Operator Algebras and Quantum Statistical Mechanics: Equilibrium States. Models in Quantum Statistical Mechanics

525 Pages·1997·18.64 MB·English

by  Professor Ola Bratteli

Texts and Monographs in Physics
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Physics and AstronOmy9:J ONUMllIIIWI'I
hnp://www.springer.de/physl
Oia Bratteli
Derek W. Robinson
Operator Aigebras
and Quantum
Statistical Mechanics 2
Equilibrium States.
Models in Quantum Statistical Mechanies
Second Edition
Springer
Professor Ola Bratteli
Universitetet i Os10
Matematisk Institutt
Moltke Moes vei 31
0316 Oslo, Norway
e-mail: [email protected]
Horne page: http://www.math.uio.no/~brattelil
Professor Derek W. Robinson
Australian National University
School of Mathematical Sciences
ACT 0200 Canberra, Australia
e-mail: [email protected]
Horne page: http://www.maths.anu.edu.aul~derekl
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Second Edition 1997. Second Printing 2002
ISSN 0172-5998
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ISBN 978-3-642-08257-3 ISBN 978-3-662-03444-6 (eBook)
DOI 10.1007/978-3-662-03444-6
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Originally published by Springer-Verlag Berlin Heidelberg New York in 1997.
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To
Trygve Bratteli,
Samuel Robinson,
and
Harold Ross
Preface to the Second Edition
Fifteen years have passed since completion of the first edition of this book
and much has happened. Any attempt to do justice to the new develop
ments would necessitate at least one new volume rather than a second
edition of the current one. Fortunately other authors have taken up the
challenge of describing these discoveries and our bibliography includes
references to a variety of new books that have appeared or are about to
appear. We consequently decided to keep the format ofthis book as a basic
reference for the operator algebraic approach to quantum statistical me
chanics and concentrated on correcting, improving, and updating the
material of the first edition. This in itself has not been easy and changes
öccur throughout the text. The major changes are a corrected presentation
of Bose-Einstein condensation in Theorem 5.2.30, insertion of a general
result on the absence of symmetry breaking in Theorem 5.3.33A, and an
extended description of the dynamics of the X - Y model in Example 6.2.14.
The discussion of phase transitions in specific models, in Sects. 6.2.6 and
6.2.7, has been expanded with the focus shifted from the classical Ising
model to genuine quantum situations such as the Heisenberg and X-Y
models. In addition the Notes and Remarks to various subsections have
been considerably augmented.
Since our interest in the subject of equilibrium states and models of
statistical mechanics has waned considerably in the last fifteen years it
VIII Preface to the Second Edition
would have been impossible to prepare this second edition without the
support and encouragement ofmany of our friends and colleagues. We are
particularly indebted to Charles Batty, Michie\ van den Berg, Tom ter EIst,
Dai Evans, Mark Fannes, Jürg Fröhlich, Taku Matsui, Andre Verbeure,
and Marinus Winnink for information and helpful advice, and we apo 1-
ogize for often ignoring the latter. We are especially grateful to Aernout
van Enter and Reinhard Werner for counselling us on recent deve\opments
and giving detailed suggestions for revisions.
Oslo and Canberra 1996 Ola Bratteli
Derek W. Robinson
Contents Volume 2
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 23
5.2.4. The Ideal Fermi Gas 45
5.2.5. The Ideal Bose Gas 57
5.3. KMS-States 76
5.3.1. The KMS Condition 76
5.3.2. The Set of KMS States 112
5.3.3. The Set of Ground States 131
5.4. Stability and Equilibrium 144
5.4.1. Stability of KMS States 144
5.4.2. Stability and the KMS Condition 176
X Contents Volume 2
5.4.3. Gauge Groups and the Chemical Potential 197
5.4.4. Passive Systems 211
Notes and Remarks 217
Models of Quantum Statistical Mechanics 235
6.1. Introduction 237
6.2 Quantum Spin Systems 239
6.2.1. Kinematical and Dynamical Descriptions 239
6.2.2. The Gibbs Condition for Equilibrium 261
6.2.3. The Maximum Entropy Principle 266
6.2.4. Translationally Invariant States 286
6.2.5. Uniqueness of KMS States 306
6.2.6. Nonuniqueness of KMS States 317
6.2.7. Ground States 338
6.3. Continuous Quantum Systems. 11 353
6.3.1. Tbe Local Hamiltonians 355
6.3.2. The Wiener Integral 366
6.3.3. The Therrnodynamic Limit. I. The Reduced Density Matrices 381
6.3.4. The Therrnodynamic Limit. 11. States and Green's Functions 395
6.4. Conclusion 422
Notes and Remarks 424
References 463
Books and Monographs 465
Articles 468
List of Symbols 487
Subject Index 499