All noncommutative spaces of $\kappa$-Poincar\'e geodesics

TL;DR

This paper classifies seven noncommutative spaces of geodesics derived from $oldsymbol{ ext{kappa-Poincaré}}$ symmetry in (3+1) dimensions, revealing unique constructions for light-like deformations and new algebraic structures for time-like and space-like cases.

Contribution

It explicitly constructs and analyzes noncommutative spaces of geodesics from $oldsymbol{ ext{kappa-Poincaré}}$ symmetry, including new algebraic structures and the role of the deformation parameter.

Findings

Only light-like deformation allows all three geodesic types.

Six new noncommutative spaces of geodesics are explicitly obtained.

Darboux generators are identified, showing the deformation parameter's algebraic role.

Abstract

Noncommutative spaces of geodesics provide an alternative way of introducing noncommutative relativistic kinematics endowed with quantum group symmetry. In this paper we present explicitly the seven noncommutative spaces of time-, space- and light-like geodesics that can be constructed from the time-, space- and light- versions of the $\kappa$-Poincar\'e quantum symmetry in (3+1) dimensions. Remarkably enough, only for the light-like (or null-plane) $\kappa$-Poincar\'e deformation the three types of noncommutative spaces of geodesics can be constructed, while for the time-like and space-like deformations both the quantum time-like and space-like geodesics can be defined, but not the light-like one. This obstruction comes from the constraint imposed by the coisotropy condition for the corresponding deformation with respect to the isotropy subalgebra associated to the given space of geodesics, since all these quantum spaces are constructed as quantizations of the corresponding classical coisotropic Poisson homogeneous spaces. The known quantum space of geodesics on the light cone is given by a five-dimensional homogeneous quadratic algebra, and the six nocommutative spaces of time-like and space-like geodesics are explicitly obtained as six-dimensional nonlinear algebras. Five out of these six spaces are here presented for the first time, and Darboux generators for all of them are found, thus showing that the quantum deformation parameter $\kappa^{-1}$ plays exactly the same algebraic role on quantum geodesics as the Planck constant $\hbar$ plays in the usual phase space description of quantum mechanics.