From Feynman-Vernon to Wiener Stochastic Path Integral

TL;DR

This work addresses the problem of connecting open-quantum-system dynamics, described by the Feynman-Vernon path integral, with classical stochastic dynamics, described by the Wiener path integral. It introduces a generalized influence functional and, in the strong decoherence limit, shows that the quantum oscillatory measure maps onto a Wiener measure, enabling a Quantum Induced Stochastic Dynamics (QISD) framework. The authors demonstrate that the Wigner function evolves under a stochastic path integral and reduces to a classical probability distribution, with the evolution governed by a stochastic propagator $P[p_\tau,x_\tau|p_0,x_0]$. They also formulate the inverse problem of quantizing a given Langevin equation by constructing an equivalent quantum influence functional, allowing non-Markovian quantum master equations to be derived from classical stochastic dynamics. This bridge provides a rigorous basis for studying decoherence, stochastic energetics, and quantum-classical transitions in non-Markovian environments, with potential applications to quantum thermodynamics and stochastic control of open systems.

Abstract

We establish a direct connection between the Feynman-Vernon path integral formalism for open quantum systems and the Wiener path integral used in classical stochastic dynamics. By considering a generalized influence functional in the strong decoherence limit, we demonstrate that integrating over the quantum coherence length leads to a derivation of stochastic Langevin dynamics. Specifically, we show that the quantum Feynman measure transforms into the stochastic Wiener measure. Applying this framework to the Wigner function representation, we show that the system follows a stochastic path interpretable via classical probability theory. Finally, we address the inverse problem: constructing an equivalent quantum influence functional from a given classical Langevin equation.