Finite dimensional approximations to Wiener measure and path integral formulas on manifolds
Loading...
Date
Authors
Journal Title
Journal ISSN
Volume Title
Publisher
Abstract
Description
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 $H^{1}$ and a $L^{2}$ type Riemannian structures $G^i_P$. It is proved that as the mesh of the partition tends to $0$,
$$ 1/Z_P^i e^{- 1/2 E(σ)} Vol_{G^i_P}(σ) \to ρ_i(σ)ν(σ) $$
where $E(σ)$ is the energy of the piecewise geodesic path $σ$, and for $i=0$ and $1$, $Z_P^i$ is a ``normalization'' constant, $Vol_{G^i_P}$ is the Riemannian volume form relative $G^i_P$, and $ν$ is Wiener measure on paths on $M$. Here $ρ_1 = 1$ and
$$ ρ_0 (σ) = \exp( -1/6 \int_0^1 Scal(σ(s))ds ) $$
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.
48 pages, latex2e using amsart and amssymb
48 pages, latex2e using amsart and amssymb