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

Abstract

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) → 0, the approximate Wiener measure converges in a L 1 sense to the measure exp{− 2+ √ 3 20 √ 3 1 0 Scal(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 [3] and followed by [34].

Heat Kernels as Path Integrals

“…In the case of the continuous L 2 metric, the pre-factor in front of the scalar curvature becomes (3 + 2 √ 3)/60 instead of 1/6 and one gets an again different τ -dependent normalization factor (see Thm. 2.1 in [Lae13]). Conceptually, the L 2 metrics are somewhat less natural compared to the H 1 metric, since the action functional S 0 is not defined on L 2 .…”

A finite dimensional approximation to pinned Wiener measure on some symmetric spaces

“…An analogous theory on general manifolds was also developed, see for example Pinsky [20], Atiyah [2], Bismut [3], Andersson and Driver [1] and references therein. In [1], followed by [19] and [18], the finite dimensional approximation problem is viewed in its full geometric form by restricting the expression in Eq. (1.3) to finite dimensional sub-manifolds of piecewise geodesic paths on M. Unlike the flat case (M = R d ) where the choice of translation invariant Riemannian metric on path spaces is irrelevant, various Riemannian metrics on approximate path spaces are explored.…”

Stochastic heat equations with values in a Riemannian manifold

“…the associated stochastic heat equation may be interpreted formally as and m i=1 σ i (u)ξ i is a suitable "white noise on loop space". By Andersson-Driver's work in [2], we know that there exists an explicit relation between the Langevin energy E(u) and the Wiener (Brownian bridge) measure (see also [42,52,61]). In [2], the Wiener (Brownian bridge) measure µ has been interpreted as the limit of a natural approximation of the measure exp(− 1 2 E(u))Du, where Du denotes a 'Lebesgue' like measure on path space.…”