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

Feynman-Kac formula in phase space with application to coherent-state transitions

Cécile DeWitt-Morette, Tian-Rong Zhang

Published Nov 15, 1983• 7 citations
Publisher

Abstract

A Feynman-Kac formula for transitions between initial and final states which are linear superpositions of momentum and position states is given by Eq. (10) (phase-space version) or equivalently by Eq. (73) (configuration-space version). The integrators are given explicity in terms of the Jacobi fields of the system (67).
DOI
10.1103/physrevd.28.2517

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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.
Inspired by the formalism that relates the star-exponential with the quantum propagator for bosonic systems, in this work we introduce the analogous extension for the fermionic case. In particular, we analyse the problem of calculating the star-exponential (i.e., the symbol of the evolution operator) for Fermi systems within the deformation quantization program. Grassmann variables and coherent states are considered in order to obtain a closed-form expression for the fermionic star-exponential in terms of its associated propagator. As a primary application, a fermionic version of the Feynman-Kac formula is derived within this formalism, thus allowing a straightforward calculation of the ground state energy in phase space. Finally, the method is validated by successfully applying it to the simple harmonic and driven Fermi oscillators, for which the results developed here provide a powerful alternative computational tool for the study of fermionic systems.
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.
In this work is introduced the harmonic oscillator process that in probability is known as Ornstein-Uhlenbeck process or velocities process of Ornstein-Uhlenbeck and is established that the mathematical expectation of certain functionals of this process coincides with the expected value of certain observables in the fundamental state or basis state to the oscillator and with it is found a relation between the spaces of classical and quantum probability. Furthermore is established a certain algebra of operators that gives rise to the dynamics of quantum harmonic oscillator and to consider certain class of perturbations of the Hamiltonian corresponding to their harmonic oscillator is obtained a version of the Feynman-Kac formula with a dilatation generated by the unitary group of translations for a co-cycle. Finally are given some generalizations of the Feynman-Kac formula.