Path Integrals on Manifolds with Boundary

TL;DR

This paper develops a method to approximate solutions to the heat equation on manifolds with boundary using time-slicing path integrals over geodesic paths, accommodating common boundary conditions.

Contribution

It introduces a novel time-slicing path integral formulation for heat equations on manifolds with boundary, including mixed boundary conditions, extending previous approaches.

Findings

Provides explicit path integral formulas for heat solutions

Shows convergence of path integrals to true solutions

Includes standard boundary conditions like Dirichlet and Neumann

Abstract

We give time-slicing path integral formulas for solutions to the heat equation corresponding to a self-adjoint Laplace type operator acting on sections of a vector bundle over a compact Riemannian manifold with boundary. More specifically, we show that such a solution can be approximated by integrals over finite-dimensional path spaces of piecewise geodesics subordinated to increasingly fine partitions of the time interval. We consider a subclass of mixed boundary conditions which includes standard Dirichlet and Neumann boundary conditions.