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Many introductory courses in quantum mechanics include Feynman's time-slicingdefinition of the path integral, with a complete derivation of the propagatorin the simplest of cases. However, attempts to generalize this, for instance tonon-quadratic potentials, encounter formidable analytic issues in showing thesuccessive approximations in fact converge to a definite expression for thepath integral. The present work describes how to carry out the analysis for aclass 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 adefinition of the path integral which agrees with the imaginary-time Feynmanpropagator. With this as the definition, the steepest-descent approximation tothe path integral for twisted $N=1/2$ supersymmetric quantum mechanics isprovably correct. These results complete a new proof of the Atiyah-Singer indextheorem for the twisted Dirac operator.

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