Operator Algebras and Quantum Statistical Mechanics Equilibrium Stat
In this chapter, and the following one, we examine various applications of C*-algebras and their states to statistical mechanics. Principally we analyze the structural properties of the equilibrium states of quantum systems con sisting of a large number
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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 Norges Tekniske HI'Jgskole Universitetet 1 Trondheim N-7034 Trondheim Norway
School of Mathematics University of New South Wales P.O. Box 1 Kensington, NSW 2033 Australia
Editors:
Wolf BeiglbOck
Maurice Goldhaber
Institut fUr Angewandte Mathematik Universitiit HeideIberg Im Neuenheimer Feld 5 D-6900 Heidelberg 1 Federal Republic of Germany
Department of Physics Brookhaven National Laboratory Associated Universities, Inc. Upton, NY 11973 USA
Elliott H. Lieb
Walter Thirring
Department of Physics Joseph Henry Laboratories Princeton University P.O. Box 708 Princeton, NJ 08540 USA
Institut fUr Theoretische Physik der Universitiit Wien Boltzmanngasse 5 A-1090Wien Austria
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
987654321
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. 5.2.2. 5.2.3. 5.2.4. 5.2.5.
The CAR and CCR Relations The CAR and CCR Algebras States and Representations The Ideal Fermi Gas The Ideal Bose Gas
5.3. KMS States 5.3.1. The KMS Condition 5.3.2. The Set of KMS States 5.3.3. The Set of Ground States
5.4. Stability and Equilibrium
5.4.1. Stability of KMS States 5.4.2. Stability and the KMS Condition
6
15 24 46
58
77 77 116 133
147 147 180 vii
viii Contents Volume II 5.4.3. Gauge Groups and the Chemical Potential 5.4.4. Passive Systems
203 217
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 6.2.2. The Gibbs Condition for Equilibrium 6~2.3. The Maximum Entropy Principle 6.2.4. Translationally Invariant States 6.2.5. Uniqueness of KMS States 6.2.6. Nonuniqueness ofKMS States 6.2.7. Ground States
6.3. Continuous Quantum Systems. II 6.3.1. 6.3.2. 6.3.3. 6.3.4.
The Local Hamiltonians The Wiener Integral The Thermodynamic Limit. I. The Reduced Density Matrices The Thermodynamic Limit. II. States and Green's Functions
243 263 269 289 307 319 334
348 350 361 376 391
6.4. Conclusion
417
Notes and Remarks
419
References
453
Books and Monograpbs
455
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