An approximation to Wiener measure and quantization of the Hamiltonian on manifolds with non-positive sectional curvature

This paper gives a rigorous interpretation of a Feynman path integral on a Riemannian manifold M with non-positive sectional curvature. An L2 Riemannian metric GP is given on the space of piecewise geodesic paths HP(M) adapted to the partition P of [0,1], whence a finite-dimensional approximation of Wiener measure is developed. It is proved that, as mesh(P)→0, the approximate Wiener measure converges in an L1 sense to the measure exp{−2+3203∫01Scal(σ(s))ds}dν(σ) on the Wiener space W(M) with Wiener measure ν. This gives a possible prescription for the path integral representation of the quantized Hamiltonian, as well as yielding such a result for the natural geometric approximation schemes originating in Andersson and Driver (1999) [3] and followed by Lim (2007) [34].