Dirac's discrete hypersurface deformation algebras

TL;DR

This paper presents a representation of Dirac's hypersurface deformation algebra within discrete 3D and 4D gravity models, demonstrating how diffeomorphism symmetry can be preserved in discretized quantum gravity frameworks.

Contribution

It introduces a novel representation of the hypersurface deformation algebra in discrete gravity, maintaining diffeomorphism symmetry in certain sectors.

Findings

Representation of Dirac algebra in discrete 3D gravity

Preservation of diffeomorphism symmetry in flat and curved sectors

Different algebra versions for simplex boundaries in arbitrary dimensions

Abstract

The diffeomorphism symmetry of general relativity leads in the canonical formulation to constraints, which encode the dynamics of the theory. These constraints satisfy a complicated algebra, known as Dirac's hypersurface deformation algebra. This algebra has been a long standing challenge for quantization. One reason is that discretizations, on which many quantum gravity approaches rely, generically break diffeomorphism symmetry. In this work we find a representation for the Dirac constraint algebra of hypersurface deformations in a formulation of discrete 3D gravity and for the flat as well as homogeneously curved sector of discrete 4D gravity. In these cases diffeomorphism symmetry can be preserved. Furthermore we present different versions of the hypersurface deformation algebra for the boundary of a simplex in arbitrary dimensions.