Title Methods of Quantum Field Theory in Statistical Physics A. A. Abrikosov Physics English 18.5 MB 729
```                            DOVER BOOKS ON PHYSICS
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AUTHORS’ PREFACE TO THE RUSSIAN EDITION
AUTHORS’ PREFACE TO THE REVISED ENGLISH EDITION
TRANSLATOR’S PREFACE
1 - GENERAL PROPERTIES OF MANY-PARTICLE SYSTEMS AT LOW TEMPERATURES
2 - METHODS OF QUANTUM FIELD THEORY FOR T=0 .
3 - THE DIAGRAM TECHNIQUE FOR T ≠ 0
4 - THEORY OF THE FERMI LIQUID
5 - SYSTEMS OF INTERACTING BOSONS
6 - ELECTROMAGNETIC RADIATION IN AN ABSORBING MEDIUM
7 - THEORY OF SUPERCONDUCTIVITY
BIBLIOGRAPHY
NAME INDEX
SUBJECT INDEX
A CATALOG OF SELECTED DOVER BOOKS IN SCIENCE AND MATHEMATICS
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where < • • • > is meant in the sense of an ordinary temperature average. It is not
hard to see that the Fourier transform of this quantity with respect to the
variables r1 — r2 and τ1 — τ2 has all the properties of a Bose temperature
Green’s function. On the other hand, this Fourier transform obviously equals

and the same applies to other vertices with small momentum transfer.
It is now clear how to go from temperature functions to time-dependent

functions. As we already know, to make this change in the case of a Green’s
function, we need only find a function which is analytic in the upper half-plane
of the variable ω and which coincides with the temperature Green’s function at
the points iωm = i2mπT. This is how the retarded function is defined. The actual
Green’s function equals the retarded function for ω > 0 and its complex
conjugate for ω < 0.
This procedure can easily be carried out for the functions , and we

denote the resulting functions by ∏e, ∏i. As can be seen from the integral
(22.11), to accomplish this we must change iωm to w + iδ sgn ω. Since in the
diagrams making up the average written above, only the factors depend on
ωm, we can obtain the corresponding actual Green’s function in the same way.
Moreover, the same obviously applies to the functions Γ(k, ω).78 Thus, the time-
dependent vertex parts Γ(k, ω) are given by the same formulas (22.8) and (22.9)
as before, where, however, iωm has been changed to ω + iδ sgn ω. The
appearance of a correction to k2 in the denominator of Γ is just due to the Debye
screening of the Coulomb interaction, which in general makes the interaction
retarded (i.e., Γ depends on w).
Next, we examine the behavior of Γ as a function of the ratio between ω and k.

In the case ω « vik, formula (22.11) gives

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(22.12)

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G’ and for bosons.

107 We have chosen F+(0+) to be real. This is always possible in the absence
of an external field, since the equations (34.13) are invariant under the
transformation

F(x - x′) → F(x - x′)exp{2iχ), F(0+) → F(0+)exp{2iχ},
F+(x – x′) → F+(x – x′)exp{ – 2iχ}, F+(0+) → F+(0+)exp{ – 2iχ},
with constant phase χ. For further details, see Sec. 34.2.

108 However, the condition Tc « ωD is always satisfied in real metals.

109 For an analysis of the electron-phonon interaction in the theory of
superconductivity, see B6, B7.

110 The functions and in (37.7) must now be understood to be the
Green’s functions of the given body, subject to the appropriate boundary
conditions.

111 In ordinary units,

112 The solution (38.7) satisfies this condition, i.e., if then

113 The titles of all papers from Russian journals are given in English. The
translations of titles given in the journal Soviet Physics, JETP (a translation of
the Journal of Experimental and Theoretical Physics of the Academy of Sciences
of the USSR, published by the American Institute of Physics, New York, N.Y.)
have been preserved, except for a few corrections of style and spelling. The
Russian original of this journal is abbreviated as Zh. Eksp. Teor. Fiz. The papers
B1, B3, B4, B7 (part I), C3, G1, G2, G3, G6, H4, L3, L4, L6 and M6 are
reprinted in D. Pines (editor), The Many-Body Problem, W. A. Benjamin, Inc.,

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New York (1961).