Kinematics of particles with quantum de Sitter symmetries

TL;DR

This paper explores the kinematics of free particles under quantum de Sitter symmetries, revealing how curvature and quantum effects influence particle motion, redshift, and travel times in a unified phase space framework.

Contribution

It introduces a detailed phase space analysis of particles with quantum de Sitter symmetries, combining effects of spacetime curvature and momentum space geometry.

Findings

Redshift effects become energy-dependent due to curvature.

Travel times are modified by both curvature and momentum space properties.

The framework unifies effects of spacetime curvature and quantum deformation on particle kinematics.

Abstract

We present the first detailed study of the kinematics of free relativistic particles whose symmetries are described by a quantum deformation of the de Sitter algebra, known as $q$-de Sitter Hopf algebra. The quantum deformation parameter is a function of the Planck length $\ell$ and the de Sitter radius $H^{-1}$, such that when the Planck length vanishes, the algebra reduces to the de Sitter algebra, while when the de Sitter radius is sent to infinity one recovers the $\kappa$-Poincar\'e Hopf algebra. In the first limit the picture is that of a particle with trivial momentum space geometry moving on de Sitter spacetime, in the second one the picture is that of a particle with de Sitter momentum space geometry moving on Minkowski spacetime. When both the Planck length and the inverse of the de Sitter radius are non-zero, effects due to spacetime curvature and non-trivial momentum space geometry are both present and affect each other. The particles' motion is then described in a full phase space picture. We find that redshift effects that are usually associated to spacetime curvature become energy-dependent. Also, the energy dependence of particles' travel times that is usually associated to momentum space non-trivial properties is modified in a curvature-dependent way.