Comparison of the Semiclassical and Quantum Dynamics of the Bianchi I Cosmology in the Polymer and GUP Extended Paradigms

TL;DR

This paper compares semiclassical and quantum dynamics of Bianchi I cosmology within Polymer Quantum Mechanics and GUP frameworks, highlighting their differences and similarities in quantum behavior and implications for cosmological singularities.

Contribution

It provides a detailed comparison of the semiclassical and quantum formulations of Polymer and GUP approaches in Bianchi I cosmology, revealing their fundamental differences and similarities.

Findings

Polymer Quantum Mechanics implies no absolute minimal position uncertainty.

GUP approach retains the classical singularity, unlike polymer formalism.

Wavepacket spreading indicates the necessity of a fully quantum state near the Planck era.

Abstract

We compare in some detail Polymer Quantum Mechanics and the Generalized Uncertainty Principle approach to clarify to what extent we can treat them on the same footing. We show that, while on a semiclassical level they may be formulated as similar modifications of the Poisson algebra, on a quantum level they intrinsically differ because PQM implies no absolute minimal uncertainty on position. Then we implement these schemes to Bianchi I cosmology on a semiclassical level deforming only the algebra of the Universe volume, searching for alternative formulations able to account for the modified Friedmann equations emerging in Brane Cosmology and Loop Quantum Cosmology. On a pure quantum level, we implement the two approaches through their original setups and reduce the two resulting Wheeler-DeWitt equations to the same morphological structure, showing how the polymer formalism is associated with a bouncing dynamics while in the Generalized Uncertainty Principle case the singularity is still present. The implications of the wavepacket spreading are also discussed in both approaches, outlining that, when the singularity survives, the Planckian era must necessarily be approached by a fully quantum (non-peaked) state of the Universe.