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

Details A Rigorous Path Integral for Supersymmetric Quantum Mechanics and the Heat Kernel A Rigorous Path Integral for Supersymmetric Quantum Mechanics and the Heat Kernel

2007-05-04

arxiv.org/abs/0705.0638v2

Commun.Math.Phys.284:79-91,2008

Physics

Mathematical Physics

14 pages, LaTeX, no figs

Scientific paper

10.1007/s00220-008-0606-2

In a rigorous construction of the path integral for supersymmetric quantum mechanics on a Riemann manifold, based on B\"ar and Pf\"affle'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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