An Approximation to Wiener Measure and Quantization of the Hamiltonian on Manifolds with Non-positive Sectional Curvature

TL;DR

This paper rigorously interprets a Feynman path integral on manifolds with non-positive curvature by approximating Wiener measure through finite-dimensional paths, leading to a measure that incorporates scalar curvature and supports quantization of the Hamiltonian.

Contribution

It provides a rigorous approximation scheme for Wiener measure on manifolds with non-positive curvature, connecting path integrals to geometric quantization.

Findings

Finite-dimensional approximation of Wiener measure converges as mesh size tends to zero.

The limiting measure incorporates scalar curvature, linking geometry to quantum measures.

Supports a path integral formulation for quantized Hamiltonians on curved manifolds.

Abstract

This paper gives a rigorous interpretation of a Feynman path integral on a Riemannian manifold M with non-positive sectional curvature. A $L^2$ Riemannian metric $G_P$ is given on the space of piecewise geodesic paths $H_P(M)$ adapted to the partition $P$ of $[0,1]$, whence a finite-dimensional approximation of Wiener measure is developed. It is proved that, as $mesh(P) \to 0$, the approximate Wiener measure converges in a $L^1$ sense to the measure $e^{-\frac{2 + \sqrt{3}}{20\sqrt{3}} \int_0^1 Scal(\sigma(s)) ds} d\nu(\sigma)$ on the Wiener space $W(M)$ with Wiener measure $\nu$. 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 [L. A. Andersson and B. K. Driver, J. Funct. Anal. 165 (1999), no. 2, 430-498] and followed by [Adrian P. C. Lim, Rev. Math. Phys. 19 (2007), no. 9, 967-1044].