Relative locality in a quantum spacetime and the pregeometry of κ-Minkowski
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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.
KeywordsQuantum Mechanic Hopf Algebra Relative Locality Minkowski Spacetime Differential Calculus
We gratefully acknowledge conversations with Daniele Oriti and Carlo Rovelli. The work of two of us (GAC and VA) was supported in part by a grant from the John Templeton Foundation. The work of one of us (GR) was supported in part by funds provided by the National Science Center under the agreement DEC-2011/02/A/ST2/00294. One of us (VA) acknowledges the hospitality of the Perimeter Institute for Theoretical Physics during parts of this work.