Rigorous path integrals for supersymmetric quantum mechanics: completing the path integral proof of the index theorem

TL;DR

This paper rigorously defines path integrals for supersymmetric quantum mechanics on Riemannian manifolds, proving convergence of approximations and completing a new proof of the Atiyah-Singer index theorem.

Contribution

It provides a rigorous analysis of path integrals for a broad class of Lagrangians, including supersymmetric cases, and completes a new proof of the index theorem.

Findings

Successive time-slicing approximations converge for the class of Lagrangians considered.

The defined path integral matches the imaginary-time Feynman propagator.

The results establish the correctness of steepest-descent approximation in supersymmetric quantum mechanics.

Abstract

Many introductory courses in quantum mechanics include Feynman's time-slicing definition of the path integral, with a complete derivation of the propagator in the simplest of cases. However, attempts to generalize this, for instance to non-quadratic potentials, encounter formidable analytic issues in showing the successive approximations in fact converge to a definite expression for the path integral. The present work describes how to carry out the analysis for a class of Lagrangians broad enough to include the evolution, in imaginary time, of spinors constrained to live on a Riemannian manifold. For these Lagrangians, the successive time-slicing approximations converge. The limit provides a definition of the path integral which agrees with the imaginary-time Feynman propagator. With this as the definition, the steepest-descent approximation to the path integral for twisted $N=1/2$ supersymmetric quantum mechanics is provably correct. These results complete a new proof of the Atiyah-Singer index theorem for the twisted Dirac operator.