Phase spaces of Doubly Special Relativity

TL;DR

This paper classifies the phase spaces in Doubly Special Relativity based on the deformation direction of the $$-Poincare9 algebra, linking them to different geometric structures of energy-momentum space.

Contribution

It establishes a direct connection between the deformation direction of the algebra and the geometric form of the energy-momentum space in Doubly Special Relativity.

Findings

Time-like deformation leads to de Sitter energy-momentum space.

Space-like deformation results in anti-de Sitter space.

Light-like deformation corresponds to flat energy-momentum space.

Abstract

We show that depending on the direction of deformation of $\kappa$-Poincar\'e algebra (time-like, space-like, or light-like) the associated phase spaces of single particle in Doubly Special Relativity theories have the energy-momentum spaces of the form of de Sitter, anti-de Sitter, and flat space, respectively.