Polymer representation of the Bianchi IX cosmology in the Misner variables

We analyze the Bianchi IX Universe in the Polymer Quantum Mechanics framework by facing both semiclassical and purely quantum effects near the cosmological singularity. We adopt Misner variables to describe the model dynamics, applying the polymer paradigm simultaneously to the isotropic one (linked to the Universe volume) and to the two anisotropy ones (characterizing the physical gravitational degrees of freedom). Setting two different cut-off scales for the two different variable sets, i.e. the geometrical volume and the gravity tensor modes, we demonstrate how the semiclassical properties of the Bianchi IX dynamics are sensitive to the ratio of the cut-off parameters. In particular, the semiclassical evolution turns out to be chaotic only if the parameter associated to the volume discretization is greater or equal to that one of the anisotropies. Concerning the chaotic case we perform a purely quantum polymer analysis, demonstrating that the original Misner result about the existence of quasi-classical states near the singularity (in the sense of high occupation numbers) is still valid in the revised approach and able to account for cut-off physics effects. The possibility for a comparison with the original study by Misner is possible because the singularity is still present in the semiclassical evolution of the cosmological model for all the parameter space. We interpret this surprising feature as the consequence of a geometrical volume discretization which does not prevent the volume from vanishing, i.e. restoring in the Minisuperspace analysis its zero value.