Methods of Quantum Field Theory in Statistical Physics

729 Pages·1975·18.45 MB·English

by  A. A. Abrikosov

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Copyright © 1963 by Richard A. Silverman. All rights reserved.

This Dover edition, first published in 1975, is an unabridged republication, with
slight corrections, of the work originally published by Prentice-Hall, Inc.,
Englewood Cliffs, New Jersey, in 1963.

Library of Congress Catalog Card Number: 75-17174
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AUTHORS’ PREFACE TO THE
RUSSIAN EDITION
In recent years, remarkable success has been achieved in statistical physics, due
to the extensive use of methods borrowed from quantum field theory. The
fruitfulness of these methods is associated with a new formulation of
perturbation theory, primarily with the application of “Feynman diagrams.” The
basic advantage of the diagram technique lies in its intuitive character: Operating
with one-particle concepts, we can use the technique to determine the structure
of any approximation, and we can then write down the required expressions with
the aid of correspondence rules. These new methods make it possible not only to
solve a large number of problems which did not yield to the old formulation of
the theory, but also to obtain many new relations of a general character. At
present, these are the most powerful and effective methods available in quantum
statistics.

There now exists an extensive and very scattered journal literature devoted to the
formulation of field theory methods in quantum statistics and their application to
specific problems. However, familiarity with these methods is not widespread
among scientists working in statistical physics. Therefore, in our opinion, the
time has come to present a connected account of this subject, which is both
sufficiently complete and accessible to the general reader.
Some words are now in order concerning the material in this book. In the first
place, we have always tried to exhibit the practical character of the new
methods. Consequently, besides a detailed treatment of the relevant
mathematical apparatus, the book contains a discussion of various special
problems encountered in quantum statistics. Naturally, the topics dealt with here
do not exhaust recent accomplishments in the field. In fact, our choice of subject
matter is dictated both by the extent of its general physical interest and by its
suitability as material illustrating the general method.

We have confined ourselves to just one of the possible formulations of quantum
statistics in field theory language. For example, we do not say anything about the
methods developed by Hugenholtz, and by Bloch and de Dominicis. From our
point of view, the simplest and most convenient method is that based on the use
of Green’s functions, and it is this method which is taken as fundamental in the
present book.

It is assumed that the reader is familiar with the elements of statistical physics
and quantum mechanics. The method of second quantization, as well as all
information needed to derive the field theory methods used here, can be found in
Chapter 1. This chapter is of an introductory character, and contains a brief
exposition of contemporary ideas on the nature of energy spectra, together with
some simple examples.

Unless the contrary is explicitly stated, we use a system of units in which both
Planck’s constant and the velocity of light c equal 1 (the latter is important in
Chapters 6 and 7). Moreover, temperature is expressed in energy units, so that k
= 1.

The authors would like to express their gratitude to L. P. Pitayevski and
Academician L. D. Landau for their valuable advice on the material discussed
here.
1961
A. A. A.
A. A. A.
L. P. G.
I. E. D.