On the relation between reduced quantisation and quantum reduction for spherical symmetry in loop quantum gravity

TL;DR

This paper explores the relationship between reduced quantisation and quantum reduction in loop quantum gravity with spherical symmetry, extending previous models by removing gauge fixing and analyzing algebraic structures.

Contribution

It generalizes the quantum reduction approach by removing gauge fixing, establishing a connection between full theory and reduced variables, and analyzing the quantum algebra in this context.

Findings

Quantum algebra reproduces reduced variables' algebra in large quantum number limit.

Embedding of reduced states into full theory states is demonstrated.

Quantitative recovery of reduced algebra is limited by simplified full theory states.

Abstract

Building on a recent proposal for a quantum reduction to spherical symmetry from full loop quantum gravity, we investigate the relation between a quantisation of spherically symmetric general relativity and a reduction at the quantum level. To this end, we generalise the previously proposed quantum reduction by dropping the gauge fixing condition on the radial diffeomorphisms, thus allowing to make direct contact between previous work on reduced quantisation. A dictionary between spherically symmetric variables and observables with respect to the reduction constraints in the full theory is discussed, as well as an embedding of reduced quantum states to a sub sector of the quantum symmetry reduced full theory states. On this full theory sub sector, the quantum algebra of the mentioned observables is computed and shown to qualitatively reproduce the quantum algebra of the reduced variables in the large quantum number limit for a specific choice of regularisation. Insufficiencies in recovering the reduced algebra quantitatively from the full theory are attributed to the oversimplified full theory quantum states we use.