Mathematical Foundations of Quantum Statistical Mechanics

MATHEMATICAL PHYSICS STUDIES

Series Editor:

M. FLATO, Universite de Bourgogne. Dijon. France

VOLUME 17

Mathematical Foundations of Quantum Statistical Mechanics Continuous Systems

by

D. Ya. Petrina Institute 0/ Mathematics, Ukrainian Academy 0/ Sciences, Kiev, Ukraine

SPRINGER SClENCE+ BUSINESS MED~ B.V.

A C.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN 978-94-010-4083-9 ISBN 978-94-011-0185-1 (eBook) DOI 10.1007/978-94-011-0185-1

The manuscript was translated from the Russian by P.V. Malyshev and D.V. Malyshev

Printed on acid-free paper

All Rights Reserved © 1995 Springer Sdence+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1995 Softcover reprint ofthe hardcover 1st edition 1995 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.

conTEnTS

Introduction xiii

CHAPTER 1. EVOLUTION OF STATES OF QUANTUM SYSTEMS OF FINITELY MANY PARTICLES 1

1. Principal Concepts of Quantum Mechanics 1

1.1. SchrOdinger Equation and the Evolution of States of Finitely Many Particles 1

1.2. Density Matrix. Equation for Density Matrix 3 1.3. Algebra of Observables. States on This Algebra 9 1.4. SchrOdinger and Heisenberg Representations 11 1.5. Bose-Einstein and Fermi-Dirac Statistics 13

2. Evolution of States of Quantum Systems with Arbitrarily Many Particles 14

2.1. The Fock Space 14 2.2. Evolution of States for the Systems with Random Number of Particles 17

2.3. Second Quantization 21 2.4. SchrMinger Equation in the Case of Second Quantization 24

3. Evolution of States in the Heisenberg Representation and in the Interaction Representation

3.1. Heisenberg Equations 3.2. Interaction Representation 3.3. Evolution Operator 3.4. Second Quantization and the Heisenberg Representation

in the Momentum Space 3.5. The Frohlich Hamiltonian

Mathematical Supplement I

1.1. Selfadjoint Operators 1.2. Kato's Criterion of Selfadjointness 1.3. Representation of Operators in Terms of Their Kernels 1.4. Solution of the SchrOdinger Equation 1.5. On the Convergence of Series (3.33) for the Evolution Operator

References

v

28

28 30 32

36 40

43

43 45 52 53 53

54

vi Contents

CHAPTER 2. EVOLUTION OF STATES OF INFINITE QUANTUM SYSTEMS 57

4. Bogolyubov Equations for Statistical Operators 57

4.1. Sequence of Statistical Operators 57 4.2. Sequence of Statistical Operators in the Grand Canonical Ensemble 60 4.3. Equations for Statistical Operators 63 4.4. Statistical Operators and Bogolyubov Equations in Terms

of the Second Quantization Operators 68

5. Solution of the Bogolyubov Equations

5.1. Statement of the Problem 5.2. Group of Evolution Operators

5.3. Infinitesimal Generator of the Group V A (t)

5.4. Feynman Integral

6. Gibbs Distributions

6.1. Stationary Solutions of Bogolyubov Equations and Equilibrium

States 6.2. Representation of the Equilibrium Statistical Operators

by the Wiener Integral. The Feynman-Kac Formula for

the System of Particles in the Entire Space A = IR 3

6.3. Feynman-Kac Formula for the System of Particles in a Bounded Region A

6.4. Representation of the S-Particle Operators in Terms of Wiener Integrals

Mathematical Supplement II

11.1. Some Properties of Nuclear Operators

11.2. Principle of Uniform Boundedness 11.3. Trotter Formula

Mathematical Supplement III

m.1 .. Necessary Information on Wiener Integrals 111.2. Justification of the Feynman-Kac Formula for the

Hamiltonian Given in the Entire Space IR 3N

111.3. Feynman-Kac Formula for the Hamiltonian Given

in a Bounded Region A

References

72

72 74 78 81

85

85

87

93

97

99

99 104 104

108

108

111

113

118

Contents

CHAPTER 3. THERMODYNAMIC LIMIT

7. Thermodynamic Limit for Statistical Operators

7.1. Kirkwood-Salsburg Equations 7.2. Solution of the Kirkwood-Salsburg Equations

7.3. Justification of the Thermodynamic Limit for the Functions pA 7 A. Justification of the Thermodynamic Limit for the Statistical

Operators FA

8. Statistical Operators in the Case of Quantum Statistics

8.1. Statement of the Problem

8.2. Representation of F/ «( x)s; (y)J in Terms of Composite

Trajectories 8.3. A Relation from the Theory of Permutation Groups

804. Integral Equations for the Functionals pA((ro)s, ().t)s)

8.5. Solutions of Integral Equations 8.6. Thermodynamic Limit

9. Bogolyubov's Principle of Weakening of Correlations

9.1. Algebraic Approach and Formulation of the Principle of Weakening

vii

123

123

123 127

130

133

136

136

137 139

143 145 149

153

of Correlations 153

9.2. Mapping of the Space E onto the Space of Formal Series 156 9.3. Kirkwood-SalsburgEquationsfor <Px 160

904. Convergence of the Series for pA(X) and FA ((x)m; (yh) 163

9.5. Proof of the Principle of Weakening of Correlations

Mathematical Supplement IV

IV.1. Estimates of the Wiener Measure for Some Sets of Trajectories IV.2. Estimates of the Conditional Wiener Measure for Some Sets

of Trajectories IV.3. Estimation of the Variable Co, ~(J3)

IVA. Estimation of the Variable ~= Ilb.~ L...j=l J J

References

CHAPTER 4. MATHEMATICAL PROBLEMS IN THE THEORY OF SUPERCONDUCTIVITY

10. Frohlich Model

10.1. Frohlich Hamiltonian

165

168

169

173 176

177

178

181

181

181

viii Contents

10.2. Canonical Transformations of the Operators \jIk,+, \jI k,+' \jIk.-,

and \jIk.- 182

10.3. State of Vacuum for the Operators ako' akO and a;;!, ak! 183 10.4. Frohlich Hamiltonian in Terms of the Operators of Creation and

Annihilation of Quasipartic1es 186 10.5. Determination of the Eigenvalues of Hamiltonian by

Perturbation Theory 187 10.6. Diagram Technique 190 10.7. Expression for the Energy in Terms of the Contributions of Diagrams 194

11. Bogolyubov's Compensation Principle for "Dangerous" Diagrams. Compensation Equations 196

11.1. "Dangerous" Denominators 11.2. Compensation Equation 11.3. Analysis of the Compensation Equation 11.4. Energy of the Ground State and Excitations 11.5. Energy of One-Particle Excitations

12. Bardeen-Cooper-SchrietTer (BCS) Hamiltonian

12.1. Hamiltonian of the Subsystem of Electrons and the Energy of the Ground State

12.2. Equation for c(k) 12.3. Excited States 12.4. Solving of the BCS Model by the Method of Approximating

Hamiltonian 12.5. Equations for a Gap at Nonzero Temperatures

13. Microscopic Theory of Superfluidity

13.1. Free Bose Gas 13.2. Bose-Einstein Condensation 13.3. Energy of Elementary Excitations 13.4. Model Bogolyubov Hamiltonian 13.5. Heisenberg Equations 13.6. Distribution Function

Mathematical Supplement V

V.l. Free Bose Gas in the Canonical Ensemble V.2. Bose Gas under General Boundary Conditions V.3. Solution of Equation for the Gap

References

196 198 200 202 204

208

208

210 212

213 216

218

218 222 226 227 233 236

238

238 242 244

249

Contents

CHAPTER 5. GREEN'S FUNCTIONS

14. Green's Functions. Equations for Green's Functions

14.1. Green's Functions at Temperature Zero 14.2. Temperature Green's Functions 14.3. Representation of Temperature Green's Functions in Terms

of the Evolution Operator 14.4. Integral Equations for Green's Functions 14.5. Representation of Euclidean Green's Functions in Terms

of the Wiener Integral

15. Investigation of the Equations for Green's Functions in the Theory of Superconductivity and Superfluidity

IX

253

253

253 256

258 261

264

268

15.1. Equations for Green's Functions in the Theory of Superconductivity 268 15.2. Structure of Solutions of the Equations for Green's Functions

in the Model of Superconductivity 270 15.3. Thermodynamic Equivalence of the Model and Approximating

Hamiltonians 273 15.4. Equations for Green's Functions in the Model of Superfluidity 278 15.5. Creation and Annihilation Operators with Isolated Condensate 284

16. Green's Functions in the Thermodynamic Limit

16.1. Statement of the Problem 16.2. Holomorphy of Green's Functions 16.3. Green's Functions in the Thermodynamic Limit

16.4. Existence of Limiting Green's Functions in l5

References

CHAPTER 6. EXACTLY SOL V ABLE MODELS

17. Description of the Hamiltonians of Model Systems

17.1. BCS (Bardeen -Cooper-Schrieffer) Hamiltonian in the Theory

290

290 292 297

301

305

307

307

of Superconductivity 307 17.2. Bogolyubov Hamiltonian in the Theory of Superfluidity 309 17.3. Huang- Yang-Luttinger (HYL) Model 312 17.4. Frohlich - Peierls Model 313

18. Functional Spaces of Translation·Invariant Functions 315

18.1. Hilbert space of Translation-Invariant Functions 315

x Contents

18.2. Spaces of Summable Translation Invariant Functions 322

19. Model Hamiltonians in the Spaces of Translation Invariant Functions 324

19.1. Hamiltonian with Pair Interaction 324

19.2. Structure of the Operator HN in h~ 326

19.3. Spectral Properties of the Operator HN 332

20. Model BCS Hamiltonian in the Space h T. Equivalence of General and Model Hamiltonians in the Space of Pairs 340

20.1. BCS Hamiltonian in h~

20.2. General Hamiltonian in h~ 20.3. General and model Hamiltonians in the Case of Bounded A 20.4. On the Coincidence of the General and Model Hamiltonians in the

340

344

349

Theory of Superfluidity in the Subspace of Pairs and Condensate 352

21. Equations for Green's Functions and Their Solutions 357

21.1. Equations for Green's Functions in the BCS Model 21.2. Equations for Green's Functions in the Model of Superfluidity 21.3. Equations for Green's Functions in the HYL Model 21.4. Equations for Green's Functions in the Frohlich-Peierls Model

358 363 371 378

Mathematical Supplement VI 383

VI.1. Representation of the Commutation Relations in the HYL Model 383 VI.2. Representation of the Anticommutation Relations in the BCS Model

of Superconductivity 388

References 393

CHAPTER 7. QUASIA VERAGES. THEOREM ON SINGULARITIES OF GREEN'S FUNCTIONS OF 11 q2 -TYPE 401

22. Quasiaverages 401

22.1. Conservation Laws and Selection Principles 401 22.2. Quasiaverages 406 22.3. States of the Pairs of Particles 411

23. Green's Functions and Their Spectral Representations 415

23.1. Advanced, Retarded, and Ordinary Green's Functions 415 23.2. Hilbert Space. Schwarz-Buniakowski Inequality 418

Contents xi

23.3. A Relation for the Variation of Averages 421 23.4. Symmetry Properties of Green's Functions 423

24. Theorem on Singularities of Green's Functions of lIq2 -Type 426

24.1. Gradient Transfonnations 426 24.2. Variations of Averages 428 24.3. Theorem on Singularities of Green's Functions of 1/ q2 -Type 431 24.4. Theorem on 1 / q2 Singularities for Fermi Systems 433

24.5. Estimates of the Number of Pairs with Momentum q"* 0 and ~"* 0 437

References 439

Subject Index 441

InTRODUCTion

This monograph is devoted to quantum statistical mechanics. It can be regarded as a continuation of the book "Mathematical Foundations of Classical Statistical Mechanics. Continuous Systems" (Gordon & Breach SP, 1989) written together with my colleagues V. I. Gerasimenko and P. V. Malyshev. Taken together, these books give a complete pre­sentation of the statistical mechanics of continuous systems, both quantum and classical, from the common point of view.

Both books have similar contents. They deal with the investigation of states of in­

finite systems, which are described by infinite sequences of statistical operators (reduced density matrices) or Green's functions in the quantum case and by infinite sequences of distribution functions in the classical case. The equations of state and their solutions are the main object of investigation in these books. For infinite systems, the solutions of the equations of state are constructed by using the thermodynamic limit procedure, accord­ing to which we first find a solution for a system of finitely many particles and then let the number of particles and the volume of a region tend to infinity keeping the density of particles constant.

However, the style of presentation in these books is quite different. The first book is standard for classical mathematical physics - the results are given

in the form of lemmas and theorems and their proofs. Clearly, this form of presentation is only possible when the problems discussed are investigated mathematically rigorously and completely. This works well in the case of classical statistical mechanics: the prob­lems discussed in the first book were studied thoroughly and it was possible to formulate them in the form of theorems and their proofs. Certainly, classical statistical mechanics also has many unsolved problems, e.g., the theory of phase transitions, the problem of deriving kinetic and transport equations, etc. However, we did not touch these problems in our book and, thus, avoided a discussion concerning the problems that are not yet in­vestigated rigorously.

The situation is quite different in the case of quantum statistical mechanics, at least in those branches of it that are considered in this monograph. From the viewpoint of rigor­

ous mathematics, our book is somewhat mosaic. This means that in some sections, the reader will find a mathematically rigorous presentation, while the others are written on the physical level of strictness. This was the main reason for choosing another style of presentation, mainly without lemmas and theorems. The second reason was the author's wish to make this book more intimate and readable. The idea was not to shock the read­er with dry proofs, without motivating the statements of the problems and the methods

Xlll

XIV Introduction

for solving them. On the contrary, we try to provide the reader with a deep understand­ing of the internal structure of quantum statistical mechanics, its ideas and problems. Private discussions with experts in theoretical physics also convinced the author that this manner of presentation is most desirable for the monographs in modem mathematical physics. The monographs of Faddeev and his colleagues served as an important addi­tional argument for the final decision concerning the choice of this style (L. A. Takhtad­zhyan and L. D. Faddeev, "Hamiltonian Approach to Soliton Theory", Nauka, Moscow (1986), S. P. Merkur'ev and L. D. Faddeev, "Quantum Scattering Theory for Systems of Several Particles", Nauka, Moscow (1985); see also V. E. Zakharov, S. V. Manakov, S. P. Novikov, and L. P. Pitaevskii, "Soliton Theory. Inverse Scattering Problem", Nau­ka, Moscow (1980)).

On the other hand, the mathematical methods used in this book are not always avail­able for experts in theoretical and mathematical physics. This is why, we find it reason­able to give a more or less complete description of these methods in the mathematical supplements. In these supplements, we use the language of lemmas and theorems.

Naturally, the choice of material depends on the author's taste and scientific interests; more precisely, the author presents the facts he knows well, and one should not reproach him with the absence of certain branches of quantum statistical mechanics. Taken as a whole, this book may be regarded as an attempt to give a landscape of quantum statis­tical mechanics, where some places are drawn quite carefully (mathematically rigorous­ly) and the others are seized in their evolution and drawn, say, with dotted lines.

After these general remarks, we describe the contents of the book. In Chapter 1, which is of rather introductory nature, we present general facts from

quantum mechanics, namely, we consider the Schrodinger equation, the Heisenberg equation, the equations for the density matrix, the second quantization method, and the representations of canonical commutation and anticommutation relations in the Fock spaces. The mathematical supplement contains the foundations of the theory of selfad­joint extensions of symmetric operators and the Kato criterion of essential selfadjointness of the SchrOdinger operator.

In Chapter 2, we define the states of finite statistical systems as sequences of statisti­cal operators (or reduced density matrices) satisfying the Bogolyubov equations. These equations are studied in the Banach space of sequences of trace-class operators. We con­struct a strongly continuous group of bounded operators whose infinitesimal generator coincides with the operator on the right-hand side of the Bogolyubov equations. Thus, we can consider these equations as an abstract evolutionary equation with the infinitesi­mal generator of the group on its right-hand side. On this way, we establish the exis­tence of a solution and its uniqueness. The solutions belonging to the Banach space of sequences of trace-class operators describe the states of finite systems. To describe a state of an infinite system, one should consider a space broader than the space of sequen­ces of trace-class operators. However, this problem is now solved only for equilibrium states. An equilibrium state is defined as a certain stationary solution of the Bogolyubov equations, namely, as a Gibbs distribution. In the mathematical supplement, we investi­gate the properties of the trace-class operators. The Trotter formula is justified and the functional Feynman and Wiener integrals are defined.

Introduction xv

In Chapter 3, we justify the thermodynamic limit procedure for equilibrium states. For this purpose, a sequence of equilibrium statistical operators describing a finite sys­tem is represented in terms of the Wiener integrals of certain functionals defined on the trajectories. It is shown that a sequence of these functionals satisfies an infinite system of linear integral Kirkwood - Salsburg equations and possesses a unique solution in the Banach space of sequences of bounded functionals at low densities and under the corres­ponding restrictions imposed on a potential. Under these conditions, we prove the exis­tence of the thermodynamic limit for the solutions of the Kirkwood - Salsburg equations. All these results are established for the Boltzmann, Bose-Einstein, and Fermi-Dirac stat­istics. The validity of the clustering principle is proved. It is shown that statistical oper­ators exist for arbitrary thermodynamic parameters; however, in this case, they are not, in general, unique. In the mathematical supplement, we present necessary information on the properties of Wiener integrals.

Chapter 4 deals with the microscopic theory of superconductivity and superfluidity. First, following Bogolyubov, we present the theory of superconductivity based on the Frohlich Hamiltonian that describes interaction between electrons and phonons. Bogo­lyubov proposed to compute the eigenvalue of the Hamiltonian corresponding to the ground state by using perturbation theory. The ground state is sought in the form of a state with randomly many pairs of electrons with opposite momenta and spins. The un­known parameters appearing in the definition of ground state are found from the prin­ciple of compensation of dangerous diagrams. By using perturbation theory, we can also find the eigenvalues of the Hamiltonian corresponding to excited states. We deduce the gap equation and show that the eigenvalues of the ground state and excited states are se­parated by a gap. Up to now, there is no rigorous justification of the Bogolyubov theory because the Hamiltonian is unbounded and this causes extremely difficult problems. Even a rigorous statement of this problem is absent.

A separate section is devoted to the Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity based on the model reduced Hamiltonian with interaction only be­tween electrons with opposite momenta and spins. This model is investigated rigorously in Chapters 5 and 6.

In Chapter 4, we also present the microscopic theory of superfluidity based on the model Bogolyubov Hamiltonian, find the spectrum of elementary excitations, and give the rigorous theory of Bose condensation. The mathematical justification of Bogolyu­bov's theory is presented in Chapters 5 and 6.

In the mathematical supplement, we prove the existence of a solution of the nonlinear integral gap equation and discuss the problem of equivalence of canonical and grand canonical ensembles in the thermodynamic limit.

In Chapter 5, we derive equations for ordinary and temperature Green's functions and show that the temperature Green's functions exist in the thermodynamic limit. By using these results and following (mainly) Ruelle's papers, we prove that ordinary Green's functions exist in the thermodynamic limit.

We investigate the equations for Green's functions in the model of superconductivity and show that systems with certain model Hamiltonians are thermodynamically equival­ent to the systems with approximating Hamiltonians. We also consider a system with the

xvi Introduction

general Hamiltonian that describes pairwise interaction of Bose particles and show that it is thermodynamically equivalent to a system with an approximating Hamiltonian in which the operators of creation and annihilation with momentum zero are replaced by

C-numbers. In Chapter 6, we investigate exactly solvable quantum statistical models, namely, the

BCS model, the Bogolyubov model of superfluidity, the Huang-Yang-Luttinger model, and the Peierls-Frohlich model. It is shown that all these models are exactly solvable because they are thermodynamically equivalent to the models with approximating Ham­iltonians reducible to the diagonal form. The Hamiltonians and equations for Green's functions are made meaningful in the functional spaces of translation invariant functions. We also study the general Hamiltonian with pairwise interaction in the Hilbert space of translation invariant functions. This Hamiltonian is not symmetric in the indicated space but its spectrum is real and consists of the union of spectra of the Hamiltonians of clus­ters. It is shown that the model Hamiltonians in the theories of superconductivity and superfluidity coincide (in a certain sense) with the general Hamiltonians in the subspaces of pairs or pairs and condensate, respectively. This implies the coincidence of the spec­tra of the general and model Hamiltonians on these subspaces. In the mathematical sup­plement, we construct the representations of commutation relations for the free Bose gas and the Huang-Yang-Luttinger model as well as the representations of the anticommu­tation relations for the BCS model.

In Chapter 7, we give a detailed proof of the famous Bogolyubov's theorem on sin­

gularities of Green's functions of the 1 / q2 type. For this purpose, we introduce a con­cept of quasiaverages and establish spectral representations for various Green's func­tions.

We also present several examples of application of this important theorem. At the end of each chapter, we give a short list of references and bibliographical

notes. The author had absolutely no idea of composing any comprehensive bibliography covering all the fields of modem quantum statistical mechanics. We only cite books and papers that are used in our presentation.