In this post, we will see the book *Methods Of Quantum Field Theory In Statistical Physics* by A. A. Abrikosov; L. P. Gorkov; I. E. Dzyaloshinski.

# About the book

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.

The book was translated from Russian by Richard Silverman was published in 1963.

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# Contents

## 1. GENERAL PROPERTIES OF MANY-PARTICLE

SYSTEMS AT LOW TEMPERATURES, Page 1.

1. Elementary Excitations. The Energy Spectrum and Properties of Liquid He* at Low Temperatures, 1.

1.1: Introduction. Quasi-particles, 1.

1.2: The spectrum of a Bose liquid, 6.

1.3: Superfluidity, 11.

2. The Fermi Liquid, 15.

2.1: Excitations in a Fermi liquid, 15.

2.2: The energy of the quasi-particles, 18.

2.3: Sound, 23.

3. Second Quantization, 28.

4. The Dilute Bose Gas, 31.

5. The Dilute Fermi Gas, 36.

## 2. METHODS OF QUANTUM FIELD THEORY FOR T = 0, Page 43.

6. The Interaction Representation, 43.

7. The Green’s Function, 51.

7.1: Definition. Free-particle Green’s functions, 51.

7.2: Analytic properties, 55.

7.3: The physical meaning of the poles, 59. 7.4: The Green’s function of a system in an external field, 63.

8. Basic Principles of the Diagram Technique, 64.

8.1: Transformation from the variable N to the variable p, 64.

8.2: Wick’s theorem, 66.

8.3: Feynman diagrams, 68.

9. Rules for Constructing Diagrams for Interactions of Various Types, 71.

9.1: The diagram technique in coordinate space. Examples, 71.

9.2: The diagram technique in momentum space. Examples, 80.

10. Dyson’s Equation. The Vertex Part. Many-Particle Green’s Functions, 85.

10.1: Sums of diagrams. Dyson’s equation, 85.

10.2: Vertex parts. Many-particle Green’s functions, 89.

10.3: The ground-state energy, 95.

## 3. THE DIAGRAM TECHNIQUE FOR T ≠ 0, Page 97.

11. Temperature Green’s Functions, 97.

11.1: General properties, 97.

11.2: Temperature Green’s functions for free particles, 102.

12. Perturbation Theory, 103.

12.1: The interaction representation, 103.

12.2: Wick’s theorem, 106.

13. The Diagram Technique in Coordinate Space. Examples, 111.

14. The Diagram Technique in Momentum Space, 120.

14.1: Transformation to momentum space, 120.

14.2: Examples, 123.

15. The Perturbation Series for the Thermodynamic Potential Q, 130.

16. Dyson’s Equation. Many-Particle Green’s Functions, 135.

16.1: Dyson’s equation, 135.

16.2: Relation between the Green’s functions and the thermodynamic potential Q, 139.

17. Time-Dependent Green’s Functions for T ≠ 0. Analytic Properties of the Green’s Functions, 144.

## 4. THEORY OF THE FERMI LIQUID, Page 154.

18. Properties of the Vertex Part for Small Momentum Transfer. Zero Sound, 154.

19. Effective Mass. Relation Between the Fermi Momentum and the Particle Number. Excitations of the Bose Type. Heat Capacity, 160.

19.1: Some useful relations, 160.

19.2: Basic relations of the theory of the Fermi liquid, 162.

19.3: Excitations of the Bose type, 165.

19.4: Another derivation of the relation between the Fermi momentum

and the particle number, 166.

19.5: Heat capacity, 169.

20. Singularities of the Vertex Part When the Total Momentum of the Colliding Particles is Small, 172.

21. Electron-Phonon Interactions, 176.

21.1: The vertex part, 176.

21.1: The phonon Green’s function, 178.

21.3: The electron Green’s function, 182.

21.4: Correction to the linear term in the electronic heat capacity, 188.

22. Some Properties of a Degenerate Plasma, 189.

22.1: Statement of the problem, 189.

22.2: The vertex part for small momentum transfer, 191.

22.3: The electron spectrum, 195.

22.4: Thermodynamic functions, 200.

## 5. SYSTEMS OF INTERACTING BOSONS, Page 204.

23. Application of Field Theory Methods to a System of Interacting Bosons for T = 0, 204.

24. The Green’s Functions, 213.

24.1: Structure of the equations, 213.

24.2: Analytic properties of the Green’s functions, 217.

24.3: Behavior of the Green’s functions for small momenta, 221.

25. The Dilute Nonideal Bose Gas, 222.

25.1: The diagram technique, 222.

25.2: Relation between the chemical potential and the self-energy parts of the one-particle Green’s functions, 224.

25.3: The low-density approximation, 228.

25.4: The effective interaction potential, 231.

25.5: The Green’s function of a Bose gas in the low-density approximation. The spectrum, 234.

26. Properties of the Spectrum of One-Particle Excitations Near the Cutoff Point, 235.

26.1: Statement of the problem, 235.

26.2: The basic system of equations, 237.

26.3: Properties of the spectrum near the threshold for phonon creation, 240.

26.4: Properties of the spectrum near the threshold for decay into two excitations with parallel nonzero momenta, 243.

26.5: Decay into two excitations emitted at an angle with each other, 245.

27. Application of Field Theory Methods to a System of Interacting Bosons for T ≠ 0, 247.

## 6. ELECTROMAGNETIC RADIATION IN AN ABSORBING MEDIUM, Page 252.

28. The Green’s Functions of Radiation in an Absorbing Medium, 252.

29. Calculation of the Dielectric Constant, 259.

30. Van der Waals Forces in an Inhomogeneous Dielectric, 263.

31. Molecular Interaction Forces, 268.

31.1: Interaction forces between solid bodies, 268.

31.2: Interaction forces between molecules in a solution, 275.

31.3: A thin liquid film on the surface of a solid body, 277.

## 7. THEORY OF SUPERCONDUCTIVITY, Page 280.

32. Background Information. Choice of a Model, 280.

32.1: The phenomenon of superconductivity, 280.

32.2: The model. The interaction Hamiltonian, 282.

33. The Cooper Phenomenon. Instability of the Ground State of a System of Noninteracting Fermions With Respect to Arbitrarily Weak Attraction Between the Particles, 284.

33.1: The equation for the vertex part, 284.

33.2: Properties of the vertex part, 287.

33.3: Determination of the critical temperature, 289.

34. The Basic System of Equations for a Superconductor 291

34.1: A superconductor at absolute zero, 291

34.2: The equations in the presence of an external electromagnetic field. Gauge invariance, 296

34.3: A superconductor at finite temperatures, 297

35. Derivation of the Equations of the Theory of Superconductivity in the Phonon Model, 299

36. The Thermodynamics of Superconductors, 303

36.1: Temperature dependence of the energy gap, 303.

36.2: Heat capacity, 304. 36.3: The critical magnetic field, 306.

37. A Superconductor in a Weak Electromagnetic Field 308.

37.1: A weak constant magnetic field, 308

37.2: A superconductor in an alternating field, 315

38. Properties of a Superconductor in an Arbitrary Magnetic Field Near the Critical Temperature, 320.

39. Theory of Superconducting Alloys 325

39.1: Statement of the problem, 325

39.2: The residual resistance of a normal metal, 327

39.3: Electromagnetic properties of superconducting alloys 334

BIBLIOGRAPHY Page 342

NAME INDEX Page 347

SUBJECT INDEX Page 349