Emergent diffeomorphism invariance in a discrete loop quantum gravity model

TL;DR

The paper addresses the problem that discretizing diffeomorphism-invariant gravity typically yields second-class constraints, complicating quantization and continuum limits. It adopts the uniform discretization framework, introducing a master constraint ${\mathbb H}$ (and its lattice version ${\mathbb H}^\epsilon$) whose evolution keeps the discrete constraints bounded and yields a well-defined continuum limit that recovers diffeomorphism invariance at the quantum level. Using a 1+1 dimensional Husain–Kuchā-like model, it constructs a spin-network basis and explicit operators $\hat D_j$ and $\hat O_j$, demonstrating that the zero-eigenvalue sector of ${\mathbb H}$ emerges in the continuum and corresponds to diffeomorphism-invariant states, with a physically meaningful inner product and reduced discretization ambiguities. The results suggest a promising route to defining a continuum quantum theory with full space-time covariance via discretization, at least in low dimensions, with potential extensions to more realistic gravitational systems.

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--Kuchař 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.