Wiener Integration for Quantum Systems: A Unified Approach to 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. Actually, in the Wiener integral the things are much simpler. [46] 1 The Feynman–Kac formula, revisited More than seventy years of Wiener’s path integration have been most gratifying for both mathematicians and theoretical physicists. Originally constructed as a mathematical model for the phenomenon of Brownian motion, it nowadays plays a major rôle also in polymer and quantum physics, and still is fundamental to the theory of general stochastic processes [39]. The importance of Wiener’s measure [45] for quantum physics became clear soon after Feynman’s stimulating paper [16] when Kac [24, 25] identified it as the key to a probabilistic representation of Schrödinger semigroups, these days called the Feynman–Kac formula [42, 20, 40]. In this section we intend to survey essentials of this formula and some of its relatives. In doing so, we leave aside most technicalities and subtleties which we believe to be of secondary importance for applications, in particular when they tend to obscure the simplicity of the underlying ideas. ∗Partially based on a plenary lecture given by H. L. at the international conference on path integrals, Dubna, Russia, May 27–31, 1996; in: Path integrals: Dubna ’96, eds. V. S. Yarunin and M. A. Smondyrev, JINR E96–321, ISBN: 5–85165–451–1, Dubna 1996, pp. 95–106.