Phase-space path integral and Brownian motion

Abstract

We formulate and examine a phase-space path integral representation of a quantum time evolution in terms of Fock–Bargmann spaces and stochastic line integrals. The Fock–Bargmann space \documentclass[12pt]{minimal}\begin{document}${{\cal H}_{\rm FB}}$\end{document}H FB is a closed subspace of \documentclass[12pt]{minimal}\begin{document}$L^2({\cal M})$\end{document}L2(M), where \documentclass[12pt]{minimal}\begin{document}${\cal M}$\end{document}M is the phase space. Let EFB be the orthogonal projection onto \documentclass[12pt]{minimal}\begin{document}${{\cal H}_{\rm FB}}$\end{document}H FB , and h the classical Hamiltonian, then H = EFBhEFB is understood as a quantization of h. The quantum time evolution e−itH is asymptotically represented by the solution of a variant of diffusion equation with the imaginary potential ih. Thus the quantum time evolution can be represented stochastically, in terms of Brownian motions and stochastic integrals.