Discrete approximations to integrals over unparametrized paths

TL;DR

This paper develops and proves the convergence of two discrete approximation methods for measures on unparametrized path spaces, with applications in one-dimensional gravity and scalar fields.

Contribution

It introduces piecewise linear and hypercubic approximations for measures on unparametrized paths and establishes their weak convergence.

Findings

Proved convergence of discrete path measure approximations

Defined sets of unparametrized paths analogous to cylinder sets

Evaluated integrals using Dirichlet propagators

Abstract

We discuss measures on spaces of unparametrized paths related to the Wiener measure. These measures arise naturally in the study of one-dimensional gravity coupled to scalar fields. Two kinds of discrete approximations are defined, the piecewise linear and the hypercubic approximations. The convergence of these approximations in the sense of weak convergence of measures is proven. We describe a family of sets of unparametrized paths that are analogous to cylinder sets of parametrized paths. Integrals over some of these sets are evaluated in terms of Dirichlet propagators in bounded regions.