Towards a quantum notion of covariance in spherically symmetric loop quantum gravity

TL;DR

This paper demonstrates that spherically symmetric loop quantum gravity models can be covariant when using Dirac observables, highlighting the importance of the quantization approach and gauge choices for maintaining covariance.

Contribution

It explicitly shows covariance in quantum space-times derived from Dirac observables, clarifying issues related to slicing dependence in loop quantum gravity.

Findings

Quantum space-times preserve the quantum line element under certain gauges.

Covariance depends on the details of the Abelianized quantization and constraints.

Semi-classical polymerization approaches may face covariance problems.

Abstract

The covariance of loop quantum gravity studies of spherically symmetric space-times has recently been questioned. This is a reasonable worry, given that they are formulated in terms of slicing-dependent variables. We show explicitly that the resulting space-times, obtained from Dirac observables of the quantum theory, are covariant in the usual sense of the way -- they preserve the quantum line element -- for any gauge that is stationary (in the exterior, if there is a horizon). The construction depends crucially on the details of the Abelianized quantization considered, the satisfaction of the quantum constraints and the recovery of standard general relativity in the classical limit and suggests that more informal polymerization constructions of possible semi-classical approximations to the theory can indeed have covariance problems. This analysis is based on the understanding of how slicing dependent quantities as the metric arise in a quantum context in terms of parameterized observables. It has implications beyond loop quantum gravity that hold for general approaches to quantum space time theories.