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On the quantum Feynmann-Kac formula

Luigi Accardi

Published Jan 1, 1978• 4 citations
OpenAlex

Abstract

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 $\\exp tH$ with suitable continuity properties into a new positivity preserving semi--group whose formal generator has the form $H+V$ and to obtain a fairly explicit representation of the latter in terms of a functional integral.

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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.
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.