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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Path-integral formulation of real-time finite-temperature field theory

R. Kobes, Kenneth L. Kowalski

Published Jul 15, 1986• 61 citations
Publisher

Abstract

A path-integral representation of the generating functional for the real-time finite-temperature Green's functions is investigated. The Feynman rules that are implied in this way are reconciled with those obtained in the thermo-field-dynamics approach for scalar fields, although the essential points involved in achieving this reconciliation hold for all spins. In addition, several aspects of the imposition and implications of the Kubo-Martin-Schwinger periodicity condition are clarified.

DOI
10.1103/physrevd.34.513

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A path-integral representation of the generating functional for the real-time finite-temperature Green's functions is investigated. The Feynman rules that are implied in this way are reconciled with those obtained in the thermo-field-dynamics approach for scalar fields, although the essential points involved in achieving this reconciliation hold for all spins. In addition, several aspects of the imposition and implications of the Kubo-Martin-Schwinger periodicity condition are clarified.
For thermal equilibrium systems it is shown, how the Kubo-Martin-Schwinger boundary condition may be used to factorize the generating functional of Green functions at least on the level of the full two-point function. Genuine non-equilibrium system exhibit correlations that one may also incorporate in the path integral. One one hand this provides a natural tool for a perturbative expansion including these correlations. On the other hand it allows to prove that in general non-equilibrium systems the generating functional does not factorize.
Thermofield Dynamics (TFD), a real time formalism for thermal quantum field theory, is formulated in terms of a path-integral approach following the Weinberg procedure.In order to assure precise rules for the development, we use a representation of Lie-algebras such that the TFD algebraic ingredients are derived, including the tilde conjugation rules.The association with the canonical formalism is obtained, and for the case of bosons, we introduce the Feynman diagrams and derive the n-point functions.
We impose the periodicity conditions corresponding to the Matsubara formalism for thermal field theory as constraints in the imaginary-time path integral. These constraints are introduced by means of time-independent auxiliary fields which, by integration of the original variables, become dynamical fields in the resulting ``dual'' representation for the theory. This alternative representation has the appealing property of involving fields that live in one dimension less than the original ones, with a quantum partition function whose integration measure is identical to the one of its classical counterpart, albeit with a different (spatially nonlocal) action.
The path-integral is a formulation of quantum mechanics equivalent to the standard one, offering a new point oh view of the subject which is more intuitive than the usual approach. In this work we firstly present the Dirac orginal idea and then we describe how Feynman developed it. Therefore we apply the path-integral approach to the case of a scalar field theory, introducing the concept of generating functional. Finally we illustrate two alternative representations of the generating functional, dual to the Schwinger formalism, discussing how they lead to a realation with the Hermite polynomials. We also introduce the Schwinger-Dyson equation and express it through the representations introduced.
The generating functional formulation of non-equilibrium thermo field dynamics (NETFD) is given. The results are compared with those obtained by Schwinger who introduced the forward and backward time axes in order to include the dissipation effect. The relation of NETFD to other closed time-path formalisms is also given. As NETFD is an operator formalism of quantum field theory for non-equilibrium situations, one can say that the usual Green's function formalisms are well founded by NETFD through the comparison given in this paper.