One of the main objectives of quantum-gravity research is to establish how the Heisenberg position-momentum uncertainty principle is modified by Planck-scale effects. We here focus on some residual gray areas of our understanding of the interplay between the uncertainty principle and special relativity, which might have been an obstacle for the further generalization needed in order to include gravitational effects. In its original formulation the uncertainty principle is inconsistent with special relativity, and in nearly a century of investigation not much progress has been made toward a satisfactory reformulation. Some partial insight has been gained in the ultra-high-velocity regime, but a full description is still missing and in particular we have no clue about the intermediate regime of particles whose speeds are much smaller than the speed of light but still high enough for tangible departures from the Heisenberg formulation. As we stress here, that intermediate regime is also our best chance for testing experimentally our understanding of the implications of special relativity for the uncertainty principle. We introduce a new approach toward facing these challenges, based mainly on the observation that the only operative notion of position of a particle at a given time involves the crossing of the worldline of that particle with the worldline of a test particle. We find that the worldline-crossing perspective opens a path toward a special-relativistic version of the uncertainty principle, which indeed could be tested experimentally.
Amelino-Camelia et al. (Wed,) studied this question.