Covariance in models of loop quantum gravity: Spherical symmetry

TL;DR

This paper examines the covariance and anomaly-freedom of loop quantum gravity models with spherical symmetry, highlighting limitations of Abelianization methods in preserving classical gauge structure, especially with matter fields.

Contribution

It clarifies that Abelianization of constraints does not automatically ensure a covariant, anomaly-free quantum theory with correct classical limit in spherical symmetry models.

Findings

Abelianized constraints can be quantized easily but may break covariance.

Matter theories with local degrees of freedom are not covariant after Abelianization.

Results agree with earlier canonical effective methods, including signature change phenomena.

Abstract

Spherically symmetric models of loop quantum gravity have been studied recently by different methods that aim to deal with structure functions in the usual constraint algebra of gravitational systems. As noticed by Gambini and Pullin, a linear redefinition of the constraints (with phase-space dependent coefficients) can be used to eliminate structure functions, even Abelianizing the more-difficult part of the constraint algebra. The Abelianized constraints can then easily be quantized or modified by putative quantum effects. As pointed out here, however, the method does not automatically provide a covariant quantization, defined as an anomaly-free quantum theory with a classical limit in which the usual (off-shell) gauge structure of hypersurface deformations in space-time appears. The holonomy-modified vacuum theory based on Abelianization is covariant in this sense, but matter theories with local degrees of freedom are not. Detailed demonstrations of these statements show complete agreement with results of canonical effective methods applied earlier to the same systems (including signature change).