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arXiv:cond-mat/9608120 · Physical Review Letters

Relations between Markov Processes via Local Time and Coordinate Transformations

Axel Pelster, Hagen M. Kleinert

Published Jan 27, 1997• 10 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevlett.78.565

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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.
Using the concepts of both Schrodinger and Feynman we perform a transformation between different classes of quantum mechanical systems. A carefully elaborated succession of three separate mappings, concerning time, space and the wave function, leads to a relationship between the potentials corresponding to the quantum mechanical systems and their related propagators. Since we admit a space transformation which may depend explicitly on time, our result contains the Duru-Kleinert-propagator transformations as a special case. Additionally we obtain as an example the transformation of the free particle to the harmonic oscillator.
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.
We solve time-sliced path integrals of one-dimensional Coulomb system in an exact manner. In formulating path integrals, we make use of the Duru-Kleinert transformation with Fujikawa’s gauge theoretical technique. Feynman kernels in the momentum representation both for bound states and scattering states will be obtained with clear pole structure that explains the exactness of the path integral. The path integrals presented here can be, therefore, evaluated exactly by making use of Cauchy’s integral theorem. 1
A new approach to quantum Markov processes is developed and the corresponding Fokker-Planck-like equation is derived. The latter was examined to reproduce known results from classical and quantum physics. It was also applied to the phase-space description of a mechanical system thus leading to a new treatment of this problem different from the Wigner presentation. The equilibrium probability density obtained in the mixed coordinate-momentum space is a reasonable extension of the Gibbs canonical distribution.
To this purpose, one uses the result that quantum phenomena in the Euclidean formulation of the theory are due to a stochastic space-time background interaction, whose essence is the time derivative of the Wiener process. The problem of calculating both the transition probability, the path integral for the systems of four particles and factorization solution of Fokker-Planck equation are then solved. The transition probability solution of Fokker-Planck equation factorizes into a first component describing the system at its ground state and a second component characterizing its transition dynamics. The path integral for these system are then solved.
In a reasonably self-contained presentation mathematical rigour is supplied to the important ideas of solving certain non-Gaussian path integrals by changes of dimension and/or path-dependent time transformations. The resulting genuine path-integral calculus neither requires discretization prescriptions nor sophisticated methods from the theory of stochastic differential equations. The power of the calculus is illustrated by two standard quantum-physics applications. First, the calculation of the time-dependent propagator corresponding to a particle on the half-line in a harmonic plus inverse-square potential is shown to be a simple exercise. Second, the first rigorous derivation of the energy-dependent Green function of the one-dimensional Morse system is given.