Coherent-State Path Integrals and Brownian Motions

Abstract

A stochastic approach to the rigorous foundation of the coherent-state (phase-space) path integral is given. Stochastic integrals and some generalizations of the Feynman–Kac theorem are used for this purpose. In this approach, quantum mechanics is described in terms of the Fock–Bargmann representation; a classical Hamiltonian is related to the corresponding quantum Hamiltonian on the Fock–Bargmann space, seen as a Hilbert subspace of \(L^{2}({\bf R}^{2})\). The coherent-state path integral is realized as a conditional expectation of a stochastic process defined by the exponential of the Fisk–Stratonovich integral of the fundamental 1-form along a path of Brownian motion on the phase space \({\bf R}^{2}\).