Quantum Mechanics at Planck's scale and Density Matrix

TL;DR

This paper introduces a deformation of quantum mechanics at Planck's scale by modifying the density matrix, preserving core principles while enabling new insights into black hole entropy and singularity issues.

Contribution

The novel approach deforms the density matrix instead of commutators, maintaining probabilistic interpretation and dynamics in quantum mechanics at fundamental length scales.

Findings

Derived deformed Liouville's equation and Schrödinger's picture.

Connected the deformation framework to black hole entropy.

Discussed implications for singularity and information loss.

Abstract

In this paper Quantum Mechanics with Fundamental Length is chosen as Quantum Mechanics at Planck's scale. This is possible due to the presence in the theory of General Uncertainty Relations. Here Quantum Mechanics with Fundamental Length is obtained as a deformation of Quantum Mechanics. The distinguishing feature of the proposed approach in comparison with previous ones, lies on the fact that here density matrix subjects to deformation whereas so far commutators have been deformed. The density matrix obtained by deformation of quantum-mechanical density one 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 the well-known measuring procedure corresponding to that interpretation. The proposed 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 definition of statistical entropy and the problem of information loss in black holes are considered. It is shown that obtained results allow to deduce in a simple and natural way the Bekenstein-Hawking's formula for black hole entropy in semiclassical approximation.