Emergent diffeomorphism invariance in a discrete loop quantum gravity model

TL;DR

This paper demonstrates that the uniform discretizations approach effectively preserves diffeomorphism invariance in a simple loop quantum gravity model, enabling a proper continuum limit and maintaining space-time covariance at the quantum level.

Contribution

It shows how uniform discretizations can overcome discretization issues in loop quantum gravity, preserving diffeomorphism invariance and enabling a consistent continuum limit in a simplified model.

Findings

Successful construction of the quantum continuum limit for the model

Preservation of space-time covariance at the quantum level

Demonstration of the approach's applicability to 1+1 dimensional quantum gravity

Abstract

Several approaches to the dynamics of loop quantum gravity involve discretizing the equations of motion. The resulting discrete theories are known to be problematic since the first class algebra of constraints of the continuum theory becomes second class upon discretization. If one treats the second class constraints properly, the resulting theories have very different dynamics and number of degrees of freedom than those of the continuum theory. It is therefore questionable how these theories could be considered a starting point for quantization and the definition of a continuum theory through a continuum limit. We show explicitly in a model that the {\em uniform discretizations} approach to the quantization of constrained systems overcomes these difficulties. We consider here a simple diffeomorphism invariant one dimensional model and complete the quantization using {\em uniform discretizations}. The model can be viewed as a spherically symmetric reduction of the well known Husain--Kucha\v{r} model of diffeomorphism invariant theory. We show that the correct quantum continuum limit can be satisfactorily constructed for this model. This opens the possibility of treating 1+1 dimensional dynamical situations of great interest in quantum gravity taking into account the full dynamics of the theory and preserving the space-time covariance at a quantum level.