Generalizing the Noether theorem for Hopf-algebra spacetime symmetries

Over these past few years several quantum-gravity research groups have been exploring the possibility that in some Planck-scale nonclassical descriptions of spacetime one or another form of nonclassical spacetime symmetries might arise. One of the most studied scenarios is based on the use of Hopf algebras, but previous attempts were not successful in deriving constructively the properties of the conserved charges one would like to obtain from the Hopf structure, and this in turn did not allow a crisp physical characterization of the new concept of spacetime symmetry. Working within the example of $\kappa$-Minkowski noncommutative spacetime, known to be particularly troublesome from this perspective, we observe that these past failures in the search of the charges originated from not recognizing the crucial role that the noncommutative differential calculus plays in the symmetry analysis. We show that, if the properties of the $\kappa$-Minkowski differential calculus are correctly taken into account, one can easily perform all the steps of the Noether analysis and obtain an explicit formula relating fields and energy-momentum charges. Our derivation also exposes the fact that an apparent source of physical ambiguity in the description of the Hopf-algebra rules of action, which was much emphasized in the literature, actually only amounts to a choice of conventions and in particular does not affect the formulas for the charges.