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Physical Review A

Path integration of a general two-time quadratic action by prodistribution approach

Dinkar C. Khandekar, Sunil Datta, S. V. Lawande

Published Jun 1, 1985• 3 citations
Publisher

Abstract

The theory of prodistributions is applied to path integrate a general two-time quadratic action characterizing memory effects. The resulting Feynman propagator is exact and is in the form of an exponential integral over a single variable. A case where the integral is explicitly evaluated, for an arbitrary memory kernel, is presented to highlight the closed analytical structure of the propagator obtained. All special cases treated in the literature follow from this general expression.

DOI
10.1103/physreva.31.3574

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The theory of prodistributions is applied to path integrate a general two-time quadratic action characterizing memory effects. The resulting Feynman propagator is exact and is in the form of an exponential integral over a single variable. A case where the integral is explicitly evaluated, for an arbitrary memory kernel, is presented to highlight the closed analytical structure of the propagator obtained. All special cases treated in the literature follow from this general expression.
Using the prodistribution theory proposed by DeWitt-Morette [C. DeWitt-Morette, Commun. Math. Phys. 28, 47 (1972); C. DeWitt-Morette, A. Maheshwari, and B. Nelson, Phys. Rep. 50, 257 (1979)], the path integration of a time-dependent forced harmonic oscillator with a two-time quadratic action has been given in terms of the solutions of some integrodifferential equations. We then evaluate explicitly both the classical path and the propagator for the specific kernel introduced by Feynman in the polaron problem. Our results include the previous known results as special cases.
Functional integrals over complex and real auxiliary fields for propagators relating to two-time (memory) actions are constructed. The integrals over fields result from partial replacement of the path variable, via functional integration, in the path integral form of the propagator by a field variable in a way that allows the remaining path integration to be performed, thus leaving a functional integral. Two new cases of actions involving memory, a quadratic and a quartic are treated explicitly. A functional integral identity is also developed. Furthermore, the method of coupling an auxiliary dynamical field for generating memory actions is also employed and formulae are developed in the case of certain quadratic actions giving the propagator exactly. The method is flexible enough embracing all explicitly known cases of two-time quadratic actions and in addition enabling further cases to be treated. An example of this nature is given explicitly.
A class of time-dependent classical Lagrangians possessing an invariant quadratic in momentum is considered from a quantal point of view. Quantum mechanics is introduced through the Feynman propagator defined as a path integral involving the classical action. It is shown, without carrying out an explicit path integration, that the propagator for such a time-dependent system is related to the propagator of an associated time-independent problem. The expansion of the propagator in terms of the eigenfunctions of the invariant operator is derived and the equivalence of the present theory to that of Lewis and Reisenfeld (1969) is discussed. Explicit analytic forms of propagators are obtained for some cases to illustrate the application of the present approach.