Quantum Space-Time Deformed Symmetries Versus Broken Symmetries

Several recent studies have concerned the faith of classical symmetries in quantum space-time. In particular, it appears likely that quantum (discretized, noncommutative,...) versions of Minkowski space-time would not enjoy the classical Lorentz symmetries. I compare two interesting cases: the case in which the classical symmetries are "broken", i.e., at the quantum level some classical symmetries are lost, and the case in which the classical symmetries are "deformed", i.e., the quantum space-time has as many symmetries as its classical counterpart but the nature of these symmetries is affected by the space-time quantization procedure. While some general features, such as the emergence of deformed dispersion relations, characterize both the symmetry-breaking case and the symmetry-deformation case, the two scenarios are also characterized by sharp differences, even concerning the nature of the new effects predicted. I illustrate this point within an illustrative calculation concerning the role of space-time symmetries in the evaluation of particle-decay amplitudes. The results of the analysis here reported also show that the indications obtained by certain dimensional arguments, such as the ones recently considered by Brustein, Eichler and Foffa, may fail to uncover some key features of quantum space-time symmetries.