Quantum space and quantum completeness

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. * Electronic address: tjuric@irb.hr

Noncommutativity and the weak cosmic censorship

“…It is plausible that the light scalar predicted by our analysis can be of relevance in early universe cosmology within the context of quintessence [52]. It may be noted that the question of quantum completeness of the BTZ spacetime using a noncommutative probe was considered in [53], which suggested that noncommutativity may smear out the singularity by allowing quantum completeness for a wider range of BTZ parameters.…”

“…In previous works by some of the authors [53,73] it was shown that the quantum completeness is achieved if the following condition is fulfilled…”

Observables in Quantum Mechanics and the Importance of Self-Adjointness

“…Self-adjointness and the theory of SA extensions are known to play important roles in a variety of physical contexts, including systems with a confined particle [21][22][23], Aharonov-Bohm effect [24][25][26][27], graphene [28], two and three dimensional delta function potentials [29], heavy atoms [30][31][32], singular potentials [33,34], Calogero models [35,36], anyons [37,38], anomalies [39][40][41], ζ-function renormalization [42], scattering theory [43], particle statistics [44], black holes [45][46][47][48][49], integrable system [50,51], Klein-Gordon equation [52], renormalons in QM [53], quasinormal modes [54], supersymmetric QM [55] and toy models for strings [56], spectral triple [57], noncommutative field theories [58][59][60], resolving the spacetime singularities [61][62][63][64][65] and even pl...…”

Quantum probing of null-singularities

“…In recent analyses [6][7][8][9][10][11], classical singularities have been probed by quantum fields within the functional Schrödinger formalism with intriguing results, e.g. Schwarzschild black-holes admit a consistent quantum field theory.…”