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 quantize 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 quantization of a Bianchi I model for the simplest choice of full theory quantum states compatible with the Bianchi I reduction. In particular, the “improved” $\overline{\mu }$ dynamics of loop quantum cosmology can be obtained by modifying the regularization of the Hamiltonian constraint with similar ideas, in turn yielding insights into the full theory dynamics.

• Received 27 October 2014

DOI:https://doi.org/10.1103/PhysRevD.91.081502