Noncommutative spaces of worldlines

Abstract

The space of time-like geodesics on Minkowski spacetime is constructed as a coset space of the Poincaré group in (3+1) dimensions with respect to the stabilizer of a worldline. When this homogeneous space is endowed with a Poisson homogeneous structure compatible with a given Poisson-Lie Poincaré group, the quantization of this Poisson bracket gives rise to a noncommutative space of worldlines with quantum group invariance. As an oustanding example, the Poisson homogeneous space of worldlines coming from the $κ$-Poincaré deformation is explicitly constructed, and shown to define a symplectic structure on the space of worldlines. Therefore, the quantum space of $κ$-Poincaré worldlines is just the direct product of three Heisenberg-Weyl algebras in which the parameter $κ^{-1}$ plays the very same role as the Planck constant $\hbar$ in quantum mechanics. In this way, noncommutative spaces of worldlines are shown to provide a new suitable and fully explicit arena for the description of quantum observers with quantum group symmetry.

15 pages