Finite dimensional approximations to Wiener measure and path integral formulas on manifolds

1998/07/19 by Lars Andersson, Andersson, Lars, Bruce K. Driver +1 · 1 citation
Mathematics · #58D30 (Primary) 58D20 (Secondary) #60H07 #Differential Geometry (math.DG) #FOS: Mathematics #Probability (math.PR) #math.DG #math.PR #msc:58D20 #msc:58D30 #msc:60H07

paper · pdf · doi:10.48550/arxiv.math/9807098

48 pages, latex2e using amsart and amssymb

arxiv created 1998/07/19 · arxiv updated 2009/11/30

Abstract

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 consisting of piecewise geodesic paths adapted to partitions P of [0,1]. The finite dimensional manifolds of piecewise geodesics carry both an H¹ and a L² type Riemannian structures Gⁱ_P. It is proved that as the mesh of the partition tends to 0, 1/Z_Pⁱ e^{- 1/2 E(σ)} Vol_GⁱP(σ) → ρ_i(σ)ν(σ) where E(σ) is the energy of the piecewise geodesic path σ, and for i=0 and 1, Z_Pⁱ is a ``normalization'' constant, Vol_GⁱP is the Riemannian volume form relative Gⁱ_P, and ν is Wiener measure on paths on M. Here ρ₁ = 1 and ρ₀ (σ) = exp( -1/6 ∫₀¹ Scal(σ(s))ds ) where Scal is the scalar curvature of M. These results are also shown to imply the well know integration by parts formula for the Wiener measure.

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• Wiener distribution on holonomy groups

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