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

TL;DR

This paper constructs a fuzzy de Sitter space from $kappa$-Minkowski space using group theory, coherent states, and operator algebra, revealing a geometric and algebraic link between non-commutative space and de Sitter geometry.

Contribution

It introduces a novel realization of $kappa$-Minkowski space as a fuzzy de Sitter space via group representations and operator methods, connecting non-commutative geometry with de Sitter spacetime.

Findings

Unique right invariant metric matches de Sitter space-time

Realization of $kappa$-Minkowski space as algebra of Hilbert-Schmidt operators

Fuzzy Laplace-Beltrami operator relates to de Sitter eigenfunctions

Abstract

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