roberto de pietri and carlo rovelli- geometry eigenvalues and the scalar product from recoupling...

8/3/2019 Roberto De Pietri and Carlo Rovelli- Geometry eigenvalues and the scalar product from recoupling theory in loop q

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Geometry eigenvalues and the scalar product from recoupling theory in loop quantum gravity

Roberto De Pietri*

Dipartimento Di Fisica, Universita di Parma and Istituto Nazionale di Fisica Nucleare, Sezione di Milano,

Gruppo Collegato di Parma, I-43100 Parma (PR), Italy

Carlo Rovelli

Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, Pennsylvania 15260

Received 13 February 1996

We summarize the basics of the loop representation of quantum gravity and describe the main aspects of the

formalism, including its latest developments, in a reorganized and consistent form. Recoupling theory, in its

graphical tangle-theoretic Temperley-Lieb formulation, provides a powerful calculation tool in this context.

We describe its application to the loop representation in detail. Using recoupling theory, we derive general

expressions for the spectrum of the quantum area and the quantum volume operators. We compute several

volume eigenvalues explicitly. We introduce a scalar product with respect to which area and volume are

symmetric operators, and the trivalent expansions of the spin network states are orthonormal.

S0556-28219604516-X

PACS numbers: 04.60.Ds, 03.70.k

I. INTRODUCTION

We start with a citation from Penrose 1: My own viewis that ultimately physical laws should find their most natural

expression in terms of essentially combinatorial principles,that is to say, in terms of finite processes such as counting orother basically simple manipulation procedures. Thus, in ac-cordance with such a view, some form of discrete or combi-natorial space time should emerge. The loop approach toquantum general relativity 2,3 seems to be leading pre-cisely to a realization of such a vision of a combinatorialspace-time, deriving it solely from a strict application of con-

ventional quantum ideas to standard general relativity.1

A number of recent advances in this direction havestrengthened this hope. First of all, there is the mathemati-cally rigorous development of the connection representation9,1416 which has lead to recovering the loop representa-tion formalism from a general quantization program. Thisapproach has sharpened various loop representation resultsusing rigorous C* algebraic and measure theoretical tech-niques, and has put them on a solid mathematical footing.For a discussion of the precise relation between the two for-mulations of loop quantum gravity, see Refs. 17,18. Fur-thermore: a simplification of the formalism due to the intro-duction in quantum gravity of the spin network basis 19see also 20,21; the result that area 22 and volume op-erators 23,24 have discrete eigenvalues; the idea that in thepresence of matter these eigenvalues might be taken asphysical predictions on quantum geometry 25; a Hamil-tonian generating clock time evolution 26 and a tentative

perturbation scheme for computing diffeomorphism invariant

transition amplitudes 27; the extension of the theory to fer-

mions 28 to the electromagnetic field 29,30. This rapiddevelopment has produced a certain amount of confusion in

the notation and the basics of the theory. A first aim of this

paper is to bring some order in the kinematics of the looprepresentation formalism, by presenting the basics formulas,notations and results in a consistent and self-contained form.This allows us to insert a sign factor into the very definitionof the loop representation, short-cutting sign complicationsof the previous formulation. In a sense, we bring to full ma-turity the insights of Ref. 19.

With this sign factor, loop states of the loop representa-tion satisfy the axioms of Penroses binor calculus 31,or, equivalently, the axioms of the tangle-theoretic formula-tion of recoupling theory 32 for the special classicalvalue A1 of the deformation parameter. This fact bringsa powerful set of computational techniques at the service ofquantum gravity. We describe here in detail how this calcu-lus can be used. Calculations in loop quantum gravity werefirst performed using the grasping operation on single loops3. It was then realized, mainly in 15, that such combina-torial techniques admitted a group theoretical interpretationsu2 representation theory admits a fully combinatorial de-scription. Recoupling theory is a furtherand far morepowerfullevel of sophistication for the same calculus.

The idea that recoupling theory plays a role in loop quan-tum gravity has been advocated by Reisenberger 33 and bySmolin. Motivated by certain physical and mathematicalconsiderations, Borissov, Major, and Smolin 34,35 haveconsidered deformations of the standard loop representationtheory. Recoupling theory with general values of the defor-mation parameter A plays a key role in the definition of thesedeformations. What we do here is very different in spirit: weremain within the framework of the standard loop-representation quantum general relativity GR, and use re-coupling theory merely as a computational tool.

*Electronic address: [email protected] address: [email protected]

1For an overview of current ideas on quantum geometry, see

4 6; for a recent overview of canonical gravity, see 7; for intro-

ductions to loop quantum gravity, see 813.

PHYSICAL REVIEW D 15 AUGUST 1996VOLUME 54, NUMBER 4

540556-2821/96/544/266427 /$10.00 2664 1996 The American Physical Society


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Using recoupling theory, we derive general formulas forarea and volume in quantum gravity. The spectrum of thearea agrees with previously published results 23. The deri-vation presented here is simpler and more elegant than theone in Ref. 23. The first of our main results is a generalformula for the volume. We present it here expressed interms of su2 6-j symbols and related quantities. We con-firm the fact that trivalent vertices have zero volume, first

pointed out by Loll 24. We explicitly compute many eigen-states for four- and five-valent vertices. Loll has computed afew of these eigenvalues in 36 using a different technique.We find agreement with the numbers published by Loll seealso 35. We show that the absolute value and the squareroot that appear in the definition of the volume operator arewell defined. Indeed, we show that the arguments of the ab-solute value are finite dimensional matrices diagonalizableand with real eigenvalues; and that the arguments of thesquare root are finite dimensional matrices diagonalizableand with real non-negative eigenvalues. We show in generalthat the eigenvalues of the volume are real and non-negative.

Finally, the technique introduced allows us to define a

scalar product in the loop representation, by requiring thatarea and volume be symmetric, and that spin network statesbe orthogonal to each other whatever the trivalent decom-position of high valence vertices we use. This is our secondmain result.

The structure of this paper is the following. In Sec. II, wereview the basics of the Ashtekar formulation of generalrelativity and we define the loop variables in the new formthat leads directly to recoupling theory. In Sec. III, we de-rive the basic equations of the loop representation. In Sec. IVwe discuss the role of recoupling theory. In Sec. V we definethe spin network basis. In Sec. VI, we discuss the area op-erator, and in Sec. VII the volume operator. In Sec. VIII wedefine the scalar product. Section IX contains our conclu-sions.

II. LOOP VARIABLES IN CLASSICAL GR

We begin by reviewing the canonical formulation of gen-eral relativity in the real Ashtekar formalism 3739. This isgiven as follows. We fix a three-dimensional manifold M

and consider two real smooth SO3 fields Aai (x) and

Eia(x) on M. We use a ,b, . . .1,2,3 for abstract spatial

indices and i,j , . . . 1,2,3 for internal SO3 indices. Weindicate coordinates on M with x. The relation between thesefields and conventional metric gravitational variables is as

follows: Eia(x) is the densitized inverse triad, related to the

three-dimensional metric g ab (x) of the constant-time surfaceby

gg abEiaEi

b , 2.1

where g is the determinant of g ab ; and

A ai x a

i x kai x , 2.2

where ai (x) is the SU2 spin connection associated to the

triad and kai (x) is the extrinsic curvature of the three surface

up to indices position. Notice the absence of the i in Eq.2.2, which yields the real Ashtekar connection. The realAshtekar connection 2.2 is the natural variable for the Rie-mannian theory, but it can be used as the basic field for theLorentzian theory as well, at the price of a more complicatedform of the Hamiltonian constraint. Recently, Thiemann andAshtekar 39 have argued that the most promising strategyfor implementing the quantum reality conditions is to start

from the real Ashtekar connection, and circumvent the diffi-culties due to the complicate form of the Lorentzian Hamil-tonian constraint by expressing it in terms of the RiemannianHamiltonian constraint via a generalized Wick transformsee also 40,41. Here, we will not discuss the dynamics.Therefore our results can be significative for the Riemanniantheory as well as for the Lorentzian theory. We will discussthe necessary modifications of the formalism for applying itto the complex Ashtekar connection at the end of Sec. VII.

It is useful for what follows to consider the dimensionalcharacter of the field with care. We set the dimension of thefields as follows:

gab L2

, E

i

a

L2

,

Aai dimensionless. 2.3

The popular choice of taking the metric dimensionless is notvery sensible in GR. It forces coordinates to have dimensionsof a length, but the freedom of arbitrary transformations onthe coordinates is hardly compatible with dimensional coor-dinates. Coordinates, for instance, can be angles, and assign-ing angles dimension of a length makes no sense. The Ein-stein action can be rewritten see, for example, 8 as

S1

G

d4xgR 1G

dx 0 d3xA a

iEiaA 0

i CiNaCaN, 2.4

where we have set

G16G Newton

c3, 2.5

G Newton being Newtons gravitational constant, and C

i ,

Ca , W the diffeomorphism, Gauss, and Hamiltonian con-

straints. It follows that the momentum canonically conjugate

to A ai is

p iax S

Aai x

1G

Ei

a 2.6

and therefore the fundamental Poisson bracket of the Hamil-tonian theory is

A ai x ,Ej

by Gabj

i

3x,y . 2.7

The spinorial version of the Ashtekar variables is given interms of the Pauli matrices i , i1,2,3, or the su2 genera-tors i (i/2)i , by

Eax iEia

x i2E

ia

x i 2.8

54 2665GEOMETRY EIGENVALUES AND THE SCALAR PRODUCT . . .


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A ax i

2A a

i x iA ai x i . 2.9

A a(x) and Ea(x) are 22 complex matrices. We use upper

case indices A ,B , . . . 1,2 for the spinor space on which thePauli matrices act. Thus the components of the gravitational

fields are AaAB(x) and EaA

B(x).

In order to construct the loop variables, we start fromsome definitions.Segment. A segment is a continuous and piecewise

smooth map from the closed interval 0,1 into M. We write:sa(s).

Loop. A loop is a segment such that a(0)a(1).Equivalently, it is a continuous, piecewise smooth map fromthe circle S1 into M

3.Free Loop Algebra. We consider formal linear combi-

nations of formal products of loops, as in

c 0i

c iijk

cjkjk , 2.10

where the cs are arbitrary complex number and the s areloops; we denote the space of such objects as the free loopalgebra AfL. See also 9.

Multiloop. We denote the monomials in AfL , namelythe elements of the form 1 . . . n as multiloops.

We indicate multiloops by a Greek letter, in the same manneras single loops: 1 . . . n .

Given a segment , we consider the parallel propagator ofAa along , This is defined by the equation

d

dU,0

da

dA aU,0 0, 2.11

with the boundary condition U(0 ,0)ABAB The formalsolution is

U,0 Pexp0

d aA a , 2.12where P indicates the path ordering of the exponential. Wealso writein a somewhat imprecise notationUU(0,1) and U(s 2 ,s 1)U(2 ,1) if s 2(2) ands1(1).

We can now define the fundamental loop variables. Givena loop and the points s1 ,s2 , . . . ,s n we define

TTrU, 2.13

Ta s Tr U s ,s Ea s 2.14

and, in general,

Ta1a2 s1 ,s 2 TrU s 1 ,s 2Ea2 s2 U s 2 ,s 1E

a1 s1 , 2.15

Ta1 . . . aN s1 , . . . ,sNTr U s 1 ,sNEaN sNU sN ,sN1 . . . . . . U s 2 ,s 1E

a1 s1 .

The function T defined in Eq. 2.13 for a single loop,can be defined over the whole free loop algebra AfL:given the generic element AfL in Eq. 2.10, we pose

T2c0i

c iTii j

c i jTiTj .

2.16

The reason for the 2 in the first term is the following. Wemay think of the first term of the sum as corresponding to thepoint loop, or a loop whose image is a point. For thisloop, the exponent in Eq. 2.12 is zero, the holonomy is theidentity in sl2,C), namely in two dimensions and T istherefore 2.

Notice that there is a sign difference between the usualloops observables 2,3 denoted T variables and these newloop observables, denoted Tvariables. This is a key techni-cality at the origin of the simplification of the formalismpresented here. The new sign takes care immediately of thesign complications extensively discussed in Ref. 19. Thesuggestion that those sign complications could be avoided byinserting a minus sign in front of the trace was considered byMajor as well 42. Let us illustrate the consequences ofhaving this sign. Consider an N component multiloop12 . . . N . We have

T 1NT1T1

Tr U1 Tr UN

1 NTr U1 Tr UN

1 NT.

This shows that the new sign choice implements in the for-malism the sign factor for the number of loops that wasrecognized in 19 as the key to transform the spinor relationinto a local relation. In fact, we have

Tr UTr UTrUUTr UU10,2.17

TTT# sT#s10, 2.18

T T # sT # s10; 2.19

namely the spinor identity one and two ) has become abinor identity all ) see Penrose 43. While the first isnonlocal, the second is local, and is the basic identity at theroots of binor calculus and A1 recoupling theory. Forthe notations and # to be used in a moment, see forinstance 8.

2666 54ROBERTO DE PIETRI AND CARLO ROVELLI


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We recall here, for later use, the retracing identity. For allloops and segments , we have 3

TT1 . 2.20

The Poisson bracket algebra of these loop variables iseasily computed. For a rigorous way of performing thesecomputations, see 44. We give here the Poisson bracket of

the Tvariables of order 0 and 1.

T ,T0, 2.21

Ta s ,T12 G

a,s T# sT# s1,2.22

where we have defined

a,s

d a3,s. 2.23

The factor 12, different than in previous papers, is due to

the new conventions.

III. THE LOOP REPRESENTATION

OF QUANTUM GRAVITY

We now define the loop representation 45 of quantumgravity as a linear representation of the Poisson algebra ofthe Tvariables. First, we define the carrier space of the rep-resentation. To this aim, we consider the linear subspace Kof the free loop algebra defined by

KAfLT0, 3.1

and we define the carrier space V of the representation by

VAfL/K. 3.2

In other words, the state space of the loop representation isdefined as the space of the equivalence classes of linear com-binations of multiloops, under the equivalence defined by theMandelstam relations

if TT, 3.3

namely by the equality of the corresponding holonomies9.2 We denote the equivalence classes defined in his way,namely the elements of the quantum state space of the theoryas Mandelstam classes, and we indicate them in Dirac nota-tion as . Clearly, the multiloop states span actually,overspan the state space V. Later we will define a scalarproduct on V, and promote it to a Hilbert space. The reasonfor preferring a bra notation over a ket notation is just his-torical at this point. We recall that the loop representation

was originally defined in terms of kets in the dual ofV. These are represented on the overcomplete basis byloop functionals

. 3.4

The principal consequences of the Mandelstam relationsare the following:

1 The element does not depends on the orientationof, 1; 2 the element does not depend onthe parametrization of, ifa()af(); 3retracing, if is a segment starting in a point of, then

1; 3.5

4 the binor identity

#s# s1 . 3.6

It has been conjectured that all Mandelstam relations canbe derived by repeated use of these identities. We expect that

the methods described below may allow to prove this con-jecture, but we do not discuss this issue here.

Next, we define the quantum operators corresponding tothe Tvariables as linear operators on V. These form a repre-sentation of the loop variables Poisson algebra. We definethe loop operators as acting on the bra states from theright. Since they act on the right, they define, more pre-cisely, an antirepresentation of the Poisson algebra. We de-

fine the T operator by

c 0i

c iii j

c i jijT

c 0i c iii j c i jij.3.7

Next, we define the Ta(s) operator. This is a derivativeoperator i.e., it satisfies Leibniz rule over the free loopalgebra such that

Ta s 12 il02a,s # s# s

1 ,3.8

where we have introduced the elementary length l0 by

l 02G

16G Newton

c316l Planck

2 . 3.9

The definition extends on the entire free loop algebra byLeibniz rule and linearity. The two operators commute withthe Mandelstam relations and are therefore well defined onV.

Notice that the factor a,s in Eq. 3.8 depends on theorientation of the loop : it changes sign if the orientation of is reversed. So does the difference in the parentheses,therefore the right-hand side rhs of Eq. 3.8 is independentfrom the orientation of, as the lhs. On the other hand, both

2T is a function on configuration space, namely a function

over the space of smooth connections. Equality between functions

means of course having the same value for any value of the inde-

pendent variable; here, for all smooth connections.



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the rhs and the lhs of Eq. 3.8 change sign if we reverse theorientation of.

The action of the Ta(s) operator on a state canbe visualized graphically. The graphical action is denoted agrasp, and it can be described as follows. i Disjoin thetwo edges of the loop and the two edges of the loop , thatenter the intersection point s . ii Pairwise join the four openends of and in the two possible alternative ways. This

defines two new states. Consider the difference betweenthese two states arbitrarily choosing one of the two as posi-tive. iii Multiply this difference by the factor

i l02a,s , where the direction of which determines

the sign of a,s) is determined as follows: it is the di-rection induced on by which is oriented in the termchosen as positive. A moment of reflection shows that thedefinition is consistent, and independent from the choice ofthe positive term. An explicit computation shows that theoperators defined realize a linear representation of the Pois-son algebra of the corresponding classical observables.

The grasping rule generalizes to higher order Tvariables.

The action of Ta1 . . . an(s1 , . . . , sn), over a single loop-

state is given as follows. First the result vanishes unless crosses all the n points s i . If it does, the action of

Ta1 . . . an(s 1 , . . . ,s n) is given by the simultaneous graspon all intersection points. This action produces 2 n terms.These terms are summed algebraically with alternate signs,

and the result is multiplied by a factor i l02a,s1 for

each grasp, where the sign of each coefficient a,s i isdetermined assuming that is oriented consistently with in the term chosen as positive. Again, a moment of reflectionshows that the definition is consistent, and independent fromthe choice of the positive terms. The generalization to arbi-trary states, using linearity and the Leibnitz rule, is straight-forward. This concludes the construction of the linear ingre-

dients of the loop representation.

IV. LOOP STATES AND RECOUPLING THEORY

A quantum state in the state space V is a Mandelstamequivalence class of elements of the form 2.10. We nowshow that because of the equivalence relation, these statesare related to tanglesin the sense of Kauffman 46 andthey obey the formal identities that define the Temperley-Lieb-Kauffman recoupling theory described in Ref. 32.This fact yields two results. First, we can write a basis inV. This basis is constructed in the next section. Second, re-coupling theory becomes a powerful calculus in loop quan-tum gravity.

Consider the element , given in Eq. 2.10, of the vectorspace AfL. We need some definitions.

Graph of a state. We denote the union in M of the imagesof all the loops in the rhs of Eq. 2.10 as the graph of , and we indicate it as . Notice that is a graph inthe sense of graph theory 47, embedded in M.

Vertex. We denote the points i where fails to be asmooth submanifold of M as vertices.

Edge. We denote the lines e of the graph connecting thevertices as edges.

Valence. We say that a vertex i has valence n, or is n

valent, if n edges are adjacent to it. A vertex can have any

positive integer valence, including 1 and 2.

Clearly, is not uniquely determined by its graph . Ifour only information about a state is its graph, then we do

not know how the state is decomposed into multiloops, nor

how many single loops run along each edge, nor how the

single loops are rooted through the vertices. We now intro-

duce a graphical technique to represent this missing informa-

tion. The technique is based on the idea of blowing up the

graph as if viewed through an infinite magnifying glass

and representing the additional information in terms of pla-

nar tangles on the blown up graph. As we will see, these

tangles obey recoupling theory.

First, draw a graph isomorphic to in the sense of graphtheory that is, the isomorphism preserves only adjacencyrelations between vertices and edges, on a two-dimensionalsurface. As usual in graph theory, we must distinguish points

representing vertices from accidental intersections between

edges generated by the fact that we are representing a non-

planar graph on a plane. Denote these accidental intersec-tions as false intersections. Next, replace each vertex notthe false intersections by the interior of a circle in theplane, and each edge by a ribbon connecting two circles. Atfalse intersections, ribbons bridge each other without merg-ing. In this way, we construct a thickened out graph: atwo-dimensional oriented surface which loosely speakinghas the topology of the graph times the 0,1 interval.

Ribbon-net. We call this two-dimensional surface theribbon net or simply the ribbon of the graph , andwe denote it as R . Notice that the graph is embedded inM, while its ribbon net R is not.

Now we can represent the missing information needed toreconstruct from

as a formal linear combination of

tangles drawn on the surface R . First, we represent eachmultiloop in Eq. 2.10 by means of a closed line over R :

Planar (representation of a) multiloop. For each loop iin a given multiloop we draw a loop i over the ribbon netR , wrapping around R in the same way in which iwraps around . We denote the drawing over R) of allthe loops of a multiloop as the planar representation ofthe multiloop , or simply as the planar multiloop. Weindicate it as P.

For technical reasons, we allow edges and vertices of theribbon net to be empty of loops as well. Thus we identify aribbon net containing a planar multiloop, with a second oneobtained from the first by adding edges and vertices empty of

loops. Finally:Planar (representation of a) state. Every state is a

formal linear combination of multiloops: jcjj upto equivalence. We denote the corresponding formal linearcombination P jcj Pj of planar multiloops on the rib-

bon net R up to equivalence, as a planar representation of

.We have split the information contained in in two

parts: determines a graph embedded in M and a planarstate P . P is a linear combinations of drawings of loopsover a surface the ribbon net R ) and codes the informationon which loops are present and how they are rooted throughintersections. This information is purely combinatorial. On



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the other hand, contains the information on how theloops are embedded into M.

Notice that a multiloop determines its planar representa-tion only up to smooth planar deformations of the lineswithin the circles and the ribbons of the ribbon net. In otherwords, we can arbitrarily deform the lines within each circleand within each ribbon, without changing . In particular,the lines of the planar representation will intersect in pointsof R , and we can apply Reidemeister 48 moves 46 to

such intersections that is, disentangle them. Under andovercrossings of loops within R are not distinguished.

Let us come to the key observation on which the possi-bility of using recoupling theory relies. Consider an element of the free vector algebra. For simplicity, let us momen-tarily assume that is formed by a single loop which may self-intersect and run over itself. Thus( , P). Consider an intersection of two lines twosegments of P) in R . Break the two lines meeting at thisintersection, and pairwise rejoin the four legs, in the twoalternative possible ways, as in Fig. 1.

We obtain two new loops on R , which we denote asP and P . Consider the element of the free vector

algebra uniquely determined by the graph , and bythe linear combination of planar representationsP PP . Notice that is different than as anelement of the free vector algebra; however, the two are inthe same Mandelstam equivalence class because of the binorrelation 3.6, and therefore they define the same element ofthe quantum state space V. Namely . We say thattwo planar representations P and P are equivalent if

. Thus, in dealing with planar representations of aquantum state , we can freely use the identity

4.1

on P without changing the quantum state. This identity isthe identity i in p. 7 of Ref. 32 Eq. B1 in Appendix B,which is the key axiom of recoupling theory with thevalue of the A parameter set to 1.

An easy to derive consequence is that every closed lineentirely contained within a circle, or within a ribbon, can bereplaced by a factor d2. Furthermore, it is easy to seethat the retracing identity 3.5 implies that the loops of Pcan be arbitrarily deformed within the entire ribbon net,without changing the state . In particular, every loop con-tractible in R can be replaced by a factor d2. This is

the second axiom of recoupling theory Eq. B2 in Appen-

dix B in the A1 case. The value A1, correspond tothe case in which the distinction between over and under-

crossings can be neglected, consistently with the fact such

distinction is irrelevant for the planar representation of aloop.

Thus P can be interpreted as a linear combination oftangles in the sense of Ref. 32. The tangles obey the axi-oms of recoupling theory. They are confined inside the ori-ented surface R with has a highly nontrivial topology. Thisis the key result of this section.

The relation between loop states and recoupling theory issubtle, and may generate confusion. A source of confusion isgiven by the fact that the relation between recouplings andknots in knot theory 32 is different from the relation be-tween recouplings and knots in quantum gravity. In bothcases recouplings enters as a consequence of a skein orbinor equation as Eq. 4.1 holding at intersections. Butin knot theory this equation is satisfied by the Kauffmanbrackets at the false intersections of the planar projectionof a loop. Contrary to this, in quantum gravity Eq. 4.1 does

not hold for the false intersections. It holds for the intersec-tions of lines within R .

On the other hand, knots play a role in quantum gravity aswell 2, because of the diffeomorphism constraint. GRs dif-feomorphism invariance identifies states that have equivalentP , and whose graphs can be deformed into each other by3D diffeomorphism of M in the connected component of theidentity. To clarify this point, let us require that the ribbonnet R is generated by a two-dimensional projection of , and let us keep track of the resulting over and under-crossings at false intersections. Then diffeomorphism invari-ance identifies all states that have equivalent P , and whoseribbon nets can be transformed into each other by Reide-

meister moves at the false intersections . Thus, as far asdiff-invariant states are concerned, Reidemeister moves canbe used at the tangles intersections within the ribbon net aswell as at the false intersections. But in the first case a skeinequation Eq. 4.1 holds, in the second it does not.3 Mixingup the two cases has generated a certain confusion in thepast.

An immediate consequence of the result is that we canwrite a basis in V following 32. Given a state , and itsribbon net R , we can use Eq. 4.1 to eliminate all inter-sections from the P of each multiloop. Next, we can retraceeach single line that returns over itself, and eliminate everyloop contractible in R . We obtain parallel lines without

intersections along each ribbon and routings without inter-sections at each vertex. No further use of the retracing orbinor identity is then possible without altering this form.This procedure defines a basis of independent states, labeledby the graph, the number of lines along each edge, and el-ementary routings at each node. An elementary routing is aplanar rooting of loops through the vertex of the ribbon net,

3This is true in general. One may wonder if there is any special

quantum state 0 for which the relation 4.1 holds at false inter-sections as well. The possibility that such a special state could exist

in quantum gravity has been explored, with various motivations, by

various authors 34,49.

FIG. 1. The binor identity.



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having no intersections. This basis is not very practical forcalculations. In the next section, we use the technology of32 to define a more useful basis.4

V. THE SPIN NETWORK BASIS

The representation ( , P ) of a state can be ex-

panded in terms of a virtual trivalent representation asfollows.Virtual graph. To every graph , we can associate a triva-

lent graph v as follows. For each n-valent vertex v of ,arbitrarily label the adjacent edges as e0 . . . e (n1) , anddisjoin them from v. Then, replace v with n2 trivalentvertices N1 . . . Nn2, denoted virtual vertices. Join thevirtual vertices with n3 virtual edges E2 . . . En2,where Ei joins Ni1 and Ni . Prolong the edgese 2 . . . e (n1) to reach the corresponding virtual verticesN1 . . . Nn2, and the edges e1 and e (n1) to reach the virtualvertices N1 and Nn2. Denote the resulting trivalent graphv as the virtual graph associated to for the chosen or-dering of edges.

Virtual ribbon net. We denote the ribbon net of v as the

virtual ribbon net R v of . We view it as a subset of R ,

namely we view the virtual circles N1 . . . Nn2 and the vir-tual ribbons E2 . . . En2 as drawn inside the circle c repre-senting v . This circle c indicates that the virtual vertices

N1 . . . Nn2 correspond all to the same point of M. Thus avirtual ribbon net is a trivalent ribbon net with strings ofadjacent intersections specified.

Virtual representation. Finally, deform P so that it lies

entirely inside Rv We indicate the deformed P as P

v and

call it the virtual planar representation of . The virtual

representation P v of a state is not unique, due to the arbi-

trariness of assigning the ordering e0 . . . e (n1) to the edgesof n-valent intersections.The above construction is more difficult to describe in

words than to visualize, and is illustrated in Fig. 2.

Consider now deformations of the tangle P v within R

v

a subset of the deformations within the full R . We can

move all intersections of deform P v away from the vertices

to the virtual or real ribbons, leaving trivalent vertices freefrom intersections. Next, we can use the binor relation toremove all intersections from the ribbons, leaving noninter-

secting tangles with n inputs and n outputs along each singleribbon e . As described in Sec 2.2 of32, tangles of this kindcan be described as elements of the tangle-theoretic

Temperley-Lieb algebras Tn(e) . A basis of this algebra is ob-

tained by using the Jones-Wenzl projectors n(e) . Since we

are here in the case A1, the n(e) are just normalized

antisymmetrizers. More precisely, given the multiloop Pwith n lines along the ribbon e, call P

(p) ,p1 . . . n! the

multiloops obtained by all possible permutations p in theway the n lines entering e are connected to the n outgoinglines, and p the parity of the permutation, then

ne P

1

n!p 1 pP

p . 5.1

Notice the 1/n! factor, which was not present in previousconventions 23. It follows from the completeness of the

Jones-Wenzel projectors that a basis for all planar loops overa given Rv is given by the linear combination of loops in

which the lines along each virtual and real edge are fullyantisymmetrized. We can therefore expand every state instates in which lines are fully antisymmetrized along eachribbon. A state in which the lines along each virtual or realribbon are fully antisymmetrized is a spin network state.Thus we recover the result of Ref. 19, to which we refer fordetails.

A spin network state is characterized by a graph inM, by the assignment of an ordering to the edges adjacent toeach vertex, and by the number p e ofantisymmetrized linesin each virtual or real edge e . We denote the integer p e as thecolor of the corresponding edge e ofv. We will use alsothe spin j e of the edge, defined as half its color: j e

12

p e .5 At each vertex, the colors p 1, p2, and p 3 of the three

adjacent edges satisfy a compatibility condition: there mustexist three positive integers a, b, and c the number of linesrooted through each pair of edges such that

p 1ab , p2bc , p 3ca . 5.2

4A basis in a linear space is a set of linearly independent vectors

that span the linear space. The fact that for the moment we are stillworking in linear spaces without fixing a scalar product we will fix

a scalar product only later, in Sec. VIII has raised some confusion

in the past. It is perhaps worthwhile recalling that the notions of

basis, eigenvalues and eigenvectors are well defined notions for

linear spaces, not just for Hilbert spaces. They do not require a

scalar product to be defined in order to make sense. Similarly, the

fact that a linear operator is diagonalizable, or has real eigenvalues

does not depend on the presence of a scalar product. Given an

arbitrary linear basis v i in a finite dimensional linear space, a linear

operator A is Hermitian in this basis if its matrix elements defined

by (Av) iA ijvj satisfy A i

jAj

i . If A is Hermitian in a basis, then

A is diagonalizable and has real eigenvalues. This is true indepen-

dently from any scalar product.

5The oscillation between the historically motivated half integer

terminology spin and the rationally motivated integer terminol-

ogy color goes back to Penroses papers on spin networks 43.

FIG. 2. Construction of virtual vertices and virtual strips

over an n-valent vertex.



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It is easy to see that this condition is equivalent to theClebsh-Gordon condition that each of the three su2 repre-sentations of spin j i1/2p i is contained in the tensor productof the other two 43.

The spin network states form a basis in V. The basis ele-ments are given as follows. For every graph embedded inM, choose an ordering of the edges at each node. This choiceassociates an oriented trivalent virtual graph v nonembed-

ded to every .Spin network. A spin network S is given by a graph S in

M, and by a compatible coloring p e of the associated ori-ented trivalent virtual graph v. Thus S(S ,pe).

Spin network state. For every spin network S, the spinnetwork quantum state S (S , P S) is the element of Vdetermined by the graph S and by the linear combinationP S of planar multiloops obtained as follows. Draw p e lines

on each ribbon e of the ribbon net RSv ; connect lines at

intersections without crossings; this gives a planar multiloop

P S(0 ) then

PS

e pee

PS0

. 5.3

We can represent a spin network state as a colored trivalent

graph over the ribbon net R Sv with a single edge along each

ribbon. This representation satisfies the identities of recou-pling theory. We describe the main ones of these identities inAppendix E. As an example, we give here the formula thatallows one to express the basis elements of a four-valentintersection in terms of the basis elements of a differenttrivalent expansion. Using the recoupling theorem of 32p. 60, we have, immediately,

5.4

where the quantities cd jab i are su2 six-j symbols normal-

ized as in 32; see Appendices.A side remark should be added. An embedded colored

trivalent graph specify a state only up to a global sign,because it does not fix the overall sign of the antisymme-trized linear combination of multiloops. To keep track of thisoverall sign, one needs oriented trivalent graphs, as in Ref.43 where Penrose considered oriented spin networks and in19. An orientation of a trivalent graph is an assignment of

a cyclic order to the edges of each node, modulo Z2 that is,identifying two orientations if they differ in an even numberof intersections. v is oriented by the order assigned to theedges entering each vertex, and ribbon-nets are orientedconsistently, we assume as graphs because they are ori-ented as two-surfaces: edges can be ordered, say, clockwise.

The action of the operators in the spin-network basis

We now describe how the T operators act on the spin

network states. From Eq. 3.7, the operator T, acting ona state simply adds a loop to . Consider the graph formed by the union in M) of the graphs of and .Since we admit empty edges, we can represent over the

ribbon net R associated to . In this representation, the ac-tion ofT consists in adding the draw of over R. Usingthe expression for the Jones-Wenzl projectors in 32 p. 96,one can expand the nonantisymmetrized lines, if any, incombinations of antisymmetrized ones.

Higher order loop operators are expressed in terms of theelementary grasp operation, Eq. 3.8. The ribbon construc-tion allows us to represent the grasp operation in a simplerform. Indeed, one easily sees that Eq. 3.8 is equivalent tothe following: acting on an edge with color 1, the graspcreates two virtual trivalent vertices inside the same circle,corresponding to the intersection point one on the spin-network state and one the loop of the operator. The twovertices are joined by a virtual strip of color 2, and the over-all multiplicative factor is determined as follows. The sign ofthe tangent of in a,s is determined by the orientationof consistent with the positive terms of the loop expansionof the spin network. The equivalence between the old defi-nition of the grasp and the new one is illustrated in Fig. 3.

A straightforward computation, using Leibnitz rule,shows that acting on an edge with color p , the grasp has thevery same action, with the multiplicative factor multiplied byp . Finally, notice that the two antisymmetrized loops form avirtual spin network edge of color 2. Therefore, we canexpress the action of the grasp in the spin network basis bythe following equation:

.5.5

This simple form of the action of the loop operators on thespin-network basis is the reason that enables us to userecoupling-theory in actual calculations involving quantumgravity operators. Notice that it is the ribbon-net constructionthat allows us to open up the intersection point and rep-

resent it by means of two vertices one over and one over) and a zero length edge connecting the two vertices.These two vertices and this edge are all in the same point ofthe three-manifold M.

Higher order loop operators act similarly, as sketched inFig. 4.

VI. THE AREA OPERATOR

A surface in M is an embedding of a two-dimensionalmanifold , with coordinates u(1,2), u,v1,2, intoM. We write S:M3,ux a(). The metric and the nor-mal one form on are given by

FIG. 3. Action of the grasp.



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g Sg, g uv

xa

ux b

vg ab ; 6.1

n a1

2uvab c

x b

uxc

v. 6.2

The area of is

Ad2detg

d21

2u u

vv

g uv g u v

d2nan bEaiEib, 6.3

where we have used

u uv v

g uv

g u v

u u

v v

x a

uxb

vgab

x a

u

x b

vg a b,

u uxa

u

x a

u

1

2

u uxa

u

x a

u aac

a a cnc

a a c,

ggc c

1

2a a

cb bc gab g a b.

On the role played by surface area in the Ashtekars formu-lation of GR, see 50. We want to construct the quantum

area operator A, namely, a function of the loop represen-tation operators whose classical limit is A. Followingconventional quantum field theoretical techniques, we deal

with operator products by defining A as a limit of regu-

larized operators A that do not contain operator prod-ucts. The difficulty in the present context is to find a regu-

larization that does not break general covariance. This can beachieved by a geometrical regularization 22,10.

Following 23, we begin by constructing a classical regu-larized expression for the area, namely a one parameter fam-ily of classical functions of the loop variables A whichconverges to the area as approaches zero. Consider a smallregion of the surface , whose coordinate area goes tozero with 2. For every s in , the smoothness of the clas-

sical fields implies that Ea(s)Ea(xI)O(), where xI is

an arbitrary fixed point in . Also, U(s ,t)AB1A

BO()

for any s, tI , and a coordinate straight segment join-ing s and t. It follows that because of Eq. A.1, to zerothorder in ,

Tab st s ,tTrEa s U s,tE

b tU t,s

2Eai xIE

ibxI. 6.4

Using this, we can write

4Eai xIE

ibxI

1

2

d2n a

d2n b

Tab ,O, 6.5

where is, say, a coordinate circular loop with the twopoints and on antipodal points. Next, consider the area ofthe full surface . By the very definition of Riemann inte-gral, Eq. 6.3 can be written as

A

d2n anbEaiEib

limN0

I

2naxInbxIEai xIEibxI 6.6

where, following Riemann, we have partitioned the surface in N small surfaces I of coordinate area

2 and xI is an

arbitrary point in I. The convergence of the limit to the

integral, and its independence from the details of the con-struction, are assured by the Riemann theorem for allbounded smooth fields. Inserting Eq. 6.5 in Eq. 6.6, weobtain the desired regularized expression for the classicalarea, suitable to be promoted to a quantum loop operator

A lim0

A , 6.7

AI AI

2 , 6.8

AI2

1

2

II

d2d2n an bTab ,. 6.9

Notice that the powers of the regulator in Eq. 6.5 and6.6 combine nicely, so that appears in Eq. 6.7 only inthe integration domains.

We are now ready to define the area operator

A lim0

A , 6.10

AI

AI2 , 6.11

AI2

1

2

II

d2d2 nanbTab ,.

6.12

The meaning of the limit in Eq. 6.10 needs to be specified.The specification of the topology in which the limit is takenis an integral part of the definition of the operator. As it isusual for limits involved in the regularization of quantumfield theoretical operators, the limit cannot be taken in the

FIG. 4. Representation of the n grasp of the

Ta1 . . . an(s1 , . . . , sn) operator.



10/27

Hilbert space topology where, in general, it does not exist.The limit must be taken in a topology that remembers thetopology in which the corresponding classical limit 6.7 istaken. This is easy to do in the present context. We say thata sequence of multi loops converges to if con-verges pointwise to ; we say that a sequence of quantumstates converges to the state if for at leastone () and one . This definition extendsimmediately to general states by linearity, and defines atopology on the state space, and the corresponding operator

topology: OO iff OO , . Notice that the

above is equivalent to say that converges to ifT converges pointwise to T , which is the topologyimplicitly used in 15 to regularize the area operator.

An important consequence of the use of this topology isthe following. Let converge to . Then the graphs converge to in the topology of M. In other words,

given a neighborhood of , there exists an such that

is included in the neighborhood for all . Visu-

ally, we can imagine that the ribbon nets R merge into

the ribbon net ex

as approaches zero. In addition, therepresentations P go to P , up to equivalence. This fact

allows us to separate the study of a limit in two steps. First,we study of the graph of the limit state. In this process, therepresentations P are merged into the ribbon net R of the

limit state. Second, we can use recoupling theory on R , inorder to express the limit representation in terms of the spinnetwork basis.

We now study the action of the area operator A givenin Eq. 6.10 on a spin network state S . Namely, we com-pute S A . Let S be the set of the points i in theintersection of S and . In other words, we label by anindex i the points where the spin network graph S and the

surface intersect. Generically S is numerable, and doesnot include vertices of S . Here we disregard spin networksthat have a vertex lying on or a continuous number ofintersection points with . It was pointed out by A. Ashtekarthat spin networks with a vertex and one -or more- of itsadjacent edges lying on are eigenstates of the area witheigenvalues that are not included in the spectrum of the op-

erator computed in 23 and derived again below. Thereforethe spectrum of the area given in 23 is not complete. Thephysical relevance of these degenerate cases is unclear tous.6

For small enough , each intersection i will lie inside adistinct I surface.

7 Let us call i the surface containing

the intersection i at every fixed ), and e i the edge through

the intersection i. Notice that S AI

2 vanishes for all sur-

faces I except the ones containing intersections. Thus thesum over surfaces I reduces to a sum over intersections.

Bringing the limit inside the sum and the square root, we canwrite

S A iS

S Ai2, 6.13

Ai2 lim

0

Ai2 . 6.14

For finite , the state S Ai2 has support on the union of

the graphs of S and the graph of the loop in the argu-ment of the operator 6.12. But the last converges to a pointon S as goes to zero. Therefore,

lim0

SAi

2S . 6.15

The operator A does not affect the graph ofS . Next, wehave to compute the planar representation of SA , whichis a tangle on R S A , namely a tangle on R S . By Eq.6.13, this is given by a sum of terms, one for each iS. Consider one of these terms. By definition of the

T loop operators and of the grasp operation Sec. III, this isobtained by inserting two trivalent intersections on the spin

network edge e i inside its ribbon, connected by a new edgeof color 2. This is because the circle has converged to a

point on e i ; in turn, this point is then expanded inside theribbon as a degenerate loop following back and forward asegment connecting the two intersections. By indicating therepresentation of the spin network simply by means of itse i edge, we thus have

6.16

where we have already taken the limit inside the integral in the state enclosed in the brackets . Notice that this does not

6Note added. The complete spectrum of the area has been obtained in the meanwhile in 16, and then reobtained in 51 using the methodsdeveloped in this paper.

7The perhaps caviling issue that an intersection may fall on the boundary between two I surfaces has been raised. This eventuality,however, does not generate difficulties for the following reason. The integrals we are using are not Lebesgue integrals, because, due to thepresence of the s, regions of zero measure of the integration domain cannot be neglected nor doubly counted. Therefore in selecting thepartition of in the I surfaces one must include each boundary in one and only one of the two surfaces which are therefore partially openand partially closed. Boundary points are then normal points that fall inside one and only one integration domain.



11/27

depend on the integration variables anymore, because the loop it contains does not represent the grasped loop for a finite , butthe a ribbon expansion of the limit state. Notice also that the two integrals are independent, and equal. Thus, we can write

. 6.17

The parenthesis is easy to compute. Using Eq. 2.23, it becomes the analytic form of the intersection number between the edge

and the surface

i

d2n aae ,

i

d2nae

d ea3e,s1, 6.18

where the sign, which depends on the relative orientation ofthe loop and the surface, becomes then irrelevant because ofthe square. Thus

6.19

where we have trivially taken the limit 6.14, since there isno residual dependence on . We have now to express thetangle inside the bracket in terms of an edge of a spinnetwork state. But tangles inside ribbons satisfy recouplingtheory, and we can therefore use the formula E.8 in theAppendix, obtaining

peAi2 l0

4pe2pe ,p e,2

2pepel0

4pep e2

4pe

l04

p e

2 pe

21 pe .

The square root in Eq. 6.13 is now easy to take because the

operator Ai2 is diagonal

peAipeAi

2l04 pe2

p e

21 pe . 6.20

Inserting in the sum 6.13, and shifting from color to spinnotation, we obtain the final result

S A l02 iS

j ij i1 S , 6.21where j i is the spin of the edge crossing in i. This resultshows that the spin network states with a finite number ofintersection points with the surface and no vertices on thesurface are eigenstates of the area operator. The correspond-

ing spectrum is labeled by multiplets j(j 1 , . . . ,j n) ofpositive half integers, with arbitrary n , and given by

Ajl02

i

j ij i1 . 6.22

The spectral values of the degenerate cases in whichS includes vertices or a continuous number of points,and a discussion on the relevance of these cases, will begiven elsewhere.

VII. THE VOLUME OPERATOR

A. The volume in terms of loop variables

Consider a three-dimensional region R. The volume ofR is given by

VRR

d3xdetg

R

d3x 13!

ab ci jkEaiEb jEck, 7.1

In order to construct a regularized form of this expression,

consider the three index three hands loop variable

Tab c s, t,rTrEa s U s ,tE

b t

U t,rEc rU r,s . 7.2

Because of Eq. A2, in the limit of the loop shrinking toa point x we have

Tab c s,t,r2i jkE

aiEb jEck2ab cdetE. 7.3

Following 23, fix an arbitrary chart of M, and consider a

small cubic region RI of coordinate volume 3

. Let xI be anarbitrary but fixed point in RI . Since classical fields are

smooth we have E(s)E(xI)O() for every sRI , and

U(s,t)AB1A

BO() for any s ,tRI and straight segment

joining s and t. Consider the quantity

WI1

163!6RI

d2RI

d2RI

d2

n an bncTab c,, , 7.4

where is a triangular loop joining the points , , and. Because of Eq. 7.3, we have, to lowest order in



12/27

WI1

83!6detExI

RI

d2RI

d2RI

d2

n anbn cab c

detExI . 7.5

Thus WI is a nonlocal quantity that approximates detg(xI)

for small . Using the Riemann theorem as in the case of thearea, we can then write the volume VR of the region R asfollows. For every , we partition of R in cubes RI of

coordinate volume 3. Then

VR lim0

VR; 7.6

VRI

3WI1/2 . 7.7

B. Quantum volume operator

We have then immediately a definition of the quantumvolume operator 23

VR lim0

VR; 7.8

VRI

3WI1/2 ; 7.9

WI

1

163! 6RI

d2RI

d2RI

d2n an bn cTab c,, .

7.10

Notice the crucial cancellation of the 6 factor. We refer tothe previous section on the area operator for the discussionon the meaning of the limit and the split of the action of theoperator in the computation of the graph and the representa-tion. We will discuss the meaning of the square root later.

For alternative definitions of the volume operators, and adiscussion on the relation between these, see 24,36 and17.

Let us now begin to compute the action of this operatoron a spin network state. The three surface integrals on thesurface of the cube and the line integrals along the loopscombineas in the case of the areato give three intersec-tion numbers, which select three intersection points betweenthe spin network and the boundary of the cube. In these threepoints, which we denote as r, s , and t, the loop of theoperator grasps the spin network.

Notice that the integration domain of the three surfaceintegrals is a six-dimensional spacethe space of the pos-

sible positions of three points on the surface of a cube. Let usdenote this integration domain as D 6. The absolute value inEq. 7.10 plays a crucial role here: contributions from dif-ferent points of D 6 have to be taken in their absolute value,while contributions from the same point of D 6 have to besummed algebraically before taking the absolute value. Theposition of each hand of the operator is integrated over thesurface, and therefore each hand grasps each of the three

points r, s , and t, producing 33 distinct terms. However,because of the absolute value, a term in which two handsgrasp the same point, say r, vanishes. This happens becausethe result of the grasp is symmetric, but the operator is anti-symmetric, in the two hands as follows from the antisym-metry of the trace of three sigma matrices. Thus, only termsin which each hand grasps a distinct point give nonvanishingcontributions. For each triple of points of intersection be-tween spin network and cubes surface r, s and t, there are3! ways in which the three hands can grasp the three points.These 3! terms have alternating signs because of the anti-symmetry of the operator, but the absolute value prevents thesum from vanishing, and yields the same contribution for

each of the 3! terms.If there are only two intersection points between the

boundary of the cube and the spin network, then there arealways two hands grasping in the same point; contributionshave to be summed before taking the absolute value, andthus they cancel. Thus the sum in Eq. 7.9 reduces to a sum

over the cubes Ii whose boundary has at least three distinct

intersections with the spin network, and the surface integra-tion reduces to a sum over the triple graspings in distinctpoints. For small enough, the only cubes whose surface hasat least three intersections with the spin network are thecubes containing a vertex i of the spin network . Therefore,the sum over cubes reduces to a sum over the vertices i

SR of the spin network, contained inside R. Let usdenote by Ii the cube containing the vertex i . We then have

S VR lim0

iSV

3S WIi ,

S WIi

i l06

163!6 s,t,r S #s,t,rs ,t,r , 7.11

where s ,t, and r are three distinct intersections between thespin network and the boundary of the box, and we have

indicated by S #

s #

t#

rst r the result of the triple grasp of thethree hands operator with loop st r on S .Let us compute one of the terms above, corresponding to

a given triple of grasps, over an n-valent intersections. Firstof all, in the limit 0 the operator does not change thegraph of the quantum state, for the same reason the areaoperator does not. Thus the computation reduces to a com-binatorial computation of the action of the operator on therepresentation of the planar state, involving recouplingtheory.

Let us represent a spin network state simply by means ofthe portion of its virtual net containing the vertex on whichthe operator is acting. We have



13/27

. 7.12

where W rt s(n) is the operator that grasps the r, t, and s edge of the the n-valent vertex as follows:

7.13

Notice that we have replaced the triangular loop with verti-ces r, s, and t by three edges of color 2 joining the threepoints r, s, and t to a trivalent vertex. This can be done asfollows. First we deform the triangle over the ribbon net.Indeed, as remarked for the case of the area, the tangle abovedoes not represent a tangle extended in M, but just the ex-pansion over the ribbon net of a rooting of lines in a singlepoint of M. Second, we notice that we can antisymmetrize

the two lines that exit from the hand of an operator by usingthe binor identity, because tracing a hand with a zero lengthloop gives a vanishing quantity.

The last equality in the last equation follows from the factthat trivalent spin network form a basis see Sec. V. From

Eq. 7.12 we see that the action of WIi splits into a multi-

plication by a numerical prefactor and a recoupling partgiven by Eq. 7.13, which does not depend on the integra-tion variables. Using Eq. 6.18 we can perform the integra-tion in Eq. 7.12. This yields the intersection number be-tween the edges r, s , and t and the surface of the cube VI .The sign of the intersection number, coming from the rela-tive orientation of the loop and the surface, is irrelevant,

because of the presence of the absolute value.Because of the symmetry properties of the three-valentnode (222), the 3! terms in Eq. 7.13 are related by

W i1i2i3n

1 pW ip1ip2ip3 n , 7.14

where p i is a permutation of 123, and p is the order of thepermutation. Thus the action the volume operator on a ge-neric spin network state S is given by

VVl03

iSV r0, . . . , n3tr1, . . . , n2st1, . . . , n1

i

16W

rt s

n i ,

7.15

where n i is the valence of the ith intersection. Equations7.13 and 7.15 completely define the volume operator.There are two remaining tasks: to find the explicit expression

for the matrix iWrs t(n)

kn2 . . . k3k2

i n2 . . . i3i2 ( P n1 , . . . , P 0), which is

defined in Eq. 7.13 only implicitly, and to show that theabsolute value and the square root in Eq. 7.15 are welldefined. Below, we complete both tasks: we provide an ex-

plicit expression for i W rs t

(n)

kn2 . . . k3k2

in2 . . . i3i2 ( Pn

1, . . . , P

0), and

we prove that the argument of the absolute value is a diago-nalizable finite dimensional matrix with real eigenvalues,and the argument of the square root is a finite dimensionaldiagonalizable matrix with positive real eigenvalues.

C. Trivalent vertices

We begin studying the case n3. It is easy to see that

W012(3 )0 from the relation

7.16

In fact, by closing the generic three-valent node with itselfwe have

7.17



14/27

Thus W012(3 )

is determined by the Wigner 9J symbol theevaluation of the hexagonal net as

W0123

P 0P 1 P 2P0 P 1 P 2

P 0 P 1 P 2

2 2 2

P 0 , P 1 , P 2 . 7.18

But the hexagonal net in the case of A1) it is antisym-metric for the exchange of two columns or of two rows.Therefore the matrix W3 vanishes, and the trivalent verticesgive no contribution to the volume. We have rederived theresult that the volume of a three-valent vertex is zero, firstobtained by Loll 24.

D. Four-valent vertices

Next, we study the n4 case:

.

7.19

Using the same technique of the three-valent node we can

compute the matrix W012(4 )

ij for a four-valent node as follows:

7.20

Using the relation

7.21

we obtain

W0124

ij

P 0 P1 P 2P 0 P 1 j

P 0 P 1 i

2 2 2 Tet i j P3P 2 P 2 2

2,j ,i

j

P 0 , P 1 ,j P 2 , P3 ,j . 7.22

We now prove that the matrix iW012(4 )

ij is diagonalizable

with real eigenvalues and, as a consequence, that its absolutevalues are well defined. To this aim, let us define the notation

A ij

P0 P 1 P2P 0 P 1 j

P 0 P 1 i

2 2 2 Tet i j P3P 2 P 2 2

2,j ,i ,

7.23

M i i P 0 , P 1 ,i P 2 , P 3 ,i

, 7.24

WijM i Mj A i

j , 7.25

S iji

jM i . 7.26

The matrix S ij can be consider as a change of basis in the

space of the four-valent vertices and the matrix iW012(4 )

ij can

be rewritten as

iW0124

ij S1 i

k iWkl S l

j , 7.27

where, because of the antisymmetry properties of the 9Jsymbol under exchange of two rows and the symmetry prop-

erty of the Tet symbol,8 the matrix Wkl is antisymmetric. We

have shown that in the basis

7.28

the action, Eq. 7.19, of the operator W 012(4 ) is given by

W

012

4 ni

j

W

012

4 ijnj , 7.29

where W012(4 )

ij a real antisymmetric matrix. Moreover, from

the admissibility condition for the three-valent node of Eq.

7.21, we see that Wkl vanishes unless kl or kl2.

Thus, we have shown that the operator iW 012(4 ) may be rep-

resented by a purely imaginary antisymmetric matrix i Wkl

with nonvanishing matrix elements only for kl2. Suchmatrix is diagonalizable and has real eigenvalues.

8For a discussion of the symmetry properties of the 9J symbol

and related quantities, see for instance 52.



15/27

Furthermore, notice the following. We write the depen-dence on the coloring of the external edges explicitly;

namely we write W012(4 )

ij( P0 , P 1 , P 2 , P 3). Using Eq. 7.20, it

is easy to see that the following relations hold between the

matrices Wi1i2i3(4 )

ij( P 0 , P 1 , P 2 , P 3)

W0134

ij P 0 , P 1 , P 2 , P 3 W012

4 ij P0 , P1 , P 3 , P 2,

7.30

W0234

ij P 0 , P 1 , P2 , P3 W123

4 ij P 3 , P 2 , P 1 , P 0 ,

W1234

ij P 0 , P 1 , P2 , P3 W012

4 ij P 3 , P 2 , P 0 , P 1 .

We have shown that there exists a basis ni in which the

four operators iW i1i2i3(4 ) that define the action of the volume

on four valent vertices, are purely imaginary antisymmetric

matrices. The eigenvalues of the four operators iW i1i2i3(4 ) are

real and, if x is an eigenvalue, so is x. Therefore, the

absolute value of the matrices i W i1i2i3 (4 ) is well defined. It is

given by a non-negative i.e., having real eigenvalues equalor greater than zero antisymmetric matrix. But the sum ofnon-negative matrices is a non-negative matrix. Thereforethe sum of the the absolute values of the four matrices

iW i1i2i3(4 ) is a non-negative antisymmetric matrix as well.

Thus the volume operator is diagonalizable on the spin net-

work basis, with positive real eigenvalues, if all the verticeshave valence 3, 4. Below, we show that these results extendto vertices of arbitrary valence.

E. The case of an n vertex

We now show that there exists a basis in which all the

operators i W i1i2i3(n) ( P0 , . . . , Pn1) are represented by a

purely imaginary antisymmetric matrix. Consider Eq. 7.13.By repeated application of the recoupling theorem, Eq.7.13 can be rewritten as

7.31

we have assumed, without loss of generality, that there is no grasp on the P 0 or P n1 edge. Closing the vertex with itself andusing the relation E8 and E9, we find

W rs t n

i2 . . . in2

k2 . . . k

n2PrP tPs

k2 P t k

3

i2 P t i3

2 2 2

1k2

i22

i4

k4in2

kn2Tet P r P r P 0k2 i

2 2

Tet i3 k3 k4Ps P s 2

k2k3 k2,2, i

2 k

3,2, i

3 P0 , P r ,k

2 k

2 , P t ,k

3 k

3 , P s ,k

4

. 7.32

We now change basis in the same fashion as we did for the

four-valent vertex see Eq. 7.28. We define a new basis in

which any edge real or virtual is multiplied by i ( i col-oring of the edge and any vertex is divided by (a ,b ,c)(a, b and c the coloring of the edges adjacent to the vertex.It is then easy to see that in this new basis the matrix on Eq.

7.32 becomes real antisymmetric. Indeed, we have simplyreduced the general problem to the case of four valent verti-

ces. Now, the key result, that we shall prove in the nextsection is that, in the basis we have defined, the recouplingtheorem is a unitary transformation. A unitary transforma-tion preserves the property of a matrix of being diagonaliz-able and having real eigenvalues. It follows that the resultswe have obtained for the four-valent vertices hold in general.

We are now ready to find an explicit expression for the

recoupling matrix iW rs t(n)

kn2 . . . k3k2

in2 . . . i3i2 ( P n1 , . . . , P 0) of Eq.

7.13 for a general valence n of the vertex. Let us begin bysketching the procedure that we follow. First, the recouplingtheorem allows us to move one of the three grasps from theexternal edge, say P r , of Eq. 7.13, and bring it to a virtualvertex. We denote this operation as move 1:

7.33

Second, we can use recoupling theorem repeatedly to movethe grasp all the way to the edge P 0. We denote this opera-tion as move 2:

7.34



16/27

In this way we can bring all three grasps to the edge P0. Thefinal step is just given by recognizing that we have Tet struc-ture on the edge P 0.

Let us begin by applying move 1 to the node r. We obtain

.

7.35

Then, using move 2 we can move the ( ir ,kr,2) node to theleft of the node ( ir1 , Pr1 ,i t):

7.36

We repeat move 2 until the first node with the 2 edge iscoupled to the P 0 edge. In this way, after a finite number ofmoves 2, we have transformed the original network to

7.37

Before repeating this procedure for each of the three grasps,it is convenient to rename the colors ka of the virtual edges

as k

a and to replace the remaining i a by ka as well; this can

be done by inserting a sum over a ka multiplied by a iaka).

Repeating the sequence of moves for the two grasps overthe edges r and s , we transform the grasped vertex to thefinal form

7.38

This it is equal to the original n-valent vertex with the i areplaced by ka and multiplied by Tet k1 ,k1 ,k1 ;2,2,2 seeEq. E9. Bringing all together, we have shown that theaction of the volume operator is described by the sum 7.15extended over all vertices of the spin network, where theexplicit form for the recoupling matrix 7.13 is given by

Wrs tn

i2 . . . i n2

k2 . . . kn2 P 0 , . . . , P n1 k1 , . . . , k

n2

k1 , . . . , k

n2

P tP rP s

Tet k1 k1 k12 2 2

P0

ar1n2

i a

k

a

M

i

r

1i

r k

r

2 r P r

a1r1

k

a12

k

a

ia P a i a1

bt1

n2

kb

kbM kr1 kr kr2 t Pt

b1

t1

kb1 2 kbkb P a k

b1

cs1

n2

kc

kc M ks1 ks ks2 s P s

c1

s1

kc1 2 kckc P a k

c1

7.39and

M

i r1 i r kr

2 r P r

1, r0;

k

r

2ir1

ir1 ir kr

2 P r Pr, 0rn1;

Pn1

2 Pn11, rn1,

7.40

where i1k1P 0 and in1kn1k

n1k

n1P1. We have used the fact that for A1, a2a1.

This formula can be specialized to the case of three-vertex ( n3) and four-vertex (n4). In the case of three-vertex wehave

W3 P 0 , P 1 , P 2k1

P0 P 1 P2 P02 k11 P 2 2 P 0

k1 P 1 P 2 P 2 P 0 k1

2 P 1 P 1

Tet P 0 k1 P 02 2 2

P0

7.41and a direct computation confirms that the volume of any three-vertex is zero. For the case of four-valent vertex, we obtain theformula



17/27

W0134

ik

k1

P 0 P1 P 31 P02 k11 i P0 k1

2 P 1 P 1 P 3 2 k

i P2 P 3 k 2 P 0

k1 P 1 i

Tet P 0 k1 P 02 2 2

P0

and the other three matrix that appear in the definition of theaction of the volume operator are easily deduced from theidentities 7.30.

F. Summary of the volumes action

Finally, let us summarize the procedure for computing theeigenvalues and eigenvectors of the volume. Consider thespin-network states S with a fixed graph and a fixed color-ing of the real edges, but with arbitrary intersections. The setof these spin networks forms a finite dimensional subspaceV of the quantum state space. The subspace V is invariantunder the action of the volume operator. We denote the va-lence of the real vertex i by n

i

. Fix a trivalent decompositionof each vertex iSR. Consider all compatible coloringsof the virtual edges. For every vertex, the number of thecompatible colorings depends on the valence of the vertex,as well as on the coloring of the external edges. Let Ni be thenumber of compatible colorings of the vertex n i . The dimen-sion N of the subspace V we are considering is N iNi .Our aim is to diagonalize the volume operator in V.

We indicate a basis in V as follows. Given a vertex i withvalence n i , we have previously denoted compatible color-ings of the internal edges by ( i 2 , . . . , in i2). It is more con-

venient here to simplify the notation by introducing a singleindex Ki1,Ni , which labels all compatible internal color-ings of the vertex i.

We now recall the basic expression we have obtained, forthe volume, namely, Eq. 7.15,

VVl03

iSVVi ,

Vi r0, . . . , n3tr1, . . . , n2st1, . . . , n1

i16

W rt s

ni , 7.42

where the first sum is over the vertices and the second sum isover the triples of edges adjacent to the vertex. We have

shown that the operators iW rt s(n i) are diagonalizable matrices

with real eigenvalues. These matrices have components

Wrt s

n i

KIi

KIiLHS of Eq. 7.39 . 7.43

Since the matrices iWrt s

(n i)

KIi

KIi are diagonalizable with real ei-

genvalues, from the spectral theorem we can write them as

iW rt s

n i

rt s

n i P rt s i , 7.44

where rt s(n

i)

are real quantities and the P rt s( i) are the spec-tral projectors of the finite dimensional matrix operator

Wrt s

(n i) acting on the ith vertexs basis.

From Eq. 7.42, we have then

Vi2

r0, . . . , n i3

tr1, . . . , n i2

st1, . . . , n i1

rt s

16P

rt s

n i . 7.45

Being the sum of Hermitian non-negative matrices, Vi2 as

well is diagonalizable with real non-negative eigenvalues,

which we denote as i2 , and spectral projectors Pi:

Vi2

i

i2

Pi7.46

with i0. Therefore we have

Vii

i Pi

7.47

and the volume is given by

VVl03

iSVi

i Pi

. 7.48

Now, the projectors acting on different vertices commute

among themselves: Pi P

j P

jPi

if ij . Therefore the

eigenvectors of V are the common eigenvectors of all Vi .They are labeled by one i for every vertex i , namely by a

multi-index (1 , . . . ,p), where p is the number of ver-tices in the region. The corresponding spectral projectors

P of V are the products over the vertices of the spectral

projectors of the vertex volume operators Vi

Pi

Pi. 7.49

It is immediate to conclude that

Vl 03

P , 7.50

where the eigenvalues of the volume are the sums of theeigenvalues of the volume of each intersection:

i

i. 7.51

The problem of the determination of the spectrum of thevolume is reduced to a well defined calculation of the eigen-



18/27

values i, which depend on the valence and coloring of

adjacent vertices of the vertex i. Let us summarize the vari-ous steps of this computation. Given an arbitrary real vertexi with coloring of adjacent edges P 0 , . . . , P n i1: i deter-

mine the set of the possible colorings of its virtual edges, andlabel them by an index Ki ; ii using Eq. 7.39 compute the

matrix elements Wrt s

(n i)

Ki

Ki iii for each of this matrices, com-

pute its spectral decomposition, i.e., the eigenvalues rt s(n i)

and the spectral projectors P rt s

(n i) iv compute the matrix

Vi from Eq. 7.45; v compute the eigenvalues of the ma-

trix Vi . The square root of these give the is. All these

steps can be fully performed using an algebraic manipulationprogram such as MATHEMATICA. We have written a MATH-EMATICA program that performs these calculations, and wewill give free access to this program on line. In Appendix Fwe give the values of the quantities i( P0 , . . . , P ni1) for

some four-valent and five-valent vertex, computed using thisprogram.

G. Complex Ashtekar connection

Before closing this section, let us discuss the modifica-tions that are necessary in order to use the complex Ashtekarconnection instead of the real one we have used here. On thissubject, see also 41. The difference is simply the appear-ance of a factor i in the commutator between the connectionand the triad. This yields to an extra i in the factor associatedto each grasp. This additional imaginary factor destroys thereality of the eigenvalues of area and volume, which is amain result here. Probably this should be taken as an indica-tion that spin networks constructed from the propagator ofthe complex Ashtekar connection are not physical states. We

can illustrate this by means of an analogy. Imagine that westudy the eigenvalue equation for the momentum operatori /x in the quantum mechanics of a single particle. For-mally, the functions (x)expkx solve the eigenvalueequation for any real k. However, the corresponding eigen-values are imaginary an indication that these states are notphysical. Indeed, they are outside the relevant Hilbert space.The physical eigenstates of the momentum are of the form(x)expikx, with an i, and these are correct generalizedphysical states. Something similar happens here. In fact, onecan check that if we insert an i in the exponent of the ho-lonomies, namely, if we replace Eqs. 2.11 and 2.12 by

d

dU,0 i

da

dA aU,0 0 7.52

and

U,0Pexpi0

d aA a 7.53

where A ai is now the complex Ashtekar connection then the

eigenvalues of area and volume result to be real. Using Eq.7.53 as the definition of the holonomy implies that the spin

network states correspond in the connection representation

to combination of parallel propagators of i times the Ash-

tekar connection. These seem therefore to be the correct

physical states related to real geometries. However, this strat-

egy explored in a previous version of this paper is notviable, at least in this form. The reason is that Eq. 7.53 isnot invariant under the internal gauge transformations gener-

ated by the Gauss constraint. Perhaps this difficulty can be

circumvented by exploiting the complexity of the group and

the nontriviality of the reality conditions, but for the moment

we have not been able to find a construction viable for theRiemannian Ashtekar connection. We leave this problem tofuture investigations.

VIII. THE SCALAR PRODUCT

The results above allow us to introduce a scalar product inthe loop representation. The original definition of the looprepresentation of quantum general relativity left the problemof fixing the scalar product undetermined: the scalar product

had to be determined by requiring quantum observables to beHermitian 3. The problem was complicated by the fact thatthe loop basis is overcomplete. Later, the introduction ofthe nonovercomplete spin network basis, and the realizationthat spin network states with suitable bases chosen on thehigh-valent vertices are eigenstates of the geometry, lead tothe natural suggestion that spin network states ought to beorthogonal. For no reason, however, these states ought to beorthonormal; namely the norm of the spin network statesremained undetermined. The methods introduced in this pa-per allow us to complete the process, suggest a norm for thespin network states, and thus yield a complete definition fora scalar product . Here, we define a scalar product , and

motivate the choice. We have no compelling argument forthe uniqueness of this scalar product, but we will show that itsatisfies all consistency requirements so far considered.Therefore, it is reasonable to take it as a first ansatz.

Let us begin by considering an n-valent vertex. This canbe arbitrarily expanded in trivalent vertices. Leti1 , . . . , i n3 be the colors of the internal edges, and let usrepresent by i 1 , . . . , in3 the n-valent vertex expanded intrivalent vertices colored i1 , . . . , i n3. We would like to de-termine an orthogonal basis from the quantitiesi1 , . . . , in3. We have two highly nontrivial requirements.First, that this works independently from the way then-valent vertex is expanded in trivalent ones. Second, thatthe volume be Hermitian in this basis. Rather remarkably, webelieve, both requirements can be satisfied.

Let us begin by considering a four-valent vertex, for sim-plicity. There are two ways in which we can expand it intrivalent vertices. Thus we have two distinct bases i andi for the four-valent vertices. If we wanted both of them tobe orthonormal, the transformation between the two had tobe given by a unitary matrix. Now, the transformation matrixbetween the two bases is provided by the recoupling theo-rem. The matrix is given by a six-J symbol, seen as a matrixin its two rightmost entries. It is easy to see that this matrixis not unitary. However, we now show that we can rescalethe length of the basis vectors i in such a way that thetransformation matrix becomes unitary indeed, orthogonal.



19/27

Indeed, let

,

.

8.1

8.2

In this basis, the recoupling theorem becomes

. 8.3

We now prove that the matrix U(a ,b ,c ,d)ji is real orthogo-

nal. The inverse transformation matrix from the ni basis tothe nj basis is given by the same expression 8.3, with areordering of the external edges colorings: i.e.,

nkk

U d,a ,b ,c iknk . 8.4

Therefore we have the relation

i

U a ,b ,c ,dji U d,a ,b ,c i

kj

k . 8.5

From direct inspection of Eq. 8.3 it is easy to see that

U(a,b,c,d) kiU(d,a ,b,c) i

k . As an immediate consequenceof Eq. 8.5 we have orthogonality. Looking at Eqs. E2 andE4 we can easily compute the sign of the argument of thesquare root, which is

sgn

1

i

1

j

1 abj /2 cdj /2 adi /2 bci/2

1 abcd1. 8.6

We have thus shown that there exists a basis in which therecoupling theorem yields a unitary transformation. Forhigher valence vertices, the transformation from one trivalentexpansion to another can be obtained by a repeated applica-tions of the recoupling theorem transformation, and thereforeby a product of orthogonal matrices. Thus the argumentabove extends immediately to higher valence.

Now, the normalization we have found is exactly the onein which the volume operator is represented by a real anti-

symmetric matrix, as shown by Eqs. 7.23 and 7.27.Therefore we have found a basis that satisfies all our require-ments.

We thus define the normalized spin-network states by thefollowing normalization: given an arbitrary spin-networkstate S , we label with an index iV all the three-valentvertices of the expanded state virtual and real and with aindex eE all its edges virtual and real. We denote the

color of the edge e by with p e and the color of the threeedges adjacent to the vertex i by a i , b i , and c i . We definethe normalized spin network state SN by

S NiV

eE

pe

a i ,b i ,c iS 8.7

and we define a scalar product on V by requiring that thesestates are orthonormal. We have immediately from the dis-cussion above that the definition does not depend on thetrivalent expansion chosen, and that the volume and areaoperators are symmetric with respect to this scalar product.

We think that the scalar product defined in this way is

precisely the one defined on the loop representation by theloop transform 3,4 of the Ashtekar-Lewandowski measure53, namely the conventional Haar measure lattice gaugetheory scalar product for each graph. The precise relation isdiscussed by Reisenberger 33 and in 18. In turn, we ex-pect that the norm derived from the scalar product we havedefined is equivalent to the evaluation of the Kauffmanbracket of the state, and to the trace of the Temperley-Liebalgebra, discussed in Appendix B.

IX. CONCLUSIONS AND FUTURE DIRECTIONS

We have reviewed the kinematics of the loop representa-

tion of quantum gravity, and presented a number of results.We have modified the definition of the theory by inserted aminus sign in the definition of the loop observables. Withthis convention, the spinor identity is transformed into thebinor identity, allowing immediately a local graphical calcu-lus for the grasping operation and the use of recouplingtheory. We have shown that the loop states obey the axiomsof recoupling theory, and the corresponding graphical for-malism provides a powerful tool for computing the action ofgeometrical operators. We have discussed in detail the wayin which recoupling theory can be used in this context.

Using recoupling theory, we have rederived known resultson the eigenstates of the area, and the volume of trivalentand four-valent vertices. We have given a general expressionfor the volume of higher valence vertices. We have proventhat the square root in the volume operator is well defined,because the relevant operator is Hermitian. We have defineda scalar product by a suitable normalization of the trivalentspin networks. We have shown that that the scalar product iswell defined and independent from the trivalent expansionchosen, and that the volume is symmetric with respect to thisscalar product.

Notice that the area and volume operators A and V do notcorrespond to physical observables: they are not gauge in-variant and do not commute with GRs constraints. The areasand volumes that we routinely measure are associated to spa-tial regions determined by matter. Indeed, the area and vol-



20/27

ume of regions determined by physical matter are repre-

sented on the phase space of the coupled gravity-matter

theory by observables which are gauge invariant see forinstance 54. However, it was suggested in Ref. 25 that itis reasonable to expect that these physical areas and volumes

of spatial regions determined by matter be still expressed

by operators unitary equivalent to A and V. See Refs. 54

and25

for the details of the argument. If this suggestion iscorrect, the spectra computed here can be taken as physical

predictions on short scale geometry, following from the loop

representation of quantum gravity 25. These predictions aretestable in principle, and could perhaps lead to indirect ob-servable consequences.

We consider the following open problems particularly im-portant for the development of the theory.

We have not explored the degenerate cases in the actionof the area operator but see 16,51.

We believe that the formalism is now well established fora precise discussion of the Hamiltonian and for computingtransition amplitudes 26,27.

Can a weave 22 be found for which not just the area butthe volume as well approximates smooth geometries? Can aweave related to a four-dimensional geometry 55 be con-structed?

A way of implementing the Lorentzian reality conditionsis, to our knowledge, still lacking for an attempt to addressthis problem, see 39.

Under the optimistic assumption that the above technicalproblems could

roberto de pietri and carlo rovelli- geometry eigenvalues and the scalar product from recoupling...

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