Perfect discretization of reparametrization invariant path integrals

TL;DR

This paper develops a method to construct discretization-independent path integrals for reparametrization invariant systems, addressing ambiguities and anomalies, with implications for quantum gravity models like spin foams.

Contribution

It introduces an iterative approach to create discretization-invariant path integrals, ensuring anomaly-free measures and preserving symmetries in reparametrization invariant systems.

Findings

Discretization invariance implies a unique continuum propagator.

The iterative method resembles a Wilsonian RG flow.

Potential applications to discrete quantum gravity models.

Abstract

To obtain a well defined path integral one often employs discretizations. In the case of gravity and reparametrization invariant systems, the latter of which we consider here as a toy example, discretizations generically break diffeomorphism and reparametrization symmetry, respectively. This has severe implications, as these symmetries determine the dynamics of the corresponding system. Indeed we will show that a discretized path integral with reparametrization invariance is necessarily also discretization independent and therefore uniquely determined by the corresponding continuum quantum mechanical propagator. We use this insight to develop an iterative method for constructing such a discretized path integral, akin to a Wilsonian RG flow. This allows us to address the problem of discretization ambiguities and of an anomaly--free path integral measure for such systems. The latter is needed to obtain a path integral, that can act as a projector onto the physical states, satisfying the quantum constraints. We will comment on implications for discrete quantum gravity models, such as spin foams.