Renormalized Integrals and a Path Integral Formula for the Heat Kernel on a Manifold

TL;DR

This paper introduces renormalized integrals as a generalization of measure integrals and applies them to derive a path integral formula for the heat kernel on compact Riemannian manifolds, extending existing methods.

Contribution

It develops a new framework of renormalized integrals and uses it to establish a path integral representation for the heat kernel on manifolds.

Findings

Defined renormalized integrals via approximation by measure spaces

Derived a path integral formula for the heat kernel of generalized Laplace operators

Extended the path integral approach to a broad class of self-adjoint operators on manifolds

Abstract

We introduce renormalized integrals which generalize conventional measure theoretic integrals. One approximates the integration domain by measure spaces and defines the integral as the limit of integrals over the approximating spaces. This concept is implicitly present in many mathematical contexts such as Cauchy's principal value, the determinant of operators on a Hilbert space and the Fourier transform of an $L^p$-function. We use renormalized integrals to define a path integral on manifolds by approximation via geodesic polygons. The main part of the paper is dedicated to the proof of a path integral formula for the heat kernel of any self-adjoint generalized Laplace operator acting on sections of a vector bundle over a compact Riemannian manifold.