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 Uspekhi Mat. Nauk, 1974, Volume 29, Issue 3(177), Pages 111–159 (Mi umn4376)

The method of diagrams in perturbation theory

E. B. Gledzer, A. S. Monin

Abstract: In this paper the mathematical methods of quantum field theory are applied to some problems that arise in the statistical description of mechanical systems with very many (in the idealized case, infinitely many) degrees of freedom. This application is based on a graphical representation of the individual terms of the formal perturbation series in powers of the coupling constant in the form of Feynman diagrams. A variety of properties of such diagrams makes it possible to sum partially the perturbation series with a view to obtaining closed integral equations that contain the required quantities as unknowns. The approach is treated in more detail in connection with the statistical hydrodynamics of a developed turbulent flow, which is similar to the theory of a quantum Bose field with strong interaction. The functional formulation of statistical hydrodynamics makes it possible to obtain integral equations of turbulence theory, which can also be derived by means of diagram methods. At the end of the paper, some closed equations of statistical hydrodynamics are considered.

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English version:
Russian Mathematical Surveys, 1974, 29:3, 117–168

Bibliographic databases:

UDC: 519.9
MSC: 76F30, 81Q30, 81Q15, 76D06

Citation: E. B. Gledzer, A. S. Monin, “The method of diagrams in perturbation theory”, Uspekhi Mat. Nauk, 29:3(177) (1974), 111–159; Russian Math. Surveys, 29:3 (1974), 117–168

Citation in format AMSBIB
\Bibitem{GleMon74} \by E.~B.~Gledzer, A.~S.~Monin \paper The method of diagrams in perturbation theory \jour Uspekhi Mat. Nauk \yr 1974 \vol 29 \issue 3(177) \pages 111--159 \mathnet{http://mi.mathnet.ru/umn4376} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=400985} \zmath{https://zbmath.org/?q=an:0362.76102|0371.76048} \transl \jour Russian Math. Surveys \yr 1974 \vol 29 \issue 3 \pages 117--168 \crossref{https://doi.org/10.1070/RM1974v029n03ABEH001286} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. V. P. Maslov, A. M. Chebotarev, “Generalized measure in Feynman path integrals”, Theoret. and Math. Phys., 28:3 (1976), 793–805
2. G. I. Babkin, V. I. Klyatskin, “Analysis of the Dyson equation for stochastic integral equations”, Theoret. and Math. Phys., 41:3 (1979), 1080–1086
3. A. S. Monin, “Geophysical turbulence”, Russian Math. Surveys, 38:4 (1983), 127–149
4. É. V. Teodorovich, “Diagram equations of the theory of fully developed turbulence”, Theoret. and Math. Phys., 101:1 (1994), 1177–1183
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