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Journal of Physics A Mathematical and Theoretical

From Feynman–Vernon to Wiener stochastic path integral

Antonio Camurati, Felipe Sobrero, Bruno B. Suassuna, Pedro V. Paraguassú

Published May 28, 2026• 2 citations
Publisher

Abstract

Abstract We establish a formal connection between the Feynman–Vernon path integral formalism for open quantum systems and the Wiener path integral used in classical stochastic dynamics, further exploring its applications. By considering a generalized influence functional in the strong decoherence limit, we demonstrate that integrating over the quantum coherence length leads to stochastic Langevin dynamics. Specifically, under our model, we show that the quantum Feynman differential transforms into an underdamped multiplicative non-Markovian Langevin dynamics. Applying this framework to the Wigner function representation, we show that the system follows a stochastic path interpretable via classical probability theory, allowing for the investigation of decoherence through classical dynamics. Finally, we address the inverse problem: constructing an equivalent quantum influence functional from a given classical Langevin equation.

DOI
10.1088/1751-8121/ae748c

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Abstract We establish a formal connection between the Feynman–Vernon path integral formalism for open quantum systems and the Wiener path integral used in classical stochastic dynamics, further exploring its applications. By considering a generalized influence functional in the strong decoherence limit, we demonstrate that integrating over the quantum coherence length leads to stochastic Langevin dynamics. Specifically, under our model, we show that the quantum Feynman differential transforms into an underdamped multiplicative non-Markovian Langevin dynamics. Applying this framework to the Wigner function representation, we show that the system follows a stochastic path interpretable via classical probability theory, allowing for the investigation of decoherence through classical dynamics. Finally, we address the inverse problem: constructing an equivalent quantum influence functional from a given classical Langevin equation.
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.
In this work, we developed a quantum algorithm for the simulation of non-Markovian quantum dynamics based on the Feynman-Vernon's path integral formulation. The algorithm performs the full path sum and proves to be polynomial either in time or in space, compared to the same classical algorithm, which is exponential in time. In addition, the algorithm has no classical overhead and is equally applicable regardless of whether the temporal entanglement due to non-Markovianity is low or high, making it a unified framework for simulating non-Markovian dynamics in open quantum systems.
The equation of motion for the reduced Wigner function of a system coupled to an external quantum system is presented for the specific case when the external quantum system can be modeled as a set of harmonic oscillators. The result is derived from the Wigner function formulation of the Feynman-Vernon influence functional theory. It is shown how the true self-energy for the equation of motion is connected with the influence functional for the path integral. Explicit expressions are derived in terms of the bare Wigner propagator. Finally, we show under which approximations the resulting equation of motion reduces to the Wigner-Boltzmann equation.
I review the generating function for quantum-statistical mechanics, known as the Feynman-Vernon influence functional, the decoherence functional, or the Schwinger-Keldysh path integral. I describe a probability-conserving iεi\varepsiloniε prescription from a path-integral implementation of Lindblad evolution. I also explain how to generalize the formalism to accommodate out-of-time-ordered correlators (OTOCs), leading to a Larkin-Ovchinnikov path integral. My goal is to provide step-by-step calculations of path integrals associated to the harmonic oscillator.
Abstract Feynman developed the path integral approach to quantum mechanics as an alternative to wave or matrix mechanics that has found fruitful implementation in quantum mechanics, quantum electrodynamics, and statistical physics. Despite its many successes, the usual Feynman path integral formalism suffers from a lack of mathematical rigor. Questions of mathematical rigor still exist, but Mark Kac provided a rigorous path integral approach to quantum mechanics. The Kac formulation for quantum systems is given in terms of Wiener measure for a path integral and employs the Brownian motion distribution. In this paper, we review the Feynman–Kac approach and other stochastic processes related to Brownian motion. We also give an account of the successes and difficulties we faced during our journey through numerical path integral problems involved in physics.
This article has to do with the derivation and solution of the Fokker-Planck equation associated to the momentum-integrated Wigner function of a particle subjected to a harmonic external field in contact with an ohmic thermal bath of quantum harmonic oscillators. The strategy employed is a simplified version of the phenomenological approach of Schramm, Jung, and Grabert of interpreting the operators as c numbers to derive the quantum master equation arising from a twofold transformation of the Wigner function of the entire phase space. The statistical properties of the random noise comes from the integral functional theory of Grabert, Schramm, and Ingold. By means of a single Wigner transformation, a simpler equation than that mentioned before is found. The Wigner function reproduces the known results of the classical limit. This allowed us to rewrite the underdamped classical Langevin equation as a first-order stochastic differential equation with time-dependent drift and diffusion terms.