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

## TL;DR

This paper introduces a finite dimensional approximation method for the Wiener measure on symmetric spaces of non-compact type, aiding in rigorous interpretation of path integrals in mathematical physics.

## Contribution

It develops a new approximation scheme for Brownian bridges on symmetric spaces, extending finite dimensional techniques to non-compact settings.

## Key findings

- Provides a convergent approximation of Wiener measure on symmetric spaces
- Enables rigorous analysis of path integrals in geometric contexts
- Facilitates applications in quantum field theory and index theorems

## Abstract

Path integrals developed by Richard Feynman have been an important tool in Physics in studying quantum field theory. In mathematics, it has also been widely used in providing formal proofs in the study of Index theorem and asymptotic behaviors of heat kernels. Finite dimensional approximations to path integral representations give a way to interpret path integrals and make the formal argument rigorous. The central idea is to restrict a path integral to smaller path spaces where everything is well defined and then to interpret the original path integral as a "limit" when smaller path spaces "exhaust" the full path space (Wiener space). In this paper I will present a finite dimensional approximation to Brownian bridge on a symmetric space of non---compact type using pinned piecewise geodesic space adapted to partitions of time.