A 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 $\mathbb{R}_{\text{Bohr}}$ 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. 
Norbert Bodendorfer
gr-qc hep-th math-ph math.MP