Methods Of Quantum Field Theory In Statistical Physics | A. A. Abrikosov, L. P. Gorkov, I. E. Dzyaloshinski

In recent years, remarkable success has been achieved in statistical physics, due to the extensive use of meth­ ods borrowed from quantum field theory. The fruitful­ ness of these methods is associated with a new formula­ tion of perturbation theory, primarily with the application of “Feynman diagrams.” The basic advantage of the diagram technique lies in its intuitive character: Operat­ ing with one-particle concepts, we can use the tech­ nique 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 power­ ful 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 applica­ tion 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. Con­ sequently, 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 lan­ guage. 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 ele­ ments of statistical physics and quantum mechanics. The method of second quantization, as well as all in­ formation 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.