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arXiv:1706.04738 · Journal of Mathematical Physics

On the effective potential of Duru-Kleinert path integrals

Seiji Sakoda

Published Jun 1, 2017• 1 citation

Abstract

We propose a new method to evaluate the effective potential in the path integral for the fixed-energy amplitude as well as for the pseudotime evolution kernel in the formalism by Duru and Kleinert. Restriction to the postpoint or the prepoint prescriptions in formulating time sliced path integrals is avoided by leaving off the use of expectation values for correction terms. This enables us to consider an arbitrary ordering prescription and to examine the ordering dependence of the effective potential. To investigate parameter dependences, we introduce the ordering parameter α in addition to the splitting parameter λ in the formulation of the time sliced path integral. The resulting path integrals are found to be independent of the ordering parameter although the explicit dependence, given by a contribution proportional to (1−2λ)2, on the splitting parameter remains. As an application, we check the relationship between path integrals for the radial oscillator and the radial Coulomb system in arbitrary dimensions.

DOI: 10.1063/1.4989986

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We propose a new method to evaluate the effective potential in the path integral for the fixed-energy amplitude as well as for the pseudotime evolution kernel in the formalism by Duru and Kleinert. Restriction to the postpoint or the prepoint prescriptions in formulating time sliced path integrals is avoided by leaving off the use of expectation values for correction terms. This enables us to consider an arbitrary ordering prescription and to examine the ordering dependence of the effective potential. To investigate parameter dependences, we introduce the ordering parameter α in addition to the splitting parameter λ in the formulation of the time sliced path integral. The resulting path integrals are found to be independent of the ordering parameter although the explicit dependence, given by a contribution proportional to (1−2λ)2, on the splitting parameter remains. As an application, we check the relationship between path integrals for the radial oscillator and the radial Coulomb system in arbitrary dimensions.

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The Duru-Kleinert method of solving unknown path integrals of quantum mechanical systems by relating them to known ones does not apply to Markov processes since the Duru-Kleinert transform of a Fokker-Planck equation is, in general, not a Fokker-Planck equation. In this Letter, we present a significant modification of the method, based again on a combination of path-dependent time and coordinate transformations, to obtain such relations after all. As an application we express unknown Green functions for a one-parameter family of Markov processes in terms of the known one for the Schenzle-Brand process.

The formulation of the relativistic spinless path integral on the general affine space is presented. For the one dimensional space, the Duru-Kleinert (DK) method and the

δ

-function perturbation technique are applied to solve the relativistic path integrals of the smooth potential and the Dirichlet boundary condition problems, respectively.