  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.
