Fuzzy de Sitter Space from kappa‐Minkowski Space in Matrix Basis

Abstract

We consider the Lie group RκD generated by the Lie algebra of κ‐Minkowski space. Imposing the invariance of the metric under the pull‐back of diffeomorphisms induced by right translations in the group, we show that a unique right invariant metric is associated with RκD. This metric coincides with the metric of de Sitter space‐time. We analyze the structure of unitary representations of the group RκD relevant for the realization of the non‐commutative κ‐Minkowski space by embedding into (2D−1)‐dimensional Heisenberg algebra. Using a suitable set of generalized coherent states, we select the particular Hilbert space and realize the non‐commutative κ‐Minkowski space as an algebra of the Hilbert‐Schmidt operators. We define dequantization map and fuzzy variant of the Laplace‐Beltrami operator such that dequantization map relates fuzzy eigenvectors with the eigenfunctions of the Laplace‐Beltrami operator on the half of de Sitter space‐time.

Noncommutative field theory from an angular twist

“…Besides QNM spectra of black holes, there are many other important concepts in physics which may require an appropriate reassessment in terms of quantum nature of spacetime, such as the geodesic and quantum completeness [74], the existence of naked singularities and related issue of validness of the cosmic censorship hypothesis [75], the renormalization of black hole entanglement entropy [76] and a description of physics in terms of fuzzy (Anti)-de Sitter space [77], [78], [79], [80] to name a few. We think that these issues should be given a considerable attention in future studies.…”