Group velocity in noncommutative spacetime

The realization that forthcoming experimental studies, such as the ones planned for the GLAST space telescope, will be sensitive to Planck-scale deviations from Lorentz symmetry has increased interest in noncommutative spacetimes in which this type of effects is expected. We focus here on $\kappa$-Minkowski spacetime, a much-studied example of Lie-algebra noncommutative spacetime, but our analysis appears to be applicable to a more general class of noncommutative spacetimes. A technical controversy which has significant implications for experimental testability is the one concerning the $\kappa$-Minkowski relation between group velocity and momentum. A large majority of studies adopted the relation $v = dE(p)/dp$, where $E(p)$ is the $\kappa$-Minkowski dispersion relation, but recently some authors advocated alternative formulas. While in these previous studies the relation between group velocity and momentum was introduced through ad hoc formulas, we rely on a direct analysis of wave propagation in $\kappa$-Minkowski. Our results lead conclusively to the relation $v = dE(p)/dp$. We also show that the previous proposals of alternative velocity/momentum relations implicitly relied on an inconsistent implementation of functional calculus on $\kappa$-Minkowski and/or on an inconsistent description of spacetime translations.