Path Integrals on a Compact Manifold with Non-negative Curvature

Abstract

A typical path integral on a manifold, $M$ is an informal expression of the form \frac{1}{Z}\int_{σ\in H(M)} f(σ) e^{-E(σ)}\mathcal{D}σ, \nonumber where $H(M)$ is a Hilbert manifold of paths with energy $E(\sigma )<\infty$, $f$ is a real valued function on $H(M)$, $\mathcal{D}\sigma$ is a \textquotedblleft Lebesgue measure \textquotedblright and $Z$ is a normalization constant. For a compact Riemannian manifold $M$, we wish to interpret $\mathcal{D}\sigma$ as a Riemannian \textquotedblleft volume form \textquotedblright over $H(M)$, equipped with its natural $G^1$ metric. Given an equally spaced partition, $\mathcal{P}$ of $[0,1],$ let $H_{\mathcal{P}%}(M)$ be the finite dimensional Riemannian submanifold of $H(M)$ consisting of piecewise geodesic paths adapted to $\mathcal{P}.$ Under certain curvature restrictions on $M,$ it is shown that \[ \frac{1}{Z_{\mathcal{P}}}e^{-{1/2}E(σ)}dVol_{H_{\mathcal{P}}% }(σ)\toρ(σ)dν(σ)\text{as}\mathrm{mesh}% ({\mathcal{P}})\to0, \] where $Z_{\mathcal{P}}$ is a \textquotedblleft normalization\textquotedblright constant, $E:H(M)\to [0,\infty )$ is the energy functional, $Vol_{H_{\mathcal{P}%}}$ is the Riemannian volume measure on $H_{\mathcal{P}}(M),$ $\nu$ is Wiener measure on continuous paths in $M,$ and $\rho$ is a certain density determined by the curvature tensor of $M.$