Relative locality in a quantum spacetime and the pregeometry of κ-Minkowski

We develop a description of the much-studied κ-Minkowski noncommutative spacetime, centered on representing on a single Hilbert space not only the κ-Minkowski coordinates, but also the κ-Poincaré symmetry generators and some suitable relativistic-transformation parameters. In this representation the relevant operators act on the kinematical Hilbert space of the covariant formulation of quantum mechanics, which we argue is the natural framework for studying the implications of the step from commuting spacetime coordinates to the κ-Minkowski case, where the spatial coordinates do not commute with the time coordinate. Within this kinematical-Hilbert-space representation we can give a crisp characterization of the “fuzziness” of points in κ-Minkowski spacetime, also allowing us to describe how the same fuzzy point is seen by different relativistic observers. The most striking finding of our analysis is a relativity of spacetime locality in κ-Minkowski. While previous descriptions of relative locality had been formulated exclusively in classical-spacetime setups, our analysis shows how relative locality in a quantum spacetime takes the shape of a dependence of the fuzziness of a spacetime point on the distance at which an observer infers properties of the event that marks the point.