Quantum reduction to Bianchi I models in loop quantum gravity

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 R-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" (mu) over bar 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.