Finite Dimensional Approximations to Wiener Measure and Path Integral Formulas on Manifolds

Certain natural geometric approximation schemes are developed for Wiener measure on a compact Riemannian manifold. These approximations closely mimic the informal path integral formulas used in the physics literature for representing the heat semi-group on Riemannian manifolds. The path space is approximated by finite dimensional manifolds HP(M) consisting of piecewise geodesic paths adapted to partitions P of [0, 1]. The finite dimensional manifolds HP(M) carry both an H1 and a L2 type Riemannian structures, G1P and G0P, respectively. It is proved that (1/ZiP) e−(1/2) E(σ) d VolGiP(σ)→ρi(σ) dν(σ) as mesh (P)→0, where E(σ) is the energy of the piecewise geodesic path σ∈HP(M), and for i=0 and 1, ZiP is a “normalization” constant, VolGiP is the Riemannian volume form relative to GiP, and ν is Wiener measure on paths on M. Here ρ1(σ)≡1 and ρ0(σ)=exp(−16 ∫10 Scal(σ(s)) ds) where Scal is the scalar curvature of M. These results are also shown to imply the well known integration by parts formula for the Wiener measure.