Doubly Special Relativity theories as different bases of $\kappa$--Poincar\'e algebra

TL;DR

This paper demonstrates that different Doubly Special Relativity theories can be viewed as specific bases of the $ppa$--Poincare9 algebra, unifying their structure and suggesting basis-independent physical predictions.

Contribution

It shows that various DSR theories are particular bases of the $ppa$--Poincare9 algebra, enabling a unified understanding and extension to the entire class of DSRs.

Findings

Different DSR theories are particular bases of the $ppa$--Poincare9 algebra.

The space-time structure in DSR theories is basis-independent.

The physical predictions of DSR are likely basis-independent.

Abstract

Doubly Special Relativity (DSR) theory is a theory with two observer-independent scales, of velocity and mass (or length). Such a theory has been proposed by Amelino--Camelia as a kinematic structure which may underline quantum theory of relativity. Recently another theory of this kind has been proposed by Magueijo and Smolin. In this paper we show that both these theories can be understood as particular bases of the $\kappa$--Poincar\'e theory based on quantum (Hopf) algebra. This observation makes it possible to construct the space-time sector of Magueijo and Smolin DSR. We also show how this construction can be extended to the whole class of DSRs. It turns out that for all such theories the structure of space-time commutators is the same. This results lead us to the claim that physical predictions of properly defined DSR theory should be independent of the choice of basis.