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arXiv · 1710.01491

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

Danijel Jurman
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Abstract

We consider the Lie group $\mathbb{R}^D_κ$ 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 $\mathbb{R}^D_κ$. This metric coincides with the metric of de Sitter space-time. We analyze the structure of unitary representations of the group $\mathbb{R}^D_κ$ 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.

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Danijel Jurman. 2019-09-23. Fuzzy de Sitter Space from kappa-Minkowski Space in Matrix Basis. https://doi.org/10.1002/prop.201800061

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