Perfect discretization of path integrals

TL;DR

This paper introduces an improvement method for discretized path integrals in reparametrization invariant systems, ensuring discretization independence and proper projection onto physical states, with implications for quantum gravity.

Contribution

It presents a novel improvement procedure for discretized path integrals that preserves reparametrization invariance and achieves discretization independence.

Findings

The improvement procedure converges to fixed points.

Discretization invariance acts as a projector onto physical states.

The method enhances the well-definedness of path integrals in gravity models.

Abstract

In order to obtain a well-defined path integral one often employs discretizations. In the case of General Relativity these generically break diffeomorphism symmetry, which has severe consequences since these symmetries determine the dynamics of the corresponding system. In this article we consider the path integral of reparametrization invariant systems as a toy example and present an improvement procedure for the discretized propagator. Fixed points and convergence of the procedure are discussed. Furthermore we show that a reparametrization invariant path integral implies discretization independence and acts as a projector onto physical states.