Representations of Extended Carroll Group

TL;DR

This paper explores the mathematical structure of the extended Carroll group by embedding Euclidean space into a higher-dimensional de Sitter space, providing a covariant formulation and analyzing its unit representations.

Contribution

It introduces a covariant formulation of the extended Carroll group and studies its unit representations, expanding understanding of its mathematical properties.

Findings

Covariant formulation of Carroll's group derived.

Unit representations of the group analyzed.

Embedding into de Sitter space elucidates group structure.

Abstract

Carroll's group is presented as a group of transformations in a 5-dimensional space ($\mathcal{C}$) obtained by embedding the Euclidean space into a (4; 1)-de Sitter space. Three of the five dimensions of $\mathcal{C}$ are related to $\mathcal{R}^3$, and the other two to mass and time. A covariant formulation of Caroll's group, analogous as introduced by Takahashi to Galilei's group, is deduced. Unit representations are studied.