Central extensions and quantum physics

The unitary implementation of a symmetry group $G$ of a classical system in the corresponding quantum theory entails unavoidable deformations $\TG$ of $G$, namely, central extensions by the typical phase invariance group U(1). The appearance of central charges in the corresponding Lie-algebra quantum commutators, as a consequence of non-trivial responses of the phase of the wave function under symmetry transformations, lead to a quantum generation of extra degrees of freedom with regard to the classical counterpart. In particular, symmetries of the Hall effect, Yang-Mills and conformally invariant classical field theories are affected when passing to the quantum realm.