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Journal of Physics A Mathematical and General

Stochastic calculus in superspace. I. Supersymmetric Hamiltonians

Alan E. E. Rogers

Published Jan 21, 1992• 15 citations
Publisher

Abstract

Various analytic results which combine fermionic Brownian motion with stochastic integration are described, and it is shown that a wide class of stochastic differential equations in superspace have solutions. Such solutions are then used to derive a Feynman-Kac formula for a supersymmetric system in terms of the supercharge whose square is the Hamiltonian of the system. This is achieved by introducing superpaths parametrized by a commuting and an anticommuting time variable.

DOI
10.1088/0305-4470/25/2/024

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Various analytic results which combine fermionic Brownian motion with stochastic integration are described, and it is shown that a wide class of stochastic differential equations in superspace have solutions. Such solutions are then used to derive a Feynman-Kac formula for a supersymmetric system in terms of the supercharge whose square is the Hamiltonian of the system. This is achieved by introducing superpaths parametrized by a commuting and an anticommuting time variable.
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.
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 paper is the third in a series devoted to constructing stochastic motions for the two-dimensional NNN-body delta-Bose gas for all integers N≥3N\geq 3N≥3 and establishing the associated Feynman-Kac-type formulas. The main results here prove the Feynman-Kac-type formulas by using the stochastic many-δδδ motions from [7] as the underlying diffusions. The associated multiplicative functionals show a new form and are derived from the analytic solutions of the two-dimensional NNN-body delta-Bose gas obtained in [4]. For completeness, the main theorem includes the formula for N=2N=2N=2, which is a minor modification of the Feynman--Kac-type formula proven in [5] for the relative motions.