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_{\sigma \in H(M)} f(\sigma) e^{-E(\sigma)}\mathcal{D}\sigma, \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 \[ \frac{1}{Z_{{\mathcal{P}}}}e^{-{1/2}E(\sigma)}dVol_{H_{{\mathcal{P}}}% }(\sigma)\to\rho(\sigma)d\nu(\sigma)\text{as}\mathrm{mesh}% ({\mathcal{P}})\to0, \] where $Z_{{\mathcal{P}}}$ is a \textquotedblleft normalization\textquotedblright constant, $E:H(M) \to\lbrack0,\infty)$ 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 $M.$