Spin-network states for the Bianchi I and IX cosmological models from quantum constrained symmetries
In this work, we implement at the quantum level the gauge-fixing conditions that relate the homogeneous SU(2) gauge theory of Ashtekar variables, describing the classical cosmological sector, to the usual minisuperspace formulation. We impose the so-called divergence constraint, which fixes the gauge to the homogeneous one and, at the classical level, recovers a finite-dimensional phase space, together with the diagonal constraint, which imposes the diagonality of the metric and has already been extensively studied in the literature. We construct the corresponding quantum operators for general cosmological models and provide a suitable regularization in terms of holonomies and fluxes. We then show that a special class of homogeneous spin-network states describing the Bianchi I and Bianchi IX models satisfies the quantum gauge-fixing constraints, using techniques developed in Reduced Quantum Loop Gravity. This provides a quantum-level link between the cosmological theory with full SU(2) gauge symmetry and standard Loop Quantum Cosmology, by reason of the effective Abelian structure of the selected states.