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

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Abstract

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.

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Notes

  1. Note that in this pregeometric setup of [22] the “pregeometric Planck constant” ħ 0 is in general unrelated to the physical Planck constant ħ.

  2. Note that \(\hat{x}_{0}^{*}\) is a good choice of κ-Minkowski time coordinate, since \([\hat{x}_{1},\hat{x}_{0}^{*}] = i \ell\hat{x}_{1}\). And one also easily verifies that κ-Minkowski described by \(\hat{x}_{1},\hat{x}_{0}^{*}\) has good properties under boosts, \([B \triangleright\hat{x}_{1}, B \triangleright\hat {x}_{0}^{*}] = i \ell B \triangleright\hat{x}_{1}\) (or \(N\triangleright[\hat{x}_{1},\hat{x}_{0}^{*}] = i\ell N \triangleright \hat{x}_{1} \)), and under translations, \([T \triangleright\hat{x}_{1}, T \triangleright\hat {x}_{0}^{*}] = i \ell T \triangleright\hat{x}_{1}\) (or \(P_{\mu}\triangleright[\hat{x}_{1},\hat{x}_{0}^{*}] = i\ell P_{\mu}\triangleright\hat{x}_{1} \)).

  3. We shall pay little attention to the fact that actually there is an exception to this “fuzziness theorem”: the interested reader can easily verify that the origin of the observer, x 0=x 1=0, can be sharp. This can be straightforwardly added as a limiting case for the discussion we offer in the following, and in particular one finds that even a point that is absolutely sharp in the origin of one observer is described by a distant observer as a fuzzy point.

  4. Of course the same results for mean values and uncertainties of κ-Minkowski coordinates can be obtained by acting with T on the pregeometric state and evaluating \(\hat{x}_{\mu}\) and \(\delta\hat{x}_{\mu}\) in the state thereby obtained. The equivalent alternative we follow, by acting with T on \(\hat{x}_{\mu}\) and evaluating the mean value and the uncertainty of \(T\triangleright\hat{x}_{\mu}\) in the original state just allows the derivation to proceed a bit more speedy.

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Acknowledgements

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.

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Authors and Affiliations

  1. Dipartimento di Fisica, Università di Roma “La Sapienza”, P.le A. Moro 2, 00185, Roma, Italy

    Giovanni Amelino-Camelia, Valerio Astuti & Giacomo Rosati

  2. INFN, Sez. Roma1, P.le A. Moro 2, 00185, Roma, Italy

    Giovanni Amelino-Camelia, Valerio Astuti & Giacomo Rosati

  3. Institute for Theoretical Physics, University of Wrocław, Pl. Maksa Borna 9, 50-204, Wrocław, Poland

    Giacomo Rosati

Authors
  1. Giovanni Amelino-Camelia
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  2. Valerio Astuti
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  3. Giacomo Rosati
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Correspondence to Giacomo Rosati.

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Amelino-Camelia, G., Astuti, V. & Rosati, G. Relative locality in a quantum spacetime and the pregeometry of κ-Minkowski. Eur. Phys. J. C 73, 2521 (2013). https://doi.org/10.1140/epjc/s10052-013-2521-8

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