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Mathematische Nachrichten

On Transformations in the Feynman-Kac Formula and Quantummechanical N-Body Systems

Michael Demuth

Published Jan 1, 1985• 1 citation
Publisher

Abstract

Coordinate transformations in the Feynman-Kac formula are considered. The contractive semigroup with transformed coordinates is expressed by an integral with a correspondingly transformed diffusion measure. This representation is used for N-body systems transformed to clustered Jacobi coordinates implying a decomposition of the Feynman-Kac formula.

DOI
10.1002/mana.19851220110

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Coordinate transformations in the Feynman-Kac formula are considered. The contractive semigroup with transformed coordinates is expressed by an integral with a correspondingly transformed diffusion measure. This representation is used for N-body systems transformed to clustered Jacobi coordinates implying a decomposition of the Feynman-Kac formula.
A generalized Feynman-Kac formula based on the Wiener measure is presented. Within the setting of a quantum particle in an electromagnetic field it yields the standard Feynman-Kac formula for the corresponding Schrödinger semigroup. In this case rigorous criteria for its validity are compiled. Finally, phase-space path-integral representations for more general quantum Hamiltonians are derived. These representations rely on a generalized Lie-Trotter formula which takes care of the operator-ordering multiplicity, but in general is not related to a path measure.
The Feynman formula is a representation of the solution of an evolutionary equation by a limit of some multiple integrals over Cartesian products of the classical configuration space, or of the classical phase space, when the multiplicity of integrals tends to infinity. From the Feynman formula one can deduce the Feynman–Kac formula, i.e. a representation of the solution by an integral over trajectories. In this paper we consider representations, by the Feynman formula, of solutions both of some Schrödinger type equations and of the corresponding diffusion equations. Introduction The first of Feynman's papers [7], on what one now calls Feynman path integrals over trajectories in the configuration space, contains three main observations. Firstly, it is shown that the solution of the Cauchy problem for a Schrödinger equation can be represented by a limit of a sequence of integrals over Cartesian products of the classical configuration space when the multiplicity of integrals tends to infinity. Secondly, the limit is interpreted as an integral over trajectories in the configuration space. And finally, it is noticed that the integrand contains the exponent of the classical action. Feynman's definition of the Feynman path integrals over trajectories in the phase space, which is formulated in his second paper on the subject, has a similar structure but, in contrast to the preceding definition, the Lagrangian in the classical action is expressed through the Hamiltonian function.
This paper is the third in a series devoted to constructing stochastic motions for the two-dimensional NNN-body delta-Bose gas for all integers N≥3N\geq 3N≥3 and establishing the associated Feynman-Kac-type formulas. The main results here prove the Feynman-Kac-type formulas by using the stochastic many-δδδ motions from [7] as the underlying diffusions. The associated multiplicative functionals show a new form and are derived from the analytic solutions of the two-dimensional NNN-body delta-Bose gas obtained in [4]. For completeness, the main theorem includes the formula for N=2N=2N=2, which is a minor modification of the Feynman--Kac-type formula proven in [5] for the relative motions.
The Feynman--Kac formula is a technique, based on functional integration, which allows to perturb a markovian (i.e. positivity and identity preserving) semi--group exptH\\exp tHexptH with suitable continuity properties into a new positivity preserving semi--group whose formal generator has the form H+VH+VH+V and to obtain a fairly explicit representation of the latter in terms of a functional integral.