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

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

1998-07-19

arxiv.org/abs/math/9807098v1

Mathematics

Differential Geometry

48 pages, latex2e using amsart and amssymb

Scientific paper

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 $0$, $$ 1/Z_P^i e^{- 1/2 E(\sigma)} Vol_{G^i_P}(\sigma) \to \rho_i(\sigma)\nu(\sigma) $$ where $E(\sigma )$ is the energy of the piecewise geodesic path $\sigma$, 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 $\nu$ is Wiener measure on paths on $M$. Here $\rho_1 = 1$ and $$ \rho_0 (\sigma) = \exp( -1/6 \int_0^1 Scal(\sigma(s))ds ) $$ 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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