Consistent discretization and loop quantum geometry

TL;DR

This paper applies consistent discretization to general relativity, resulting in a constraint-free theory with a well-defined quantum dynamics, exemplified in 2+1 dimensions, advancing loop quantum gravity approaches.

Contribution

It introduces a consistent discretization method that simplifies the constraints in general relativity, enabling a clearer quantum dynamics formulation within loop quantum gravity.

Findings

Constraint-free discretized theory of gravity

Exact preservation of diffeomorphism constraint under evolution

Explicit demonstration in 2+1 dimensional gravity

Abstract

We apply the ``consistent discretization'' approach to general relativity leaving the spatial slices continuous. The resulting theory is free of the diffeomorphism and Hamiltonian constraints, but one can impose the diffeomorphism constraint to reduce its space of solutions and the constraint is preserved exactly under the discrete evolution. One ends up with a theory that has as physical space what is usually considered the kinematical space of loop quantum geometry, given by diffeomorphism invariant spin networks endowed with appropriate rigorously defined diffeomorphism invariant measures and inner products. The dynamics can be implemented as a unitary transformation and the problem of time explicitly solved or at least reduced to as a numerical problem. We exhibit the technique explicitly in 2+1 dimensional gravity.