Functional Integrals and Collective Excitations

224 Pages·1988·6.809 MB·English

by  Victor Nikolaevich Popov

CAMBRIDGE MONOGRAPHS ON
MATHEMATICAL PHYSICS
General editors: P. V. Landshoff, D. W. Sciama, S. Weinberg
FUNCTIONAL INTEGRALS AND
COLLECTIVE EXCITATIONS

FUNCTIONAL INTEGRALS
AND COLLECTIVE
EXCITATIONS
V. N. POPOV
Leningrad Branch of V. A. Steklov Mathematical Institute of the
Academy of Sciences of the USSR
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© Cambridge University Press 1987
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First published 1987
First paperback edition (with corrections) 1990
A catalogue record for this publication is available from the British Library
Library of Congress Cataloguing in Publication Data
Popov, V. N. (Viktor Nikolaevich)
Functional integrals and collective excitations.
"Partly based on my lectures at the Institut für
Theoretische Physik, Freie Universitat, West
Berlin" - Pref.
Bibliography
Includes index.
1. Integration, Functional. 2. Collective
excitations. I. Title.
QC20.7.F85P65 1987 53O.i'55 86-28398
ISBN 978-0-521-30777-2 Hardback
ISBN 978-0-521-40787-8 Paperback
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Contents
Preface vii
Part I: Functional integrals and diagram techniques in statistical
physics 1
1 Functional integrals in statistical physics 3
2 Functional, integrals and diagram techniques for Bose particles 5
3 Functional integrals and diagram techniques for Fermi
particles 16
4 Method of successive integration over fast and slow variables 20
Part II: Superfluid Bose systems 23
5 Superfluidity 25
6 Low-density Bose gas 31
7 The modified perturbation scheme for superfluid Bose systems 40
8 Quantum vortices in superfluids 49
Part III: Plasma and superfluid Fermi systems 57
9 Plasma theory 59
10 Perturbation theory for superconducting Fermi systems 68
11 Superconductivity of the second kind 76
12 Collective excitations in superfluid Fermi systems 90
13 Bose spectrum of superfluid Fermi gas 96
14 Superfluid phases in 3He 105
15 Collective excitations in the B-phase of 3He 112
16 Collective excitations in the A-phase of 3He 130
17 Superfluidity and Bose excitations in 3He films 138
Part IV: Crystals, heavy atoms, model Hamiltonians 145
18 Functional integral approach to the theory of crystals 147
19 Effective interaction of electrons near the Fermi surface 158
20 Crystal structure of a dense electron-ion system 163
21 Quantum crystals 173
22 The theory of heavy atoms 185
23 Functional integral approach to the theory of model
Hamiltonians 195
References 212
Index 215

Preface
Nowadays functional integrals are used in various branches of theoret-
ical physics, and may be regarded as an 'integral calculus' of modern
physics. Solutions of differential or functional equations arising in diffusion
theory, quantum mechanics, quantum field theory and quantum statistical
mechanics can be written in the form of functional integrals.
Functional integral methods are widely applied in quantum field theory,
especially in gauge fields. There exist numerous interesting applications of
functional integrals to the study of infrared and ultraviolet asymptotic
behaviour of Green's functions in quantum field theory and also to the
theory of extended objects (vortex-like excitations, solitons, instantons).
In statistical physics functional methods are very useful in problems
dealing with collective modes (long-wave phonons and quantum vortices in
superfluids and superconductors, plasma oscillations in systems of charged
particles, collective modes in 3He-type systems and so on).
This book is devoted to some applications of functional integrals for
describing collective excitations in statistical physics. The main idea is to go
in the functional integral from the initial variables to some new fields
corresponding to 'collective' degrees of freedom. The choice of specific
examples is to a large extent determined by the scientific interests of the
author.
We dwell on modifications of the functional integral scheme developed
for the description of collective modes, such as longwave phonons and
quantum vortices in superfluids, superconductors and 3He, plasma oscill-
ations in systems of charged particles. The last chapters are devoted to the
functional integral approach to the theory of crystals, the theory of many
electron atoms and also to the theory of some model systems of statistical
physics, exactly integrable in the sense of N. N. Bogoliubov (the BCS model
and the Dicke model).
Problems of rigorous justification of functional integral formalism are
almost not touched upon in this book. Here functional methods are used as
a powerful tool to build up perturbation theory or to go over from one
perturbative scheme to another. The only exclusion is the last chapter
viii Preface
devoted to the BCS and Dicke models. Here the results concerning the
Dicke model are obtained in the mathematically rigorous way. These
results, up to now, are more strong than that obtained by any other
methods.
This book is partly based on my lectures at the Institut fur Theoretische
Physik, Freie Universitat, West Berlin, FRG (Popov, 1978).
I am very grateful to A. G. Reiman, for many remarks on the style of the
book.
Parti
Functional integrals and diagram techniques
in statistical physics