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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 The right of the University of Cambridge to print and ttlt all mamtr of books s y Henry Vm in 1534. The Uniwjiiy hat printed and pubiishtd continuously since 1SS4. CAMBRIDGE UNIVERSITY PRESS Cambridge New York Port Chester Melbourne Sydney CAMBRIDGE university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Dubai, Tokyo, Mexico City Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521407878 © Cambridge University Press 1987 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. 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 Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Information regarding prices, travel timetables, and other factual information given in this work is correct at the time of first printing but Cambridge University Press does not guarantee the accuracy of such information thereafter. 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 oscillations 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