Texts and Monographs in Physics W. BeiglbOck M. Goldhaber E. H. Lieb W. Thirring Series Editors Ola Bratteli Derek W. Robinson Operator Algebras and Quantum Statistical Mechanics II Equilibrium States Models in Quantum Statistical Mechanics I Springer Science+ Business Media, LLC Ola Bratteli Derek W. Robinson Institutt for Matematikk School of Mathematics Norges Tekniske HI'Jgskole University of New South Wales Universitetet 1 Trondheim P.O. Box 1 N-7034 Trondheim Kensington, NSW 2033 Norway Australia Editors: Wolf BeiglbOck Maurice Goldhaber Institut fUr Angewandte Mathematik Department of Physics Universitiit HeideIberg Brookhaven National Laboratory Im Neuenheimer Feld 5 Associated Universities, Inc. D-6900 Heidelberg 1 Upton, NY 11973 Federal Republic of Germany USA Elliott H. Lieb Walter Thirring Department of Physics Institut fUr Theoretische Physik Joseph Henry Laboratories der Universitiit Wien Princeton University Boltzmanngasse 5 P.O. Box 708 A-1090Wien Princeton, NJ 08540 Austria USA ISBN 978-3-662-09091-6 ISBN 978-3-662-09089-3 (eBook) DOI 10.1007/978-3-662-09089-3 Library of Congress Cataloging in Publication Data Bratteli, Ola. Operator algebras and quantum statistical mechanics. (Texts and monographs in physics) Bibliography: p. Includes index. QA326.B74 512'.55 78-27159 All rights reserved. No part of this book may be translated or reproduced in any form without written permission from Springer Science+Business Media, LLC. © 1981 by Springer Science+Business Media New York Originally published by Springer-Verlag New York Inc. in 1981 Softcover reprint ofthe hardcover Ist edition 1981 9 8 7 6 5 4 3 2 1 To Trygve Bratteli, Samuel Robinson, and Harold Ross Contents Volume II States in Quantum Statistical Mechanics 5.1. Introduction 3 5.2. Continuous Quantum Systems. I 6 5.2.1. The CAR and CCR Relations 6 5.2.2. The CAR and CCR Algebras 15 5.2.3. States and Representations 24 5.2.4. The Ideal Fermi Gas 46 5.2.5. The Ideal Bose Gas 58 5.3. KMS States 77 5.3.1. The KMS Condition 77 5.3.2. The Set of KMS States 116 5.3.3. The Set of Ground States 133 5.4. Stability and Equilibrium 147 5.4.1. Stability of KMS States 147 5.4.2. Stability and the KMS Condition 180 vii viii Contents Volume II 5.4.3. Gauge Groups and the Chemical Potential 203 5.4.4. Passive Systems 217 Notes and Remarks 223 Models of Quantum Statistical Mechanics 239 6.1. Introduction 241 6.2. Quantum Spin Systems 243 6.2.1. Kinematical and Dynamical Descriptions 243 6.2.2. The Gibbs Condition for Equilibrium 263 6~2.3. The Maximum Entropy Principle 269 6.2.4. Translationally Invariant States 289 6.2.5. Uniqueness of KMS States 307 6.2.6. Nonuniqueness ofKMS States 319 6.2.7. Ground States 334 6.3. Continuous Quantum Systems. II 348 6.3.1. The Local Hamiltonians 350 6.3.2. The Wiener Integral 361 6.3.3. The Thermodynamic Limit. I. The Reduced Density Matrices 376 6.3.4. The Thermodynamic Limit. II. States and Green's Functions 391 6.4. Conclusion 417 Notes and Remarks 419 References 453 Books and Monograpbs 455 Articles 457 List of Symbols 471 Subject Index 483 Corrigenda to Volume I 503 Contents Volume I Introduction Notes and Remarks 16 C*-Aigebras and von Neumann Algebras 17 2.1. C*-Algebras 19 2.1.1. Basic Definitions and Structure 19 2.2. Functional and Spectral Analysis 25 2.2.1. Resolvents, Spectra, and Spectral Radius 25 2.2.2. Positive Elements 32 2.2.3. Approximate Identities and Quotient Algebras 39 2.3. Representations and States 42 2.3.1. Representations 42 2.3.2. States 48 2.3.3. Construction of Representations 54 2.3.4. Existence of Representations 58 2.3.5. Commutative C*-Algebras 61 IX x Contents Volume I 2.4. von Neumann Algebras 65 2.4.1. Topologies on .!if(t;) 65 2.4.2. Definition and Elementary Properties ofvon Neumann Algebras 71 2.4.3. Normal States and the Predual 75 2.4.4. Quasi-Equivalence of Representations 79 2.5. Tomita-Takesaki Modulu Theory and Staadanl Forms of von Neumann Algebras 83 2.5.1. a-Finite von Neumann Algebras 84 2.5.2. The Modular Group 86 2.5.3. Integration and Analytic Elements for One-Parameter Groups of lsometries on Banach Spaces 97 2.5.4. Self-Dual Cones and Standard Forms 102 2.6. Quasi-Local Algebras 118 2.6.1. Cluster Properties 118 2.6.2. Topological Properties 129 2.6.3. Algebraic Properties 133 2.7. MisceUaneous Results and Structure 136 2. 7. I. Dynamical Systems and Crossed Products 136 2.7.2. Tensor Products of Operator Algebras 142 2.7.3. Weights on Operator Algebras; Self-Dual Cones of General von Neumann Algebras; Duality and Classification of Factors; Classification of c•-A1gebras 145 Notes and ReiiW'k$ 152 Groups, Semigroups, and Generators 157 3.1. Banach Space Theory 159 3.1.1. Uniform Continuity 161 3.1.2. Strong, Weak, and Weak• Continuity 163 3.1.3. Convergence Properties 183 3.1.4. Perturbation Theory 189 3.1.5. Approximation Theory 198 3.2. Algebraic: Theory 205 3.2.1. Positive Linear Maps and Jordan Morphisms 205 3.2.2. General Properties of Derivations 228 Contents Volume I xi 3.2.3. Spectral Theory and Bounded Derivations 244 3.2.4. Derivations and Automorphism Groups 259 3.2.5. Spatial Derivations and Invariant States 263 3.2.6. Approximation Theory for Automorphism Groups 285 Notes and Rellllllb 298 Decomposition Theory 309 4.1. Geaeni Theory 311 4.1.1. Introduction 311 4.1.2. Barycentric Decompositions 315 4.1.3. Orthogonal Measures 333 4.1.4. Borel Structure of States 344 4.2. Extremal, Central, and SubceDtral Decompositions 353 4.2.1. Extremal Decompositions 353 4.2.2. Central and Subcentral Decompositions 362 4.3. IDvariant States 367 4.3.1. Ergodic Decompositions 367 4.3.2. Ergodic States 386 4.3.3. Locally Compact Abelian Groups 400 4.3.4. Broken Symmetry 416 4A. Spatial Decomposition 432 4.4.1. General Theory 433 4.4.2. Spatial Decomposition and Decomposition of States 442 Notes and Rellllllb 451 References 459 Books and Moaograpbs 461 Articles 464 List of Symbols 481 Subject Index 487 States in Quantum Statistical Mechanics