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

Abstract

We consider the Lie group ${\mathbb{R}}_{\kappa }^D$ generated by the Lie algebra of $\kappa$-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 ${\mathbb{R}}_{\kappa }^D$. This metric coincides with the metric of de Sitter space-time. We analyze the structure of unitary representations of the group ${\mathbb{R}}_{\kappa }^D$ relevant for the realization of the non-commutative $\kappa$-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 $\kappa$-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.

21 pages, v3 differs from version published in Fortschritte der Physik by a note and references added and adjusted

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Cited by in corpus (1)

• The Momentum Spaces of $\kappa$-Minkowski noncommutative spacetime