Understand
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.