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Journal of Mathematical Physics

Path integrals in N=2 supersymmetric quantum mechanics

R. A. Corns

Published Jul 1, 1993• 0 citations
Publisher

Abstract

By use of path integrals in N=2 supersymmetric quantum mechanics, a formula for the kernel of the supersymmetric time evolution operator is found. Structurally it is similar to the Feynman–Kac formula used in the path integrals of quantum mechanics. Crucial to the development of this formula is a generalization of the Wiener measure based upon the free supercharges rather than the free Hamiltonian.

DOI
10.1063/1.530092

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By use of path integrals in N=2 supersymmetric quantum mechanics, a formula for the kernel of the supersymmetric time evolution operator is found. Structurally it is similar to the Feynman–Kac formula used in the path integrals of quantum mechanics. Crucial to the development of this formula is a generalization of the Wiener measure based upon the free supercharges rather than the free Hamiltonian.
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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.
Adapting ideas of Daubechies and Klauder [J. Math. Phys. 26, 2239 (1985)] we derive a rigorous continuum path-integral formula for the semigroup generated by a spin Hamiltonian. More precisely, we use spin coherent vectors parametrized by complex numbers to relate the coherent representation of this semigroup to a suitable Schrödinger semigroup on the Hilbert space L2(R2) of Lebesgue square-integrable functions on the Euclidean plane R2. The path-integral formula emerges from the standard Feynman–Kac–Itô formula for the Schrödinger semigroup in the ultradiffusive limit of the underlying Brownian bridge on R2. In a similar vein, a path-integral formula can be constructed for the coherent representation of the unitary time evolution generated by the spin Hamiltonian.
Adapting ideas of Daubechies and Klauder [J. Math. Phys. 26 (1985) 2239] we derive a rigorous continuum path-integral formula for the semigroup generated by a spin Hamiltonian. More precisely, we use spin-coherent vectors parametrized by complex numbers to relate the coherent representation of this semigroup to a suitable Schrödinger semigroup on the Hilbert space L2 (R2) of Lebesgue squareintegrable functions on the Euclidean plane R2. The path-integral formula emerges from the standard Feynman-Kac-Itô formula for the Schrödinger semigroup in the ultra-diffusive limit of the underlying Brownian bridge on R2. In a similar vein, a path-integral formula can be constructed for the coherent representation of the unitary time evolution generated by the spin Hamiltonian.
Techniques for integration in function spaces which are not necessarily vector spaces are presented in the light of DeWitt-Morette’s recent redefinition of path integrals, which does not involve the usual limiting process, and is therefore free from the ambiguities inherent in this approach. General translated Gaussian measures defined by their Fourier transforms are introduced on various path spaces and used to derive generalized moments formulas, and what is essentially the Feynman–Kac formula for the expression of solutions of the Schrödinger equation as functional integrals.