Semiclassical and quantum behavior of the Mixmaster model in the polymer approach for the isotropic Misner variable

TL;DR

This paper investigates the semiclassical and quantum dynamics of the Bianchi IX Universe using the Polymer Quantum Mechanics framework applied to the Misner variable, finding that the singular and chaotic nature persists despite the reformulation.

Contribution

It demonstrates that the Polymer approach does not significantly alter the asymptotic chaotic and singular behavior of the Bianchi IX Universe at quantum and semiclassical levels.

Findings

The chaotic behavior remains unchanged in the Polymer framework.

Quantum numbers of the universe are conserved, allowing quasi-classical states.

Polymer reformulation does not resolve the singularity or chaos.

Abstract

We analyze the semiclassical and quantum behavior of the Bianchi IX Universe in the Polymer Quantum Mechanics framework, applied to the isotropic Misner variable, linked to the space volume of the model. The study is performed both in the Hamiltonian and field equations approaches, leading to the surprising result of a still singular and chaotic cosmology, whose Poincar\'e return map asymptotically overlaps the standard Belinskii-Khalatnikov-Lifshitz one. In the quantum sector, we reproduce the original analysis due to Misner, within the revised Polymer approach and we arrive to demonstrate that the quantum numbers of the point-Universe still remain constants of motion. This issue confirms the possibility to have quasi-classical states up to the initial singularity. The present study clearly demonstrates that the asymptotic behavior of the Bianchi IX Universe towards the singularity is not significantly affected by the Polymer reformulation of the spatial volume dynamics both on a pure quantum and a semiclassical level.