A Rigorous Path Integral for Supersymmetric Quantum Mechanics and the Heat Kernel

TL;DR

This paper rigorously constructs the path integral for supersymmetric quantum mechanics on a Riemann manifold, linking it to the heat kernel of the Laplacian on forms through a geometric and analytical framework.

Contribution

It provides a rigorous mathematical formulation of the path integral for supersymmetric quantum mechanics using geodesic paths and Thom forms, connecting it to the heat kernel.

Findings

Path integral approximated by integrals of Thom forms on geodesic path spaces

Limit of the integral yields the supertrace of the heat kernel

Establishes a rigorous link between path integrals and heat kernels in supersymmetric QM

Abstract

In a rigorous construction of the path integral for supersymmetric quantum mechanics on a Riemann manifold, based on B\"ar and Pf\"affle's use of piecewise geodesic paths, the kernel of the time evolution operator is the heat kernel for the Laplacian on forms. The path integral is approximated by the integral of a form on the space of piecewise geodesic paths which is the pullback by a natural section of Mathai and Quillen's Thom form of a bundle over this space. In the case of closed paths, the bundle is the tangent space to the space of geodesic paths, and the integral of this form passes in the limit to the supertrace of the heat kernel.