Wiener Integration for Quantum Systems: A Unified Approach to the Feynman-Kac Formula
Gaussian linearization on the basis of Wiener integration over operator-valued functionals provides a unifying approach to the probabilistic representation of certain operator semigroups [B. Bodmann, H. Leschke, S. Warzel, in: Path integrals: Dubna’ 96, eds. V. S. Yarunin, M. A. Smondyrev, Dubna 1996, pp. 95–106]. Within the setting of a quantum particle in an electromagnetic field it naturally yields a basis independent version of the standard Feynman-Kac(-Ito) formula for the corresponding Schrödinger semigroup. In this framework even semigroups generated by non-standard Hamiltonians such as for a quantum particle with a spatially dependent mass can be represented by conventional Wiener integrals — in contrast to [B. Gaveau, L. S. Schulman: J. Math. Phys. 30, (1989) 2019]. The approach offers a convenient starting point for estimations and calculations.