Discretizing parametrized systems: the magic of Ditt-invariance

TL;DR

This paper explores how discretizing reparametrization-invariant systems leads to a unique continuum limit characterized by Ditt-invariance, which differs from traditional lattice approaches and has implications for quantum gravity.

Contribution

It introduces the concept of Ditt-invariance in discretized systems and demonstrates its significance for understanding continuum limits in quantum gravity.

Findings

Continuum limit does not require tuning parameters to critical values.

A regime exists where discretization points approximate transition amplitudes well.

Ditt-invariance emerges as an asymptotic topological invariance in the continuum limit.

Abstract

Peculiar phenomena appear in the discretization of a system invariant under reparametrization. The structure of the continuum limit is markedly different from the usual one, as in lattice QCD. First, the continuum limit does not require tuning a parameter in the action to a critical value. Rather, there is a regime where the system approaches a sort of asymptotic topological invariance ("Ditt-invariance"). Second, in this regime the expansion in the number of discretization points provides a good approximation to the transition amplitudes. These phenomena are relevant for understanding the continuum limit of quantum gravity. I illustrate them here in the context of a simple system.