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A Feynman-Kac Formula for Anticommuting Brownian Motion

S Leppard, A Rogers

Published Jan 1, 2000• 4 citations
OpenAlex

Abstract

Motivated by application to quantum physics, anticommuting analogues of Wiener measure and Brownian motion are constructed. The corresponding Ito integrals are defined and the existence and uniqueness of solutions to a class of stochastic differential equations is established. This machinery is used to provide a Feynman-Kac formula for a class of Hamiltonians. Several specific examples are considered.

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Motivated by application to quantum physics, anticommuting analogues of Wiener measure and Brownian motion are constructed. The corresponding Ito integrals are defined and the existence and uniqueness of solutions to a class of stochastic differential equations is established. This machinery is used to provide a Feynman-Kac formula for a class of Hamiltonians. Several specific examples are considered.
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 stochastic calculus constructed in [2] for fermion Brownian motion is augmented through the inclusion of stochastic integration with respect to the gauge process. The solutions of certain non-commutative stochastic differential equations are used to construct dilations of contraction semigroups on a Hilbert space \mathfrak h_0 and of uniformly continuous, completely positive semigroups on \boldsymbol B(\mathfrak h_0) . Finally we construct a fermion analogue of the classical Poisson process and investigate some of its properties.
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.