  The space of time-like geodesics on Minkowski spacetime is constructed as a
coset space of the Poincar\'e 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 Poincar\'e
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 $\kappa$-Poincar\'e
deformation is explicitly constructed, and shown to define a symplectic
structure on the space of worldlines. Therefore, the quantum space of
$\kappa$-Poincar\'e worldlines is just the direct product of three
Heisenberg-Weyl algebras in which the parameter $\kappa^{-1}$ 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.
