This comprehensive introduction to the many-body theory was written by three renowned physicists and acclaimed by American Scientist as "a classic text on field theoretic methods in statistical physics." Methods of Quantum Field Theory in Statistical Physics By A. A. Abrikosov, L. P. Gorkov, I. E. Dzyaloshinki, Richard A. Silverman Dover Publications, Inc. Copyright © 1963 Richard A. Silverman All rights reserved. ISBN: 978-0-486-63228-5 Contents 1 GENERAL PROPERTIES OF MANY-PARTICLE SYSTEMS AT LOW TEMPERATURES, Page 1, 2 METHODS OF QUANTUM FIELD THEORY FOR T = 0, Page 43, 3 THE DIAGRAM TECHNIQUE FOR T ≠ 0, Page 97, 4 THEORY OF THE FERMI LIQUID, Page 154, 5 SYSTEMS OF INTERACTING BOSONS, Page 204, 6 ELECTROMAGNETIC RADIATION IN AN ABSORBING MEDIUM, Page 252, 7 THEORY OF SUPERCONDUCTIVITY, Page 280, BIBLIOGRAPHY, Page 342, NAME INDEX, Page 347, SUBJECT INDEX, Page 349, CHAPTER 1 GENERAL PROPERTIES OF MANY-PARTICLE SYSTEMS AT LOW TEMPERATURES I. Elementary Excitations. The Energy Spectrum and Properties of Liquid He4 at Low Temperatures 1.1. Introduction. Quasi-particles. Statistical physics studies the behavior of systems consisting of a very large number of particles. In the last analysis, the macroscopic properties of liquids, gases and solids are due to microscopic interactions between the particles making up the system. Obviously, a complete solution of the problem, involving determination of the behavior of each individual particle, is out of the question. Fortunately, however, the overall macroscopic characteristics are determined only by certain average properties of the system. To be explicit, we now consider some thermodynamic properties. The macroscopic state of a system is specified by giving three independent thermodynamic variables, e.g., the pressure P, the temperature T, and the average number N of particles in the system. From a quantum-mechanical point of view, a closed system of N particles is characterized by its energy levels En. Suppose that from the system we single out a volume (subsystem) which can still be regarded as macroscopic. The number of particles in such a subsystem is still very large, whereas the interaction forces between particles act at distances whose order of magnitude is that of atomic dimensions. Therefore, apart from boundary effects, we can regard the subsystem itself as closed, and characterized by certain energy levels (for a given number of particles). Since the subsystem actually interacts with other parts of the closed system, it does not have a fixed energy and a fixed number of particles, and, in fact, it has a nonzero probability of occupying any energy state. As is familiar from statistical physics (see e.g., L8), the microscopic derivation of thermodynamic formulas is based on the Gibbs distribution, which gives the following probability of finding the subsystem in the energy state with a number of particles equal to N: (1.1) [MATHEMATICAL EXPRESSION OMITTED] In this formula, T denotes the absolute temperature, μ the chemical potential, and Z a normalization factor which is determined from the condition (1.2) [MATHEMATICAL EXPRESSION OMITTED] According to (1.1), we have (1.3) [MATHEMATICAL EXPRESSION OMITTED] The quantity Z is called the grand partition function. If the energy levels EnN are known, the partition function can be calculated. This immediately determines the thermodynamic functions as well, since the formula (1.4) Ω = -T ln Z relates the quantity Z to the thermodynamic potential Ω (involving the variables V, T and μ). Obviously, the simplest use that can be made of these formulas is to calculate the thermodynamic functions of ideal gases, since in this case the energy is just the sum of the energies of the separate particles. However, in general, it is impossible to determine the energy levels of a system consisting of a large number of interacting particles. Therefore, so far, interactions between particles in quantum statistics have been successfully taken into account only when the interactions are sufficiently weak, and perturbation-theory calculations of thermodynamic quantities have been carried out only to the first or second approximations. In the majority of physical problems, where the interaction is far from small, an approach based on the direct use of formulas (1.1)–(1.4) is unrealistic. The case of very low temperatures is somewhat exceptional. As T -> 0, the important energy levels in the partition function are the weakly excited states, whose energies differ only very little from the energy of the ground state. The character of the energy spectrum of the system in this region of energies can be ascertained in some detail, by using very general considerations which are valid regardless of the magnitude and specific features of the interaction between the particles. As an example illustrating the subsequent discussion, consider the excitation of