Abelianized structures in spherically symmetric hypersurface deformations

TL;DR

This paper explores an Abelian substructure within hypersurface deformations in spherically symmetric gravity, simplifying quantization but highlighting issues with maintaining full covariance in quantum gravity models.

Contribution

It identifies an Abelian substructure in hypersurface deformations for spherical symmetry and discusses its implications for quantization and covariance in loop quantum gravity.

Findings

Abelian substructure can be quantized more easily.

Symmetries generated do not directly correspond to hypersurface deformations.

Recent Abelianized models may not resolve covariance issues.

Abstract

In canonical gravity, general covariance is implemented by hypersurface-deformation symmetries on phase space. The different versions of hypersurface deformations required for full covariance have complicated interplays with one another, governed by non-Abelian brackets with structure functions. For spherically symmetric space-times, it is possible to identify a certain Abelian substructure within general hypersurface deformations, which suggests a simplified realization as a Lie algebra. The generators of this substructure can be quantized more easily than full hypersurface deformations, but the symmetries they generate do not directly correspond to hypersurface deformations. The availability of consistent quantizations therefore does not guarantee general covariance or a meaningful quantum notion thereof. In addition to placing the Abelian substructure within the full context of spherically symmetric hypersurface deformation, this paper points out several subtleties relevant for attempted applications in quantized space-time structures. In particular, it follows that recent constructions by Gambini, Olmedo and Pullin in an Abelianized setting fail to address the covariance crisis of loop quantum gravity.