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 [Formula: see text] where H(M) is a Hilbert manifold of paths with energy E(σ) < ∞, f is a real-valued function on H(M), [Formula: see text] is a "Lebesgue measure" and Z is a normalization constant. For a compact Riemannian manifold M, we wish to interpret [Formula: see text] as a Riemannian "volume form" over H(M), equipped with its natural G1 metric. Given an equally spaced partition, [Formula: see text] of [0, τ], let [Formula: see text] be the finite dimensional Riemannian submanifold of H(M) consisting of piecewise geodesic paths adapted to [Formula: see text]. Under certain curvature restrictions on M, it is shown that [Formula: see text] where [Formula: see text] is a "normalization" constant, E : H(M) → [0,∞) is the energy functional, [Formula: see text] is the Riemannian volume measure on [Formula: see text], ν is Wiener measure on continuous paths in M, and ρ is a certain density determined by the curvature tensor of M.