An Approximation of Wiener Measure on Manifolds with Non-positive Sectional Curvature

An interpretation is given for an informal path integral expression 1Zs∈ WMf se-12 EsDs , where Z is a "normalization" constant, W(M) consists of continuous paths σ on M parametrized on [0, 1], E(σ) is the energy of the path σ, and Ds is Lebesgue type measure. Approximating the path space by finite dimensional manifolds HP (M) consisting of the piecewise geodesic paths adapted to a partition of P of [0, 1], it is proved that when M has non-positive sectional curvature, then as mesh( P ) → 0, 1ZPe- 1/2EsdVol GPs →exp- 2+3203 01Scals s dsdn s. Here ZP is a normalization constant, GP is an L2-metric on HP (M), VolGP is the Riemannian volume measure induced from GP , Scal is the scalar curvature on M, and ν is the Wiener measure on W(M). This allows one to rigorously interpret the heuristic measure given above by Z-1exp-12 Es Ds as the measure exp-2+3 2030 1Scalss ds dns .