# Noncommutative spaces of worldlines

## TL;DR

This paper constructs a noncommutative space of worldlines in Minkowski spacetime using quantum group theory, specifically the $ ext{kappa}$-Poincaré deformation, providing a new framework for quantum observers with symmetry.

## Contribution

It explicitly constructs a noncommutative space of worldlines from the $ ext{kappa}$-Poincaré deformation, linking quantum group symmetry with the geometry of worldlines.

## Key findings

- The space of worldlines can be quantized to form a noncommutative space with quantum group invariance.
- The $ ext{kappa}$-Poincaré deformation leads to a symplectic structure on worldlines.
- The quantum space is a product of three Heisenberg-Weyl algebras with $ ext{kappa}^{-1}$ as the quantum parameter.

## Abstract

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.