Quantum Feynman-Kac perturbations

TL;DR

This paper develops noncommutative Feynman-Kac formulas using quantum stochastic processes, expanding the mathematical framework for quantum perturbations and generalizing classical and unitarily implemented flow results.

Contribution

It introduces a theory for perturbing quantum stochastic flows with multiplier cocycles constructed via quantum stochastic differential equations.

Findings

Established theory for perturbing quantum stochastic flows.

Characterized multiplier cocycles under separability or Markov regularity.

Generalized classical Brownian motion and unitarily implemented flow results.

Abstract

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 characterised under alternative assumptions of separability or Markov regularity. Our results generalise those obtained using classical Brownian motion on the one hand, and results for unitarily implemented flows on the other.