Mathematical Foundations of Quantum Statistical Mechanics Continuous
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 colleagu
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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. Principal Concepts of Quantum Mechanics 1.1. 1.2. 1.3. 1.4. 1.5.
The Fock Space Evolution of States for the Systems with Random Number of Particles Second Quantization SchrMinger Equation in the Case of Second Quantization
3. Evolution of States in the Heisenberg Representation and in the Interaction Representation 3.1. 3.2. 3.3. 3.4.
Heisenberg Equations Interaction Representation Evolution Operator Second Quantization and the Heisenberg Representation in the Momentum Space 3.5. The Frohlich Hamiltonian
14 14 17 21 24
28 28 30 32 36 40
43
Mathematical Supplement I 1.1. 1.2. 1.3. 1.4. 1.5.
1
SchrOdinger Equation and the Evolution of States of Finitely Many Particles 1 Density Matrix. Equation for Density Matrix 3 9 Algebra of Observables. States on This Algebra 11 SchrOdinger and Heisenberg Representations Bose-Einstein and Fermi-Dirac Statistics 13
2. Evolution of States of Quantum Systems with Arbitrarily Many Particles 2.1. 2.2. 2.3. 2.4.
1
Selfadjoint Operators Kato's Criterion of Selfadjointness Representation of Operators in Terms of Their Kernels Solution of the SchrOdinger Equation On the Convergence of Series (3.33) for the Evolution Operator
References
43
45 52 53 53
54 v
Contents
vi
CHAPTER 2. EVOLUTION OF STATES OF INFINITE QUANTUM SYSTEMS
57
4. Bogolyubov Equations for Statistical Operators
57
4.1. 4.2. 4.3. 4.4.
57 60 63
Sequence of Statistical Operators Sequence of Statistical Operators in the Grand Canonical Ensemble Equations for Statistical Operators Statistical Operators and Bogolyubov Equations in Terms of the Second Quantization Operators
68
5. Solution of the Bogolyubov Equations
72
5.1. Statement of the Problem 5.2. Group of Evolution Operators
72
74 78
5.3. Infinitesimal Generator of the Group V A (t) 5.4. Feynman Integral
81
6. Gibbs Distributions
85
6.1. Stationary Solutions of Bogolyubov Equations and E
