalized hamiltonian constraint operator inloop quantum gravity and its simplest euclidean matrix elements Classand Quantum Grav 18 (2001) 1593ndash1624 gr-qc0011106

[211] T Thiemann QSD V Quantum gravity as the natural regulator of thehamiltonian constraint of matter quantum field theories Class and Quan-tum Grav 15 (1998) 1281ndash1314 gr-qc9705019

[212] S Alexandrov SO(4C)-covariant AshtekarndashBarbero gravity and theImmirzi parameter Class and Quantum Grav 17 (2000) 4255ndash4268S Alexandrov and ER Livine SU(2) loop quantum gravity seen fromcovariant theory Phys Rev D67 (2003) 044009 gr-qc0209105

[213] S Alexandrov and DV Vassilevich Area spectrum in Lorentz covariantloop gravity gr-qc0103105

[214] BS DeWitt Quantum theory of gravity I the canonical theory PhysRev 160 (1967) 1113

[215] B Brugmann R Gambini and J Pullin Jones polynomials for intersectingknots as physical states of quantum gravity Nucl Phys B385 (1992) 587Knot invariants as nondegenerate quantum geometries Phys Rev Lett 68(1992) 431 C Di Bartolo R Gambini J Griego and J Pullin Consistentcanonical quantization of general relativity in the space of Vassiliev knotinvariants Phys Rev Lett 84 (2000) 2314

[216] K Ezawa Nonperturbative solutions for canonical quantum gravity anoverview Phys Repts 286 (1997) 271ndash348 gr-qc9601050

[217] C Rovelli and L Smolin The physical hamiltonian in nonperturbativequantum gravity Phys Rev Lett 72 (1994) 44

[218] C Rovelli Outline of a general covariant quantum field theory and a quan-tum theory of gravity J Math Phys 36 (1995) 6529

[219] T Thiemann Quantum spin dynamics (QSD) Class and Quantum Grav15 (1998) 839ndash73 gr-qc9606089 QSD II The kernel of the WheelerndashDeWitt constraint operator Class and Quantum Grav 15 (1998) 875ndash905 gr-qc9606090 QSD III Quantum constraint algebra and physicalscalar product in quantum general relativity Class and Quantum Grav15 (1998) 1207ndash1247 gr-qc9705017 QSD IV 2 + 1 euclidean quantumgravity as a model to test 3 + 1 lorentzian quantum gravity Class andQuantum Grav 15 (1998) 1249ndash1280 gr-qc9705018 QSD VI QuantumPoincare algebra and a quantum positivity of energy theorem for canon-ical quantum gravity Class and Quantum Grav 15 (1998) 1463ndash1485gr-qc9705020 QSD VII Symplectic structures and continuum lattice for-mulations of gauge field theories Class and Quantum Grav 18 (2001)3293ndash3338 hep-th0005232

[220] L Smolin Quantum gravity with a positive cosmological constant hep-th0209079 L Freidel and L Smolin The linearization of the Kodamastate hep-th0310224

[221] T Thiemann Loop Quantum Gravity An Inside View hep-th0608210

References 439

[222] C Rovelli and H Morales-Tecotl Fermions in quantum gravity Phys RevLett 72 (1995) 3642 Loop space representation of quantum fermions andgravity Nucl Phys B451 (1995) 325

[223] J Baez and K Krasnov Quantization of diffeomorphism-invariant theorieswith fermions J Math Phys 39 (1998) 1251ndash1271 hep-th9703112

[224] T Thiemann Kinematical Hilbert spaces for fermionic and Higgs quan-tum field theories Class and Quantum Grav 15 (1998) 1487ndash1512 gr-qc9705021

[225] M Montesinos and C Rovelli The fermionic contribution to the spec-trum of the area operator in nonperturbative quantum gravity Class andQuantum Grav 15 (1998) 3795ndash3801

[226] S O Bilson-Thompson F Markopoulou L Smolin hep-th0603022[227] A Alekseev AP Polychronakos and M Smedback On area and entropy

of a black hole Phys Lett B574 (2003) 296 AP Polychronakos Areaspectrum and quasinormal modes of black holes hep-th0304135

[228] S Major and L Smolin Quantum deformations of quantum gravity NuclPhys B473 (1996) 267ndash290 gr-qc9512020 R Borissov S Major andL Smolin The geometry of quantum spin networks Class and QuantumGrav 13 (1996) 3183ndash3196

[229] LH Kauffman Map coloring q-deformed spin networks and TuraevndashViroinvariants for three manifolds Int J Mod Phys B6 (1992) 1765ndash1794Erratum B6 (1992) 3249

[230] N Reshetikhin and V Turaev Ribbon graphs and their invariants derivedfrom quantum groups Comm Math Phys 127 (1990) 1ndash26

[231] K Noui and Ph Roche Cosmological deformation of Lorentzian spin foammodels Class and Quantum Grav 20 (2003) 3175ndash3214 gr-qc0211109

Chapter 8 Applications

[232] M Bojowald and HA Morales-Tecotl Cosmological applications of loopquantum gravity to appear in Proc 5th Mexican School (DGFM) TheEarly Universe and Observational Cosmology gr-qc0306008

[233] M Domagala L Lewandowski Black-hole entropy from quantum geome-try Class Quant Grav 21 (2004) 52335243

[234] K A Meissner Black-hole entropy in loop quantum gravity Class QuantGrav 21 (2004) 52455252

[235] A Ashtekar An Introduction to Loop Quantum Gravity Through Cos-mology gr-qc0702030

[236] L Modesto Disappearance of Black Hole Singularity in Quantum GravityPhys Rev D70 (2004) 124009

[237] A Ashtekar and M Bojowald Quantum geometry and Schwarzschild sin-gularity Class Quant Grav 23 (2006) 391ndash411

440 References

[238] A Ashtekar J Baez A Corichi and K Krasnov Quantum geometryand black hole entropy Phys Rev Lett 80 (1998) 904 gr-qc9710007A Ashtekar JC Baez and K Krasnov Quantum geometry of isolatedhorizons and black hole entropy Adv Theor Math Phys 4 (2001) 1ndash94gr-qc0005126

[239] C Rovelli Black hole entropy from loop quantum gravity Phys Rev Lett14 (1996) 3288 Loop quantum gravity and black hole physics Helv PhysActa 69 (1996) 583

[240] GH Hardy and S Ramanujan Proc London Math Soc 2 (1918) 75[241] G Amelino-Camelia Are we at dawn with quantum gravity phenomenol-

ogy Lectures given at 35th Winter School of Theoretical Physics FromCosmology to Quantum Gravity Polanica Poland 2ndash12 Feb 1999 Lec-ture Notes in Physics 541 (2000) 1ndash49 gr-qc9910089

[242] C Rovelli and S Speziale Reconcile Planck-scale discreteness and theLorentzndashFitzgerald contraction Phys Rev D67 (2003) 064019

[243] M Bojowald Absence of singularity in loop quantum cosmology PhysRev Lett 86 (2001) 5227ndash5230 gr-qc0102069

[244] M Bojowald Inflation from quantum geometry Phys Rev Lett 89 (2002)261301 gr-qc0206054

[245] SW Hawking Black holes in general relativity Comm Math Phys 25(1972) 152

[246] JM Bardeen B Carter and SW Hawking The four laws of black holemechanics Comm Math Phys 31 (1973) 161

[247] JD Bekenstein Black holes and the second law Nuovo Cimento Lett 4(1972) 737ndash740 Black holes and entropy Phys Rev D7 (1973) 2333ndash2346Generalized second law for thermodynamics in black hole physics PhysRev D9 (1974) 3292ndash3300

[248] SW Hawking Black hole explosions Nature 248 (1974) 30 Particle cre-ation by black holes Comm Math Phys 43 (1975) 199

[249] K Krasnov Geometrical entropy from loop quantum gravity Phys RevD55 (1997) 3505 On statistical mechanics of gravitational systems GenRel Grav 30 (1998) 53ndash68 gr-qc9605047 On statistical mechanics of aSchwarzschild black hole Gen Rel Grav 30 (1998) 53

[250] JW York Dynamical origin of black hole radiance Phys Rev D28 (1983)2929

[251] G rsquot Hooft Horizon operator approach to black hole quantization gr-qc9402037 L Susskind Some speculations about black hole entropy in stringtheory hep-th9309145 L Susskind L Thorlacius and J Uglum PhysRev D48 (1993) 3743 C Teitelboim Statistical thermodynamics of ablack hole in terms of surface fields Phys Rev D53 (1996) 2870ndash2873 ABuonanno M Gattobigio M Maggiore L Pilo and C Ungarelli EffectiveLagrangian for quantum black holes Nucl Phys B451 (1995) 677

[252] A Ashtekar C Beetle O Dreyer et al Isolated horizons and their appli-cations Phys Rev Lett 85 (2000) 3564ndash3567 gr-qc0006006 A Ashtekar

References 441

Classical and quantum physics of isolated horizons Lect Notes Phys 541(2000) 50ndash70

[253] L Smolin Linking topological quantum field theory and nonperturbativequantum gravity J Math Phys 36 (1995) 6417ndash6455

[254] AP Balachandran L Chandar and A Momen Edge states in gravity andblack hole physics Nucl Phys B461 (1996) 581ndash596 A Momen Edgedynamics for BF theories and gravity Phys Lett 394 (1997) 269ndash274 SCarlip Black hole entropy from conformal field theory in any dimensionPhys Rev Lett 82 (1999) 2828ndash2831

[255] S Hod Bohrrsquos correspondence principle and the area spectrum of quantumblack holes Phys Rev Lett 81 (1998) 4293 Gen Rel Grav 31 (1999)1639 Kerr black hole quasinormal frequencies Phys Rev D67 (2003)081501

[256] O Dreyer Quasinormal modes the area spectrum and black hole entropyPhys Rev Lett 90 (2003) 081301

[257] A Corichi On quasinormal modes black hole entropy and quantum ge-ometry Phys Rev D67 (2003) 087502 gr-qc0212126

[258] JD Bekenstein and VF Mukhanov Spectroscopy of the quantum blackhole Phys Lett B360 (1995) 7ndash12

[259] L Smolin Macroscopic deviations from Hawking radiation In Matters ofGravity 7 gr-qc9602001

[260] M Barreira M Carfora and C Rovelli Physics with loop quantum gravityradiation from quantum black hole Gen Rel Grav 28 (1996) 1293

[261] R Gambini and J Pullin Nonstandard optics from quantum spacetimePhys Rev D59 (1999) 124021 gr-qc9809038 Quantum gravity experi-mental physics Gen Rel Grav 31 (1999) 1631ndash1637

[262] J Alfaro HA Morales-Tecotl and LF Urrutia Quantum gravity cor-rections to neutrino propagation Phys Rev Lett 84 (2000) 2318ndash2321gr-qc9909079 Loop quantum gravity and light propagation Phys RevD65 (2002) 103509 hep-th0108061

[263] C Kozameh and F Parisi Lorentz invariance and the semiclassical ap-proximation of loop quantum gravity gr-qc0310014

[264] J Collins A Perez D Sudarsky L Urrutia H Vucetich Lorentz in-variance An Additional fine tuning problem Phys Rev Lett 93 (2004)191301

[265] G Amelino-Camelia J Ellis NE Mavromatos DV Nanopoulos andS Sarkar Potential sensitivity of gamma-ray buster observations to wavedisperion in vacuo Nature 393 (1998) 763 astro-ph9712103 G Amelino-Camelia An interferometric gravitational wave detector as a quantumgravity apparatus Nature 398 (1999) 216

[266] J Ellis J Hagelin D Nanopoulos and M Srednicki Search for vio-lations of quantum mechanics Nucl Phys B241 (1984) 381 J EllisNE Mavromatos and DV Nanopoulos Testing quantum mechanics inthe neutral kaon system Phys Lett B293 (1992) 142 IC Percival

442 References

and WT Strunz Detection of space-time fluctuations by a model mat-ter interferometer quant-ph9607011 G Amelino-Camelia J Ellis NEMavromatos and DV Nanopoulos Distance measurement and wave dis-persion in a Liouville string approach to quantum gravity Int J ModPhys A12 (1997) 607

Chapter 9 Spinfoams

[267] J Barrett Quantum gravity as topological quantum field theory J MathPhys 36 (1995) 6161ndash6179 gr-qc9506070

[268] ME Peskin and DV Schroeder An Introduction to Quantum Field The-ory (Reading Addison Wesley 1995)

[269] JW Barrett and L Crane Relativistic spin networks and quantum grav-ity J Math Phys 39 (1998) 3296ndash3302

[270] M Bojowald and A Perez Spin foam quantization and anomalies gr-qc0303026

[271] JC Baez JD Christensen TR Halford and DC Tsang Spin foammodels of riemannian quantum gravity Class and Quantum Grav 19(2002) 4627ndash4648 gr-qc0202017

[272] G Roepstorff Path Integral Approach to Quantum Physics An Introduc-tion (Berlin Springer-Verlag 1994)

[273] AS Wightman Quantum field theory in terms of vacuum expectationvalues Phys Rev 101 (1956) 860 RF Streater and AS WightmanPCT Spin and Statistics and All That Mathematical Physics MonographSeries (Reading MA Benjamin-Cummings 1964) K Osterwalder and RSchrader Axioms for euclidean Greenrsquos functions Comm Math Phys 31(1973) 83 Axioms for euclidean Greenrsquos functions 2 42 (1975) 281

[274] A Perez and C Rovelli Observables in quantum gravity gr-qc0104034[275] A Ashtekar D Marolf J Mourao and T Thiemann Constructing Hamil-

tonian quantum theories from path integrals in a diffeomorphism invari-ant context Class and Quantum Grav 17 (2000) 4919ndash4940 quant-ph9904094

[276] J Baez Strings loops knots and gauge fields In Knots and QuantumGravity ed J Baez (Oxford Oxford University Press 1994)

[277] J Iwasaki A reformulation of the PonzanondashRegge quantum gravity modelin terms of surfaces gr-qc9410010 A definition of the PonzanondashReggequantum gravity model in terms of surfaces J Math Phys 36 (1995)6288

[278] M Reisenberger Worldsheet formulations of gauge theories and gravitytalk given at the 7th Marcel Grossmann Meeting Stanford July 1994 gr-qc9412035 A lattice worldsheet sum for 4-d Euclidean general relativitygr-qc9711052

[279] M Reisenberger and C Rovelli Sum over surfaces form of loop quantumgravity Phys Rev D56 (1997) 3490ndash3508 gr-qc9612035

References 443

[280] C Rovelli Quantum gravity as a lsquosum over surfacesrsquo Nucl Phys (ProcSuppl) B57 (1997) 28ndash43

[281] C Rovelli The projector on physical states in loop quantum gravity PhysRev D59 (1999) 104015 gr-qc9806121

[282] F Markopoulou Dual formulation of spin network evolution gr-qc9704013

[283] G Ponzano and T Regge Semiclassical limit of Racah coefficients InSpectroscopy and Group Theoretical Methods in Physics ed F Bloch(Amsterdam North-Holland 1968)

[284] L Crane and D Yetter On the classical limit of the balanced state sumgr-qc9712087 JW Barrett The classical evaluation of relativistic spinnetworks Adv Theor Math Phys 2 (1998) 593ndash600 JW Barrett andRM Williams The asymptotics of an amplitude for the 4-simplex AdvTheor Math Phys 3 (1999) 209ndash214 L Freidel and K Krasnov Simplespin networks as Feynman graphs J Math Phys 41 (2000) 1681ndash1690

[285] T Regge General relativity without coordinates Nuovo Cimento 19(1961) 558ndash571

[286] VG Turaev and OY Viro State sum invariants of 3-manifolds and quan-tum 6j symbols Topology 31 (1992) 865 VG Turaev Quantum Invari-ants of Knots and 3-Manifolds (New York de Gruyter 1994)

[287] D V Boulatov A model of three-dimensional lattice gravity Mod PhysLett A7 (1992) 1629ndash1648

[288] E Brezin C Itzykson G Parisi and J Zuber Comm Math Phys 59(1978) 35 F David Nucl Phys B257 (1985) 45 J Ambjorn B Durhuusand J Frolich Nucl Phys B257 (1985) 433 VA Kazakov IK Kos-tov and AA Migdal Phys Lett 157 (1985) 295 DV Boulatov VAKazakov IK Kostov and AA Migdal Nucl Phys B275 (1986) 641 MDouglas and S Shenker Nucl Phys B335 (1990) 635 D Gross and AAMigdal Phys Rev Lett 64 (1990) 635 E Brezin and VA Kazakov PhysLett B236 (1990) 144 O Alvarez E Marinari and P Windey RandomSurfaces and Quantum Gravity (New York Plenum Press 1991)

[289] H Ooguri Topological lattice models in four dimensions Mod Phys LettA7 (1992) 2799

[290] VG Turaev Quantum Invariants of Knots and 3-manifolds (New Yorkde Gruyter 1994)

[291] AS Schwartz The partition function of degenerate quadratic function-als and RayndashSinger invariants Lett Math Phys 2 (1978) 247ndash252G Horowitz Exactly soluble diffeomorphism-invariant theories CommMath Phys 125 (1989) 417ndash437 D Birmingham M Blau M Rakowskiand G Thompson Topological field theories Phys Rep 209 (1991) 129ndash340

[292] L Crane and D Yetter A categorical construction of 4-D topological quan-tum field theories In Quantum Topology ed L Kauffman and R Baadhio(Singapore World Scientific 1993) pp 120ndash130 hep-th9301062 L

444 References

Crane L Kauffman and D Yetter State-sum invariants of 4-manifoldsI J Knot Theor Ramifications 6 (1997) 177ndash234 hep-th9409167 JDRoberts Skein theory and the TuraevndashViro invariants Topology 34 (1995)771ndash787

[293] C Rovelli Basis of the PonzanondashReggendashTuraevndashVirondashOoguri quantumgravity model is the loop representation basis Phys Rev D48 (1993)2702

[294] R De Pietri and L Freidel SO(4) Plebanski action and relativisticstate sum models Class and Quantum Grav 16 (1999) 2187ndash2196 gr-qc9804071

[295] M Reisenberger Classical Euclidean GR from left-handed area = right-handed area Class and Quantum Grav 16 (1999) 1357ndash1371 gr-qc9804061 On relativistic spin network vertices J Math Phys 40 (1999)2046ndash2054 gr-qc9711052

[296] A Perez Spin foam quantization of Plebanskirsquos action Adv Theor MathPhys 5 (2002) 947ndash968 gr-qc0203058

[297] A Barbieri Quantum tetrahedron and spin networks Nucl Phys B518(1998) 714ndash728

[298] J Baez and J Barrett The quantum tetrahedron in 3 and 4d Adv TheorMath Phys 3 (1999) 815 gr-qc9903060

[299] J Baez Spin foam models Class and Quantum Grav 15 (1998) 1827ndash1858 gr-qc9709052

[300] R De Pietri L Freidel K Krasnov and C Rovelli BarrettndashCrane modelfrom a BoulatovndashOoguri field theory over a homogeneous space NuclPhys B574 (2000) 785ndash806 hep-th9907154

[301] A Perez and C Rovelli A spinfoam model without bubble divergencesNucl Phys B599 (2001) 255ndash282

[302] D Oriti and RM Williams Gluing 4-simplices a derivation of theBarrettndashCrane spinfoam model for Euclidean quantum gravity Phys RevD63 (2001) 024022

[303] M Reisenberger and C Rovelli Spinfoam models as Feynman diagramsgr-qc0002083 Spacetime as a Feynman diagram the connection formu-lation Class and Quantum Grav 18 (2001) 121ndash140 gr-qc0002095

[304] A Mikovic Quantum field theory of spin networks gr-qc0102110[305] A Perez Finiteness of a spinfoam model for euclidean GR Nucl Phys

B599 (2001) 427ndash434[306] L Freidel Group field theory An Overview Int J Theor Phys 44 (2005)

1769ndash1783[307] D Oriti The group field theory approach to quantum gravity gr-

qc0607032[308] L Freidel ER Levine 3d Quantum Gravity and Effective Non-

Commutative Quantum Field Theory Phys Rev Lett 96 (2006) 221301[309] L Freidel and D Louapre Nonperturbative summation over 3d discrete

topologies hep-th0211026

References 445

[310] L Freidel A PonzanondashRegge model of lorentzian 3-dimensional gravityNucl Phys Proc Suppl 88 (2000) 237ndash240 gr-qc0102098

[311] JW Barrett and L Crane A lorentzian signature model for quantumgeneral relativity Class and Quantum Grav 17 (2000) 3101ndash3118 gr-qc9904025

[312] A Perez and C Rovelli Spin foam model for lorentzian general relativityPhys Rev D63 (2001) 041501 gr-qc0009021

[313] L Crane A Perez and C Rovelli Perturbative finiteness in spin foamquantum gravity Phys Rev Lett 87 (2001) 181301 A finiteness proof forthe lorentzian state sum spin foam model for quantum general relativitygr-qc0104057

[314] A Perez and C Rovelli 3 + 1 spin foam model of quantum gravity withspacelike and timelike components Phys Rev D64 (2001) 064002 gr-qc0011037

[315] L Freidel ER Livine and C Rovelli Spectra of length and area in 2 + 1lorentzian loop quantum gravity Class and Quantum Grav 20 (2003)1463ndash1478 gr-qc0212077

[316] L Freidel and K Krasnov Spin foam models and the classical actionprinciple Adv Theor Phys 2 (1998) 1221ndash1285 hep-th9807092

[317] J Ambjorn J Jurkiewicz and R Loll Lorentzian and euclidean quan-tum gravity analytical and numerical results hep-th0001124 A non-perturbative lorentzian path integral for gravity Phys Rev Lett 85 (2000)924 hep-th0002050 J Ambjorn A Dasgupta J Jurkiewicz and R LollA lorentzian cure for euclidean troubles Nucl Phys Proc Suppl 106(2002) 977ndash979 hep-th0201104 R Loll and A Dasgupta A proper timecure for the conformal sickness in quantum gravity Nucl Phys B606(2001) 357ndash379 hep-th0103186

[318] ER Livine and D Oriti Implementing causality in the spin foam quantumgeometry Nucl Phys B663 (2003) 231ndash279 gr-qc0210064

[319] M Arnsdorf Relating covariant and canonical approaches to triangulatedmodels of quantum gravity Class and Quantum Grav 19 (2002) 1065ndash1092 gr-qc0110026

[320] ER Livine Projected spin networks for lorentz connection linking spinfoams and loop gravity Class and Quantum Grav 19 (2002) 5525ndash5542gr-qc0207084

[321] K Noui A Perez Three-dimensional loop quantum gravity Physicalscalar product and spin foam models Class Quant Grav 22 (2005) 1739ndash1762

[322] R Capovilla M Montesinos VA Prieto and E Rojas BF gravity andthe Immirzi parameter Class and Quantum Grav 18 (2001) L49ndashL52

[323] T Thiemann and O Winkler Coherent states for canonical quantum gen-eral relativity and the infinite tensor product extension Nucl Phys B606(2001) 401ndash440 gr-qc0102038

[324] F Markopoulou An insiderrsquos guide to quantum causal histories NuclPhys Proc Suppl 88 (2000) 308ndash313 hep-th9912137

446 References

[325] A Mikovic Spin foam models of matter coupled to gravity Class andQuantum Grav 19 (2002) 2335 Spinfoam models of YangndashMills theorycoupled to gravity Class and Quantum Grav 20 (2003) 239ndash246

[326] D Oriti and H Pfeiffer A spin foam model for pure gauge theory coupledto quantum gravity Phys Rev D66 (2002) 124010

[327] NJ Vilenkin Special Functions and the Theory of Group Representations(Providence Rhode Island American Mathematical Society 1968)

[328] W Ruhl The Lorentz Group and Harmonic Analysis (New YorkWA Benjamin Inc 1970)

[329] L Modesto C Rovelli Particle scattering in loop quantum gravity PhysRev Lett 95 (2005) 191301

[330] C Rovelli Graviton propagator from background-independent quantumgravity Phys Rev Lett 97 151301 (2006) E Bianchi L ModestoC Rovelli and S Speziale Graviton propagator in loop quantum grav-ity Class Quant Grav 23 6989 (2006)

[331] S Speziale Towards the graviton from spinfoams The 3-D toy modelJHEP (2006) 0605039 ER Livine S Speziale JL Willis Towards thegraviton from spinfoams Higher order corrections in the 3-D toy modelPhys Rev D75 (2007) 024038 ER Livine S Speziale Group IntegralTechniques for the Spinfoam Graviton Propagator gr-qc0608131

Chapter 10 Conclusion and Appendices

[332] G Galilei Dialogo dei massimi system (Firenze 1632)[333] DM Brink and GR Satchler Angular Momentum (Oxford Clarendon

Press 1968)[334] R Penrose In Combinatorial Mathematics and its Application

ed D Welsh (New York Academic Press 1971)[335] LH Kauffman Int J Mod Phys A5 (1990) 93[336] LH Kauffman In Knots Topology and Quantum Field Theories

ed L Lusanna (Singapore World Scientific 1991)[337] JP Moussoris in Advances in Twistor Theory Research Notes in Mathe-

matics ed JP Huston and RS Ward (Boston Pitman 1979) pp 308ndash312

[338] I Levinson Liet TSR Mokslu Acad Darbai B Ser 2 (1956) 17[339] AP Yutsin JB Levinson and VV Vanagas Mathematical Apparatus of

the Theory of Angular Momentum (Jerusalem Israel Program for ScientificTranslation 1962)

[340] A Einstein Naeherungsweise Integration der Feldgleichungen derGravitation Preussische Akademie der Wissenschaften (Berlin) Sitzungs-berichte (1916) p 688

[341] O Klein Zur Funfdimensionalen Darstellung der RelativitaetstheorieZ fur Physik 46 (1927) 188

References 447

[342] L Rosenfeld Zur Quantelung der Wellenfelder Ann der Physik 5 (1930)113 Uber die Gravitationswirkungen des Lichtes Z fur Physik 65 (1930)589

[343] M Fierz Hel Physica Acta 12 (1939) 3 W Pauli and M Fierz On rela-tivistic field equations of particles with arbitrary spin in an electromagneticfield Hel Physica Acta 12 (1939) 297

[344] DI Blokhintsev and FM Galrsquoperin Pod Znamenem Marxisma 6 (1934)147

[345] W Heisenberg Z fur Physik 110 (1938) 251[346] J Stachel Early history of quantum gravity (1916ndash1940) Presented at

the HGR5 Notre Dame July 1999 Early history of quantum gravity InBlack Holes Gravitational Radiation and the Universe ed BR Iyer andB Bhawal (Netherlands Kluwer Academic Publishers 1999)

[347] MP Bronstein Quantentheories schwacher Gravitationsfelder Physika-lische Z der Sowietunion 9 (1936) 140

[348] GE Gorelik First steps of quantum gravity and the planck values InStudies in the History of General Relativity [Einstein Studies Vol 3]ed J Eisenstaedt and AJ Kox (Boston Birkhauser 1992) pp 364ndash379GE Gorelik and VY Frenkel Matvei Petrovic Bronstein and the SovietTheoretical Physics in the Thirties (Boston Birkhauser-Verlag 1994)

[349] PG Bergmann Non-linear field theories Phys Rev 75 (1949) 680 Non-linear field theories II canonical equations and quantization Rev ModPhys 21 (1949)

[350] PG Bergmann Nuovo Cimento 3 (1956) 1177[351] ET Newman and PG Bergmann Observables in singular theories by

systematic approximation Rev Mod Phys 29 (1957) 443[352] S Gupta Proc Phys Soc A65 (1952) 608[353] C Misner Feynman quantization of general relativity Rev Mod Phys 29

(1957) 497[354] P Bergmann The canonical formulation of general relativistic theories the

early years 1930ndash1959 In Einstein and the History of General Relativityed D Howard and J Stachel (Boston Birkhauser 1989)

[355] R Feynman Quantum theory of gravitation Acta Physica Polonica 24(1963) 697

[356] B DeWitt In Conference Internationale sur les Theories Relativistes dela Gravitation ed Gauthier-Villars (Warsaw Editions Scientifiques dePologne 1964)

[357] PG Bergmann and A Komar The coordinate group symmetries of gen-eral relativity Int J Theor Phys 5 (1972) 15

[358] A Peres Nuovo Cimento 26 (1962) 53[359] JA Wheeler Geometrodynamics and the issue of the final state In Rel-

ativity Groups and Topology ed C DeWitt and BS DeWitt (New Yorkand London Gordon and Breach 1964) p 316

448 References

[360] BS DeWitt Phys Rev Lett 12 (1964) 742 In Dynamical Theory ofGroups and Fields (New York Wiley 1965)

[361] LD Faddeev and VN Popov Feynman diagrams for the YangndashMills fieldPhys Lett 25B (1967) 30

[362] M Veltman in Proc 6th Int Symp Electron and Photon Interactions atHigh Energies ed H Rollnik and W Pfeil (Amsterdam North Holland1975)

[363] LD Faddeev and VN Popov Perturbation theory for gauge invariantfields Kiev Inst Theor Phys Acad Sci 67-036 (Fermilab Publication72-057-T)

[364] BS DeWitt Quantum theory of gravity II The manifestly covariant the-ory Phys Rev 162 (1967) 1195 Quantum theory of gravity III Applica-tions of the covariant theory Phys Rev 162 (1967) 1239

[365] R Penrose Twistor theory J Math Phys 8 (1967) 345[366] C Misner Quantum cosmology Phys Rev 186 (1969) 1319[367] B Zumino Effective lagrangians and broken symmetries In Brandeis Uni-

versity Lectures On Elementary Particles And Quantum Field Theory Vol2 ed S Deser (MIT Press Cambridge MA 1971) pp 437ndash500

[368] G rsquot Hooft Renormalizable lagrangians for massive YangndashMills fieldsNucl Phys B35 (1971) 167 G rsquot Hooft and M Veltman Regularizationand renormalization of gauge fields Nucl Phys B44 (1972) 189

[369] D Finkelstein Space-time code Phys Rev 184 (1969) 1261ndash1279[370] G trsquoHooft An algorithm for the poles at dimension four in the dimensional

regularization Nucl Phys B62 (1973) 444 G trsquoHooft and M VeltmanOne-loop divergencies in the theory of gravitation Ann Inst Poincare 20(1974) 69 S Deser and P Van Nieuwenhuizen One loop divergences ofthe quantized EinsteinndashMaxwell fields Phys Rev D10 (1974) 401 Non-renormalizability of the quantized DiracndashEinstein system Phys Rev D10(1974) 411

[371] WG Unruh Notes on black hole evaporation Phys Rev D14 (1976) 870[372] G Parisi The theory of non-renormalizable interactions 1 the large-N

expansion Nucl Phys B100 (1975) 368[373] S Ferrara P van Nieuwenhuizen and DZ Freedman Progress toward

a theory of supergravity Phys Rev D13 (1976) 3214 S Deser and PZumino Consistent supergravity Phys Lett B62 (1976) 335 For a reviewsee P van Nieuwenhuizen Supergravity Physics Reports 68 (1981) 189

[374] L Brink P Di Vecchia and P Howe A locally supersymmetric andreparameterization-invariant action for the spinning string Phys LettB65 (1976) 471ndash474 S Deser and B Zumino A complete action for thespinning string Phys Lett B65 (1976) 369

[375] KS Stelle Renormalization of higher derivatives quantum gravity PhysRev D16 (1977) 953

[376] JB Hartle and SW Hawking Path integral derivation of the black holeradiance Phys Rev D13 (1976) 2188

References 449

[377] AM Polyakov Quantum geometry of the bosonic string Phys Lett 103B(1981) 207 Quantum geometry of the fermionic string Phys Lett 103B(1981) 211

[378] G Horowitz and A Strominger in 10th Int Conf on General Relativityand Gravitation ndash Contributed Papers ed B Bertotti F de Felice and APascolini (Padova Universita di Padova 1983)

[379] CJ Isham Quantum logic and the histories approach to quantum theoryJ Math Phys 35 (1994) 2157 gr-qc9308006

[380] RD Sorkin Posets as lattice topologies In General Relativity and Grav-itation Proceedings of the GR10 Conference Volume I ed B BertottiF de Felice and A Pascolini (Rome Consiglio Nazionale Delle Ricerche1983) p 635

[381] MB Green and JH Schwarz Anomaly cancellation in supersymmetricd = 10 gauge theory requires SO(32) Phys Lett 149B (1984) 117

[382] DJ Gross JA Harvey E Martinec and R Rohm The heterotic stringPhys Rev Lett 54 (1985) 502ndash505

[383] P Candelas GT Horowitz A Strominger and E Witten Vacuum con-figurations for superstrings Nucl Phys B258 (1985) 46

[384] AA Belavin AM Polyakov and AB Zamolodchikov Infinite conformalsymmetry in two-dimensional quantum field theory Nucl Phys B241(1984) 333

[385] R Penrose Gravity and state vector reduction In Quantum Concepts inSpace and Time ed R Penrose and CJ Isham (Oxford Clarendon Press1986) p 129

[386] GT Horowitz J Lykken R Rohm and A Strominger A purely cubicaction for string field theory Phys Rev Lett 57 (1986) 283

[387] K Fredenhagen and R Haag Generally covariant quantum field theoryand scaling limits Comm Math Phys 108 (1987) 91

[388] H Sato and T Nakamura eds Marcel Grossmann Meeting on GeneralRelativity (Singapore World Scientific 1992)

[389] E Witten Topological quantum field theory Comm Math Phys 117(1988) 353

[390] E Witten Quantum field theory and the Jones polynomial Comm MathPhys 121 (1989) 351

[391] MF Atiyah Topological quantum field theories Publ Math Inst HautesEtudes Sci Paris 68 (1989) 175 The Geometry and Physics of Knots edAccademia Nazionale dei Lincei (Cambridge Cambridge University Press1990)

[392] GT Horowitz Exactly soluble diffeomorphism invariant theories CommMath Phys 125 (1989) 417

[393] E Witten (2 + 1)-dimensional gravity as an exactly soluble system NuclPhys B311 (1988) 46

[394] S Carlip Lectures on (2 + 1)-dimensional gravity (lecture given at theFirst Seoul Workshop on Gravity and Cosmology February 24ndash25 1995)

450 References

[395] S Deser and R Jackiw Three-dimensional cosmological gravity dynam-ics of constant curvature Ann Phys 153 (1984) 405 S Deser R Jackiwand G rsquot Hooft Three-dimensional Einstein gravity dynamics of flatspace Ann Phys 152 (1984) 220 A Achucarro and PK Townsend AChernndashSimon action for three-dimensional antidesitter supergravity theo-ries Phys Lett B180 (1986) 89


[397] D Gross and A Migdal Nonperturbative two-dimensional quantum grav-ity Phys Rev Lett 64 (1990) 635 M Douglas and S Shenker Nucl PhysB335 (1990) 635 E Brezin and VA Kazakov Phys Lett B236 (1990)144 Random Surfaces and Quantum Gravity ed O Alvarez E Marinariand P Windey (New York Plenum Press 1991)

[398] CG Callan BS Giddings JA Harvey and A Strominger Evanescentblack holes Phys Rev D45 (1992) 1005

[399] CW Misner KS Thorne and JA Wheeler Gravitation (San FranciscoFreeman 1973)

[400] G rsquotHooft Dimensional reduction in quantum gravity Utrecht PreprintTHU-9326 gr-qc9310026 L Susskind The world as a hologram JMath Phys 36 (1995) 6377

[401] AH Chamseddine and A Connes Universal formula for noncommutativegeometry actions unification of gravity and the standard model PhysRev Lett 24 (1996) 4868 The spectral action principle Comm MathPhys 186 (1997) 731

[402] J Polchinski Dirichlet branes and RamonndashRamon charges Phys RevLett 75 (1995) 4724

[403] CM Hull and PK Townsend Unity of superstring dualities Nucl PhysB438 (1995) 109

[404] T Banks W Fischler SH Shenker and L Susskind M-theory as a matrixmodel a conjecture Phys Rev D55 (1997) 5112

[405] MJ Duff M-Theory (the theory formerly known as strings) Int J ModPhys A11 (1996) 5623

[406] S Frittelli C Kozameh and ET Newman GR via characteristic surfacesJ Math Phys 5 (1995) 4984 5005 6397 T Newman in On EinsteinrsquosPath ed A Harvey (New York Berlin Heidelberg Springer-Verlag 1999)

[407] A Strominger and G Vafa Microscopic origin of the BekensteinndashHawkingentropy Phys Lett B379 (1996) 99 G Horowitz and A Strominger Blackstrings and p-branes Nucl Phys B360 (1991) 197 J Maldacena and AStrominger Black hole grey body factor and D-brane spectroscopy PhysRev D55 (1997) 861 G Horowitz Quantum states of black holes In ProcSymp Black Holes and Relativistic Stars in Honor of S ChandrasekharDecember 1996 gr-qc9704072

[408] A Connes MR Douglas and A Schwarz Noncommutative geometry andmatrix theory compactification on tori JHEP 9802 (1998) 003

References 451

[409] J Ambjorn M Carfora and A Marzuoli The Geometry of DynamicalTriangulations Lecture Notes in Physics (Berlin Springer-Verlag 1997)

[410] JM Maldacena The large-N limit of superconformal field theories and su-pergravity Adv Theor Math Phys 2 (1998) 231 Int J Theor Phys 38(1999) 1113 E Witten Anti-deSitter space and holography Adv TheorMath Phys 2 (1998) 253

[411] M Gasperini and G Veneziano Pre-Big Bang in string cosmology As-tropart Phys 1 (1993) 317

[412] P Bergmann in Conference Internationale sur les Theories Relativistesde la Gravitation ed Gauthier-Villars (Warsaw Scientifiques de Pologne1964)

[413] P van Nieuwenhuizen in Proc First Marcel Grossmann Meeting on Gen-eral Relativity ed R Ruffini (Amsterdam North Holland 1977)

[414] P Bergmann in Cosmology and Gravitation ed P Bergmann and V DeSabbata (New York Plenum Press 1980)

Index

n-j symbols 377SO(2 1) 359SO(3) 36 146 331 360SO(3 1) 34 359SO(4) 341 343 390

Casimirs 349 355SO(n) 390SU(2) 227 377

local gauge invariance 146 234so(3) basis xxiiso(3 1C) 36su(2) basis xxii 380

4-simplex 332 343

Achucarro 409action-at-a-distance 49 50ADM formalism 400Alexandrov 295Alfaro 315Amati 409ambient isotopic 384Anderson 96arc 282area

definition 43 150in spinfoams 350operator 249 different orderings

293spectrum degenerate sector 259

main sequence 249Aristotle 53 77Arnold 98

Arnowit 163 400Ashtekar xxv 163 275 318 408AshtekarndashLewandowski measure 231Atiyah 409

background 12background independence 7ndash31

263ndash265 281 371in strings 8 412

background-independent QFT 13Baez 274 364Barbero connection 156 163 276Barbieri 364Barbour 96Bardeen 301 318Barrett 349 364 411BarrettndashCrane model 330 348Bekenstein 301 312 404BekensteinndashMukhanov effect 293

312Belavin 407Bergmann 162 399ndash401 413BF theory 340 350binor calculus 383bivector 354black hole

entropy 301 308 372 404 411extremal 411ringing modes 312

Bohr 28 398Bojowald 318 353Boltzmann 29

452

Index 453

boosted geometrical operators 318Boulatov 364boundary data space 123 134BrinkndashSatcheler diagrams 390Bronstein 398bubble 338 342

CalabyndashYau manifolds 407Cartan 60 96

structure equation 35 46 331 333Carter 301 318ChernndashSimon

functional 409theory 411

ChoquetndashBruhat 96Christoffel connection 46chromatic evaluation 386Ciafaloni 409Ciufolini 96classical limit 103 179 268 373ClebschndashGordan condition 379color 252coloring 234 252 326configuration space

nonrelativistic 98relativistic 101 105 107

conformal field theory 407Connes 10 32 144 206 410contiguity 77 220Copernicus 419correlation 105 175 218cosmic-ray energy thresholds 316cosmological constant 35 293 330

343 371covariant derivative 34 37 38 46Crane 349 355 411CranendashYetter model 330 342curvature 34curve xxicycle 345cylindrical functions 227

DeDonder 144hamiltonian 132

degenerate sector 253De Pietri 275 364 365Descartes 28 48 53 77

Deser 163 400 404 409DeWitt 4 17 399 401 402DeWittndashMorette 96diffeomorphisms 41 146 229 238

active and passive 62extended 232 266

DillardndashBleick 96Dirac 4 144 162 211 221 400 417divergences 359

infrared 293 339 342ultraviolet 277 282 291 339 407

Earman 96edge 325 329 332Egan 32Einstein 10 28 33 47 48 50 51

55 65 66 68 71 74 209 415416

equations 35 36 38 39 47EinsteinndashSchrodinger equation 226

402energy 203energy-momentum 39event 105

space 105evolution equation 106extended loop representation 294eyeglasses graph 238

face 325 329 332Faddeev 402FaddeevndashPopov ghosts 402Faraday 17 28 49 415

lines 16 49Fermi theory 7 8fermion 36 38 287Feynman 4 320 321 401 402

expansion 343 348Feynman rules for GR 403Fierz 399finiteness 280 289 359Finkelstein 5 222 404flat 35Fleischhack 274Fock space 190 272four-simplex 329 341Fraser 96

454 Index

Fredenhagen 408Freidel 364ndash365Friedmann

equation 57 297models 296

functional representation 186

Galileo 28 29 374 416Gambini 274 294 315 408gamma-ray bursts 316 412Gelfand spectrum 231Gelfand triple 167GelfandndashNeimarkndashSiegal

construction 362GFT (group field theory) 330 343

356lorentzian 359

Goedelrsquos solution 76Gorelik 398Goroff 31 407GPS coordinates 88grains of space 18graph 227

4-strand 345null 231subspaces 230 proper 231

grasp 244 245 253 282gravitational electric field 148 243graviton 272 398Green 32 407 408Gupta 399

Haag 408Hamilton 104

equations 99 111function 103 120 in field theory

135 of GR 151HamiltonndashJacobi

formalism 102 in field theory 137in GR 146

function characteristic 103principal 103

relativistic formalism 113hamiltonian

nonrelativistic 99relativistic 108 177

hand 250harmonic oscillator 104

Hartle 164 407Hawking 4 32 301 302 318 320

404 405 407Heisenberg 28 398 415higher-derivative theories 7 406Hodge star xxihole argument 68 80holographic principle 412holonomy 44 227 242Holst 36 156Horowitz 406 409Huygens 416

Immirzi parameter 19 156 163 250303 365

Inertia 57Inertial frame 57ndash61inflation 297 301inflaton 297 301information 218initial singularity 298 372intertwiners 199 237 381Isham 32 275 408Iwasaki 411

Jackiw 409Jacobson 274 408JacobsonndashSmolin solutions 408Jones polynomial 409

Kauffman brackets 384KauffmanndashLins diagrams 389Kepler 419Klein 398KMS 205knot 241Komar 401Krasnov 318Kretschmann 78 96Kuchar 32

Lagrange 98 129lagrangian 98Landau 33Lapse function 111lattice YangndashMills theory 16 198

227 228 229 230 269 271Leibnitz 55

Index 455

lengthdefinition 43operator 275

LensendashThirring effect 76Lewandowski 274 275linear connection 46link 234 329

virtual 257 258 267 342 352 381Livine 365local Schrodinger equation 194Loll 274loop

operator 250state 15 228 236transform 228 409

Lorentzlocal transformations 41

Lorentz invariance 316lorentzian theory 292

M-theory 412Mach 55

principle 75Major 275Maldacena 412Mandelstam 16 17Markopoulou 364 365Maxwell 28 416measurement 210

quantum gravitational 368metric 46metric geometry 61Minkoswski solution 40Misner 163 400 403 409 413Modesto 318Morales-Tecotl 315motion 99 105 107motion absolute or relative 53moving frame 60Mukhanov 312multiloop 228multiplicity 234

new variable 408Newman 399 411Newton 28 30 48 54 415 417

bucket 54 76constant 35

NewtonndashWigner operator 190node 234 329Norton 96Noui 363 365null surface formulation 411

observablecomplete 178gravitational 367nonrelativistic 105partial 105 172 177 in field

theory 130relativistic 105

observablespartial 107

observer 210Oeckl 221Okolow 274Ooguri 364 409

model 330orbit 101Oriti 364

Palandri 32Palatini 96Parisi 405particle

global 196in GR 39local 196scattering 363

path xxiPauli 399Pauli matrices 380pendulum 104 105

timeless double 109 112 113 122181

Penrose 4 5 274 383 402 403407

evaluation 386Peres 401Perez 363ndash365PeterndashWeyl theorem 230phase space

nonrelativistic 99of GR 150relativistic 98 102 106 107

Planck 419

456 Index

Plebanskiconstraints 37 159 353 355two-form 36 96 147 160

Poincare one-form 100Polchinksi 32Polyakov 16 17 406 407PonzanondashRegge

ansatz 336model 330 334 364 403

presymplectic mechanics 100110

projective family 230projector 170 178 226propagator 167 185 321Pullin 274 294 315pulsar velocities 316

quadritangent 131quanta of area 19 249quantum cosmology 296 403

loop 299quantum event 211quantum gravity phenomenology

412quantum group 293 340 343quantum tetrahedron 355

reality conditions 128 145 208recoupling

theorem 257 388theory 251 383

reduction formulae 388Regge 5

calculus 335 401Reisenberger 365 411relationalism 77 220relativistic particle 118 122Ricci

scalar 35tensor 35 47

Riemannconnection 46geometry 47tensor 47

rigged Hilbert space 167Robinson 32Rosenfeld 398 399

s-knot 241 263Sagnotti 31 407Sahlmann 271scale factor 297Schrodinger 211

equation 165 168 370local equation 194picture 370representation 191

Schwarz 32 407 408Schwarzschild solution 269 302 304Schwinger

equations 400function 192

selfdualconnection 36projector 36

Sen 96 408seperating 205simple representation 349 355 390simplicity relation 390Smolin 32 274 365 408soldering form 60Sorkin 5 407Souriau 98space entity or relation 52spacetime coincidences 70 74 95

220spacetime foam 28spectral action 410spherical harmonics 390spherical vector 391spike 339spin 230spin connection 34spin network 16 17 200 234 274

347 372 402abstract 19embedded 21lorentzian 294state 200 234 236

spinfoam 26 325 412GFT duality 343

spinor 377spinor calculus 383Stachel 398

Index 457

standard model 5 38 276 286 291state 210n-particle 189boundary 174 176 178empty 203 273equilibrium 141 204

Gibbs 204instantaneous 102 116kinematical 169ndash178 193 229one-particle 188particle 195relativistic 106spacetime 168statistical 141 204

state-sum 320Stelle 405string field theory 407string theory 7 406Strominger 406substantivalism 77sum-over-geometries 28 320 322sum-over-surfaces 320 338supergravity 7 405 406superstrings 408supersymmetry 7 13Susskind 412symplectic mechanics 99

rsquot Hooft 4 5 404 409 412tangles 384target space 130tetrad 34 60tetrahedral net 387thermal clock 143thermal fluctuations of the geometry

305thermal time hypothesis 143 206Thiemann 163 274 284 295 310

365 411Thorne 409time

clock 84coordinate 83cosmological 31 86newtonian 85parameter 86

proper 84thermal 142 205thermodynamical 85

TOCY model 330 344Tomita flow 205topological field theory 332 340 409torsion 34 46

free spin connection 34Townsend 409transition amplitudes 178 200 323

346triangulation 328 332

dual 332independence 340

Turaev 409TuraevndashViro model 330 340twistor theory 403two-complex 326two-skeleton 332

unitarity 369Unruh 404Urrita 315

vacuum 202covariant 23 176 178 197dynamical 174 175 193Minkowski 175 188 192 271 273

363 374Van Nieuwenhuizen 404variational principle 102 108 131Veltman 402 404Veneziano 409vertex 325 329 332

amplitude 328Viro 409volume

definition 44 150operator 260 different

regularizations 294quanta of 263

Wald 96wave function of the Universe 407weave 268Weinberg 4 32 96 405Weyl 96 144

458 Index

whale 9 75Wheeler 4 17 28 32 96 310 400

401 403 409WheelerndashDeWitt equation 169 178

185 226 276 295 403Wilson 16 17 405

loop 15

Winkler 365Witten 32 408ndash410

YangndashMills field 37 286

Zamolodchikov 407Zumino 403

Cover

Front matter

Title13

Copyright13

Contents13

Foreword13

Preface13


Acknowledgements


Part I Relativistic foundations








121 Why loops







21 Formalism







221 Einsteinrsquos first problem a field theory for the newtonian interaction


223 The key idea



23 Interpretation



24 Complements

241 Mach principles


243 Has general covariance any physical content




3 Mechanics

31 Nonrelativistic mechanics





324 Mechanics is about relations between observables




33 Field theory











421 Euclidean GR



43 Hamiltonian GR






5 Quantum mechanics





514 Boundary state space K and covariant vacuum |0gt13


52 Relativistic QM









535 Boundary state space K and covariant vacuum |0gt


54 Quantum gravity



55 Complements







563 Information


Part II Loop quantum gravity

6 Quantum space


62 The kinematical state space K

621 Structures in K



63 Internal gauge invariance The space Ko



64 Diffeomorphism invariance The space K subscript diff



65 Operators



66 Operators on K subscript 0


662 Quanta of area




67 Quantum geometry




711 Finiteness

712 Matrix elements

713 Variants


721 YangndashMills

722 Fermions

723 Scalars




741 Variants

8 Applications


811 Inflation










921 Boundaries

93 Models


932 BF theory


934 BC models





10 Conclusion


1011 GR and QM





Part III Appendices

Appendix A Groups and recoupling theory





A23 Normalizations


Appendix B History


B2 Five periods






B3 The divide

Appendix C On method and truth


C2 On realism

C3 On truth

References

Index

the eye carlo... · 2020. 1. 17. · quantum gravity quantum gravity is perhaps the most important...

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