Quantum Mechanics of the Early Universe and its Limiting Transition

TL;DR

This paper develops a deformation of quantum mechanics incorporating a fundamental length, enabling a consistent description of the early universe, and explores implications for singularities, entropy, and black hole information loss.

Contribution

It introduces a novel deformation of density matrices and commutators in quantum mechanics to include a fundamental length, preserving probabilistic interpretation and enabling dynamic descriptions.

Findings

Deformed Liouville's equation derived for early universe conditions.

Density pro-matrix maintains probabilistic interpretation.

Implications for singularity resolution and black hole information loss discussed.

Abstract

In this paper Quantum Mechanics with Fundamental Length is chosen as the theory for describing the early Universe. This is possible due to the presence in the theory of General Uncertainty Relations from which unavoidable it follows that in nature a fundamental length exits. Here Quantum Mechanics with Fundamental Length is obtained as a deformation of Quantum Mechanics. The distinguishing feature of the proposed in this paper approach in comparison with previous ones, lies on the fact that here density matrix subjects to deformation as well as so far commutators had been deformed. The deformed density matrix mentioned above, is named throughout this paper density pro-matrix. Within our approach two main features of Quantum Mechanics are conserved: the probabilistic interpretation of the theory and exact predefined measuring procedure corresponding to that interpretation. The proposed here approach allows to describe dynamics. In particular, the explicit form of deformed Liouville's equation and the deformed Shr\"odinger's picture are given. Some implications of obtained results are discussed. In particular, the problem of singularity, the hypothesis of cosmic censorship, a possible improvement of the statistical entropy definition and the problem of information loss in black holes are considered.