The noncommutative space of light-like worldlines

The noncommutative space of light-like worldlines that is covariant under the light-like (or null-plane) $\kappa$-deformation of the (3+1) Poincar\'e group is fully constructed as the quantization of the corresponding Poisson homogeneous space of null geodesics. This new noncommutative space of geodesics is five-dimensional, and turns out to be defined as a quadratic algebra that can be mapped to a non-central extension of the direct sum of two Heisenberg-Weyl algebras whose noncommutative parameter is just the Planck scale parameter $\kappa^{-1}$. Moreover, it is shown that the usual time-like $\kappa$-deformation of the Poincar\'e group does not allow the construction of the Poisson homogeneous space of light-like worldlines. Therefore, the most natural choice in order to model the propagation of massless particles on a quantum Minkowski spacetime seems to be provided by the light-like $\kappa$-deformation.