Doubly Special Relativity versus $\kappa$-deformation of relativistic kinematics

TL;DR

The paper demonstrates that doubly special relativity (DSR) models with certain deformations are essentially classical special relativity in disguise, and clarifies their relation to $ abla$-deformed kinematics and noncommutative spacetime.

Contribution

It shows that DSR with symmetric composition laws is a nonlinear basis of classical Poincaré algebra, not leading to noncommutative spacetime, and constructs two classes of DSR theories.

Findings

DSR models with symmetric composition laws are classical Poincaré algebra in nonlinear basis.

Older $ abla$-deformed kinematics lead to noncommutative $ abla$-deformed Minkowski space.

Constructed two classes of DSR theories generalizing previous models.

Abstract

We argue that recently proposed by Amelino-Camelia et all [1,2] so-called doubly special relativity (DSR), with deformed boost transformations identical with the formulae for $\kappa$-deformed kinematics in bicrossproduct basis is a classical special relativity in nonlinear disguise. The choice of symmetric composition law for deformed fourmomenta as advocated in [1, 2] implies that DSR is obtained by considering nonlinear fourmomenta basis of classical Poincar\'{e} algebra and it does not lead to noncommutative space-time. We also show how to construct large two classes of doubly special relativity theories - generalizing the choice in [1,2] and the one presented by Magueijo and Smolin [3]. The older version of deformed relativistic kinematics, differing essentially from classical theory in the coalgebra sector and leading to noncommutative $\kappa$-deformed Minkowski space is provided by quantum $\kappa$-deformation of Poincar\'e symmetries.