Finite Dimensional Approximations to Wiener Measure and Path Integral Formulas on Manifolds
✍ Scribed by Lars Andersson; Bruce K Driver
- Publisher
- Elsevier Science
- Year
- 1999
- Tongue
- English
- Weight
- 408 KB
- Volume
- 165
- Category
- Article
- ISSN
- 0022-1236
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✦ Synopsis
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 H P (M) consisting of piecewise geodesic paths adapted to partitions P of [0, 1]. The finite dimensional manifolds H P (M) carry both an H 1 and a L 2 type Riemannian structures, G 1 P and G 0 P , respectively. It is proved that
as mesh(P) Ä 0, where E(_) is the energy of the piecewise geodesic path _ # H P (M), and for i=0 and 1, Z i P 6 1 0 Scal(_(s)) ds) where Scal is the scalar curvature of M. These results are also shown to imply the well known integration by parts formula for the Wiener measure.