Fuzzy worldlines with κ-Poincaré symmetries

Abstract

A novel approach to study the properties of models with quantum-deformed relativistic symmetries relies on a noncommutative space of worldlines rather than the usual noncommutative spacetime. In this setting, spacetime can be reconstructed as the set of events, that are identified as the crossing of different worldlines. We lay down the basis for this construction for the κ-Poincaré model, analyzing the fuzzy properties of κ-deformed time-like worldlines and the resulting fuzziness of the reconstructed events.

The noncommutative space of light-like worldlines

“…At this point we recall that the 6D noncommutative space of time-like lines constructed in [37] was shown to be generated by the direct sum of three Heisenberg-Weyl algebras, and this enabled the analysis presented in [38] of the fuzziness properties of events defined as the crossing of light-like quantum geodesics. In the light-like case we are dealing with a 5D quantum space, whose fuzziness could be studied in a similar manner by taking into account the mapping (32).…”

“…Obviously, the linearization of (38) gives (x ± , x i ) ≡ (y ± , y i ). Note also that if we compute the Poisson brackets for (x ± , x i ) by starting from (27) we recover the linear Poisson structure for the Poisson spacetime (37), as it should be.…”

“…Note also that if we compute the Poisson brackets for (x ± , x i ) by starting from (27) we recover the linear Poisson structure for the Poisson spacetime (37), as it should be. Therefore, the nonlinear map (38) will also provide the starting point for the study of the link between the noncommutative spaces of points (37) and light-like geodesics (28) arising from the light-like κ-deformation.…”

“…In fact, the construction of "quantum geodesics" on a given noncommutative spacetime is the essential problem underlying the definition of "quantum observers". The explicit construction of the noncommutative space of time-like worldlines associated to the κ-Minkowski spacetime has been recently given in [25,37], and the fuzziness of events has also been analysed in this quantum worldline setting [38].…”

Noncommutative spacetimes versus noncommutative spaces of geodesics

“…Therefore, W κ can be expressed as three copies of the Heisenberg-Weyl algebra of quantum mechanics where the deformation parameter κ −1 replaces the Planck constant . We also recall that a first phenomenological analysis for W κ (17) has been performed in [19], which may have relevance in a quantum gravity setting [20].…”