Covariant four dimensional differential calculus in κ-Minkowski

Covariant four dimensional differential calculus in κ-Minkowski

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It is generally believed that it is not possible to have a four dimensional differential calculus in κ-Minkowski spacetime, with κ-Poincaré relativistic symmetries, covariant under (κ-deformed) Lorentz transformations.Thus, one usually introduces a fifth differential form, whose physical interpretation is still challenging, and defines berness white sneakers a covariant five dimensional calculus.Nevertheless, the four dimensional calculus is at the basis of several works based on κ-Minkowski/κ-Poincaré framework that led to meaningful insights on its physical interpretation and phenomenological implications.

We here revisit the argument against the covariance of the four dimensional calculus, and find that it depends crucially on an incomplete characterization of Lorentz transformations in this framework, which neglected opi the color that keeps on giving a feature, still uncovered at the time, that turns out to be fundamental for the consistency of the relativistic framework: the noncommutativity of the Lorentz transformation parameters.This suggests to revise the notion of covariance to accommodate the action of the full infinitesimal Lorentz transformation.Once this is taken into account, the four dimensional calculus is found to be fully Lorentz covariant.

The result we obtain extends naturally to the whole κ-Poincaré algebra of transformations, showing the close relation between its relativistic nature and the properties of the differential calculus.