Fuzzy de Sitter space

Maja Burić,Luka Nenadović,Duško Latas

doi:10.1140/epjc/s10052-018-6432-6

Maja Burić, Luka Nenadović + Show 1 more

Open Access

https://doi.org/10.1140/epjc/s10052-018-6432-6

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Abstract

We discuss properties of fuzzy de Sitter space defined by means of algebra of the de Sitter group text {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{1}{2}) unitary irreducible representations of the principal continuous series of the text {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.

Highlights

Understanding of the structure of spacetime at very small scales is one of the most challenging problems in theoretical physics: more so as it is, as we commonly believe, related to the properties of gravity at small scales, that is to quantization of gravity
Derivatives are usually given by commutators; once they are defined, one can proceed more or less straightforwardly to differential geometry
We shall in the following use a variant of noncommutative differential geometry which was introduced by Madore, known as the noncommutative frame formalism [1]

Summary

Introduction

Understanding of the structure of spacetime at very small scales is one of the most challenging problems in theoretical physics: more so as it is, as we commonly believe, related to the properties of gravity at small scales, that is to quantization of gravity. We shall in the following use a variant of noncommutative differential geometry which was introduced by Madore, known as the noncommutative frame formalism [1] It is a noncommutative generalization of the Cartan moving frame formalism and gives a very natural way to describe gravity on curved noncommutative spacetimes. General features of the noncommutative frame formalism and many applications to gravity are known [2,3,4,5]; the aim of our present investigation is to construct four-dimensional noncommutative spacetimes which correspond to known classical configurations of gravitational field This means, to find algebras and differential structures which are noncommutative versions of, for example, black holes or cosmologies.

Metric and scaling limits

Coordinates

Group-theoretic view

Summary and outlook