A quantum reduction to Bianchi I models in loop quantum gravity

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.

5 pages. v2: partly rewritten to clarify derivation, gauge group enlarged from U(1) to R_Bohr, relation to old and new LQC dynamics discussed, previous results unchanged. v3: journal version, minor clarifications

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