The noncommutative space of light-like worldlines

TL;DR

This paper constructs a new five-dimensional noncommutative space of light-like worldlines covariant under a specific quantum deformation of the Poincaré group, revealing its algebraic structure and implications for modeling massless particles.

Contribution

It introduces a fully constructed noncommutative space of light-like geodesics based on the light-like ppa-deformation, highlighting its algebraic properties and physical relevance.

Findings

The noncommutative space is five-dimensional and quadratic.

It can be mapped to a non-central extension of two Heisenberg-Weyl algebras.

Time-like ppa-deformation does not support light-like worldline construction.

Abstract

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.