A quantum reduction to Bianchi I models in loop quantum gravity

TL;DR

This paper develops a quantum reduction of loop quantum gravity to Bianchi I spacetimes, simplifying the dynamics and connecting to loop quantum cosmology, providing new insights into the full theory.

Contribution

It introduces a quantum symmetry reduction to Bianchi I models within loop quantum gravity using a diagonal metric gauge and demonstrates the resulting dynamics align with polymer quantisation.

Findings

The reduced model's dynamics match those of a polymer quantised Bianchi I model.

The approach yields a simple, manifestly consistent quantum dynamics.

Insights into full theory dynamics are gained through modified regularisation techniques.

Abstract

We propose a quantum symmetry reduction of loop quantum gravity to Bianchi I spacetimes. To this end, we choose the diagonal metric gauge for the spatial diffeomorphism constraint at the classical level, leading to an $\mathbb{R}_{\text{Bohr}}$ gauge theory, and quantise the resulting theory via loop quantum gravity methods. Constraints which lead classically to a suitable reduction are imposed at the quantum level. The dynamics of the resulting model turn out to be very simple and manifestly coincide with those of a polymer quantisation of a Bianchi I model for the simplest choice of full theory quantum states compatible with the Bianchi I reduction. In particular, the "improved" $\bar{\mu}$ dynamics of loop quantum cosmology can be obtained by modifying the regularisation of the Hamiltonian constraint with similar ideas, in turn yielding insights into the full theory dynamics.