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The Feynman-Kac Formula

Gerald W Johnson, Michel L Lapidus

Published Mar 16, 2000• 0 citations
Publisher

Abstract

Abstract Be this as it may, the formal analogy between [formula (7.4.2) above] and integrals appearing in Wiener’s theory is striking and since Wiener’s theory is rigorously founded Feynman’s heuristic connection between the Schri:idinger equation and the path integral can be made into an unassailable theorem [namely, the Feynman–Kac formula].

DOI
10.1093/oso/9780198535744.003.0012

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Abstract Be this as it may, the formal analogy between [formula (7.4.2) above] and integrals appearing in Wiener’s theory is striking and since Wiener’s theory is rigorously founded Feynman’s heuristic connection between the Schri:idinger equation and the path integral can be made into an unassailable theorem [namely, 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.
Abstract Feynman developed the path integral approach to quantum mechanics as an alternative to wave or matrix mechanics that has found fruitful implementation in quantum mechanics, quantum electrodynamics, and statistical physics. Despite its many successes, the usual Feynman path integral formalism suffers from a lack of mathematical rigor. Questions of mathematical rigor still exist, but Mark Kac provided a rigorous path integral approach to quantum mechanics. The Kac formulation for quantum systems is given in terms of Wiener measure for a path integral and employs the Brownian motion distribution. In this paper, we review the Feynman–Kac approach and other stochastic processes related to Brownian motion. We also give an account of the successes and difficulties we faced during our journey through numerical path integral problems involved in physics.