2016/05/31 by Dana Fine, Stephen Sawin · 1 citation
Mathematics · Physics and Astronomy · #hep-th #math-ph #math.MP #msc:51P05 #msc:58J20 #msc:81T60
paper · pdf · doi:10.1063/1.4973368
published as Journal of Mathematical Phsyics, Volume 58, number 1. Jan 2017 · 36 pages, hyperref, To Appear in Journal of Mathematical Physics
arxiv created 2017/01/05 · arxiv updated 2017/01/11
Feynman's time-slicing construction approximates the path integral by a product, determined by a partition of a finite time interval, of approximate propagators. This paper formulates general conditions to impose on a short-time approximation to the propagator in a general class of imaginary-time quantum mechanics on a Riemannian manifold which ensure these products converge. The limit defines a path integral which agrees pointwise with the heat kernel for a generalized Laplacian. The result is a rigorous construction of the propagator for supersymmetric quantum mechanics, with potential, as a path integral. Further, the class of Laplacians includes the square of the twisted Dirac operator, which corresponds to an extension of N=1/2 supersymmetric quantum mechanics. General results on the rate of convergence of the approximate path integrals suffice in this case to derive the local version of the Atiyah-Singer index theorem.