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

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 semiclassical 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 parametrized observables. It has implications beyond loop quantum gravity that hold for general approaches to quantum space time theories.