Quantum space and quantum completeness

TL;DR

This paper investigates whether quantum gravity effects can eliminate classical singularities by analyzing a noncommutative BTZ black hole, demonstrating that quantum effects extend the parameter range for which the space is quantum complete, indicating a form of singularity resolution.

Contribution

The study introduces a model of quantum space with a noncommutative BTZ black hole and shows quantum effects enlarge the domain of parameters ensuring quantum completeness, suggesting singularity smearing.

Findings

Quantum completeness is achieved for a larger parameter range.

Quantum effects extend the self-adjointness domain of the wave operator.

Quantum space exhibits smearing out of the classical singularity.

Abstract

Motivated by the question whether quantum gravity can "smear out" the classical singularity we analyze a certain quantum space and its quantum-mechanical completeness. Classical singularity is understood as a geodesic incompleteness, while quantum completeness requires a unique unitary time evolution for test fields propagating on an underlying background. Here the crucial point is that quantum completeness renders the Hamiltonian (or spatial part of the wave operator) to be essentially self-adjoint in order to generate a unique time evolution. We examine a model of quantum space which consists of a noncommutative BTZ black hole probed by a test scalar field. We show that the quantum gravity (noncommutative) effect is to enlarge the domain of BTZ parameters for which the relevant wave operator is essentially self-adjoint. This means that the corresponding quantum space is quantum complete for a larger range of BTZ parameters rendering the conclusion that in the quantum space one observes the effect of "smearing out" the singularity.