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

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Abstract

In a rigorous construction of the path integral for supersymmetric quantum mechanics on a Riemann manifold, based on Bär and Pfäffle’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.

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Authors and Affiliations

  1. Department of Mathematics, University of Massachusetts Dartmouth, North Dartmouth, MA, 02747, USA

    Dana S. Fine

  2. Department of Mathematics and Computer Science, Fairfield University, Fairfield, CT, 06824, USA

    Stephen F. Sawin

Authors
  1. Dana S. Fine
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  2. Stephen F. Sawin
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Correspondence to Dana S. Fine.

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Communicated by A. Connes

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Fine, D.S., Sawin, S.F. A Rigorous Path Integral for Supersymmetic Quantum Mechanics and the Heat Kernel. Commun. Math. Phys. 284, 79–91 (2008). https://doi.org/10.1007/s00220-008-0606-2

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  • DOI: https://doi.org/10.1007/s00220-008-0606-2

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