Path Integrals on Manifolds with Boundary

Matthias Ludewig

doi:10.1007/s00220-017-2915-9

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.

Highlights

First let M be a compact Riemannian manifold without boundary and let V ∈ C∞(M)
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
Our proof works in the case that ∂ M = ∅. This gives a new proof of the path integral formula (1.4) in the closed case, which seems to be simpler than the ones existing in the literature: It neither uses stochastic analysis nor knowledge of the short-time asymptotics of the heat kernel, only basic properties of the geodesic flow

Summary

Introduction

First let M be a compact Riemannian manifold without boundary and let V ∈ C∞(M). For u0 ∈ L2(M), let u be the solution to the heat equation. Our proof works in the case that ∂ M = ∅ This gives a new proof of the path integral formula (1.4) in the closed case, which seems to be simpler than the ones existing in the literature: It neither uses stochastic analysis nor knowledge of the short-time asymptotics of the heat kernel, only basic properties of the geodesic flow (to which the broken billiard flow reduces in the closed case). We first review some basic results on vector-valued Laplace type operators acting on vector bundles, and we introduce the class of boundary conditions considered in this paper. Afterwards, we introduce the broken billiard flow on a manifold with boundary, reflected geodesics and the space Hxre;τfl(M). We formulate and prove our results on time-slicing path integrals

Involutive Boundary Conditions and the Heat Equation

Reflected geodesics and the broken billiard flow

Reflected path spaces

Reflecting path integrals