Fuzzy de Sitter Space

TL;DR

This paper explores the properties of fuzzy de Sitter space using the algebra of the de Sitter group, analyzing spectra of embedding coordinates in specific unitary irreducible representations to understand its geometric structure.

Contribution

It determines the spectra of embedding coordinates for principal continuous series representations of the de Sitter group, extending previous work on fuzzy de Sitter space.

Findings

Spectra of embedding coordinates are derived for specific representations.

Results are obtained in the Hilbert space framework and generalized via representation theory.

The fuzzy de Sitter space retains local metric properties similar to classical de Sitter space.

Abstract

We discuss properties of fuzzy de Sitter space defined by means of algebra of the de Sitter group $\mathrm{SO}(1,4)$ in unitary irreducible representations. It was shown before that this fuzzy space has local frames with metrics that reduce, in the commutative limit, to the de Sitter metric. Here we determine spectra of the embedding coordinates for $(\rho,s=\frac 12)$ unitary irreducible representations of the principal continuous series of the $\mathrm{SO}(1,4)$. The result is obtained in the Hilbert space representation, but using representation theory it can be generalized to all representations of the principal continuous series.