Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1202.5241 · Communications on Stochastic Analysis

A vacuum-adapted approach to quantum Feynman-Kac formulae

Alexander C. R. Belton, J. Martin Lindsay, Adam Skalski

Published Mar 1, 2012• 1 citation
arXivHTMLPDF

Abstract

The vacuum-adapted formulation of quantum stochastic calculus is employed to perturb expectation semigroups via a Feynman-Kac formula.This gives an alternative perspective on the perturbation theory for quantum stochastic flows that has recently been developed by the authors.
DOI
10.31390/cosa.6.1.09

Related papers

Quantum Feynman-Kac perturbations

Alexander C. R. Belton, J. Martin Lindsay, Adam Skalski

We develop fully noncommutative Feynman–Kac formulae by employing quantum stochastic processes. To this end, we establish some theory for perturbing quantum stochastic flows on von Neumann algebras by multiplier cocycles. Multiplier cocycles are constructed via quantum stochastic differential equations whose coefficients are driven by the flow. The resulting class of cocycles is characterized under alternative assumptions of separability or Markov regularity. Our results generalize those obtained using classical Brownian motion on the one hand, and results for unitarily implemented flows on the other.

arXiv:1202.6489•Journal of the London Mathematical Society•Oct 30, 2013•3 citations

Feynman–Kac perturbation of $$C^*$$ quantum stochastic flows

Alexander C. R. Belton, Stephen J. Wills

arXiv:2407.06732•Indian Journal of Pure and Applied Mathematics•Jul 6, 2024•1 citation

A review on Feynman–Kac path integrals and their connection to various stochastic processes

Sumita Datta

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.

Monte Carlo Methods and Applications•Oct 23, 2026•0 citations

Wiener Integration for Quantum Systems: A Unified Approach to the Feynman-Kac formula

Bernhard G. Bodmann, Hajo Leschke, Simone Warzel

A generalized Feynman-Kac formula based on the Wiener measure is presented. Within the setting of a quantum particle in an electromagnetic field it yields the standard Feynman-Kac formula for the corresponding Schrödinger semigroup. In this case rigorous criteria for its validity are compiled. Finally, phase-space path-integral representations for more general quantum Hamiltonians are derived. These representations rely on a generalized Lie-Trotter formula which takes care of the operator-ordering multiplicity, but in general is not related to a path measure.

arXiv:quant-ph/9703031•Mar 18, 1997•0 citations

On the quantum Feynmann-Kac formula

Luigi Accardi

The Feynman--Kac formula is a technique, based on functional integration, which allows to perturb a markovian (i.e. positivity and identity preserving) semi--group $\\exp tH$ with suitable continuity properties into a new positivity preserving semi--group whose formal generator has the form $H+V$ and to obtain a fairly explicit representation of the latter in terms of a functional integral.

•Jan 1, 1978•4 citations

The Feynman-Kac Formula

Gerald W Johnson, Michel L Lapidus

Abstract Be this as it may, the formal analogy between [formula (7.4.2) above] and integrals appearing in Wiener’s theory is striking and since Wiener’s theory is rigorously founded Feynman’s heuristic connection between the Schri:idinger equation and the path integral can be made into an unassailable theorem [namely, the Feynman–Kac formula].

•Mar 16, 2000•0 citations
We develop fully noncommutative Feynman–Kac formulae by employing quantum stochastic processes. To this end, we establish some theory for perturbing quantum stochastic flows on von Neumann algebras by multiplier cocycles. Multiplier cocycles are constructed via quantum stochastic differential equations whose coefficients are driven by the flow. The resulting class of cocycles is characterized under alternative assumptions of separability or Markov regularity. Our results generalize those obtained using classical Brownian motion on the one hand, and results for unitarily implemented flows on the other.
Feynman–Kac perturbation of C∗C^*C∗ quantum stochastic flows
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.
A generalized Feynman-Kac formula based on the Wiener measure is presented. Within the setting of a quantum particle in an electromagnetic field it yields the standard Feynman-Kac formula for the corresponding Schrödinger semigroup. In this case rigorous criteria for its validity are compiled. Finally, phase-space path-integral representations for more general quantum Hamiltonians are derived. These representations rely on a generalized Lie-Trotter formula which takes care of the operator-ordering multiplicity, but in general is not related to a path measure.
The Feynman--Kac formula is a technique, based on functional integration, which allows to perturb a markovian (i.e. positivity and identity preserving) semi--group exptH\\exp tHexptH with suitable continuity properties into a new positivity preserving semi--group whose formal generator has the form H+VH+VH+V and to obtain a fairly explicit representation of the latter in terms of a functional integral.