Localization and symmetries

Abstract

The violation of the Noether relation between symmetries and charges is reduced to the time dependence of the charge associated with a conserved current. For the U(1) gauge symmetry a non-perturbative control of the charge commutators is obtained by an analysis of the Coulomb charged fields. From this, in the unbroken case we obtain a correct expression for the electric charge on the Coulomb states, its superselection and the presence of massless vector bosons; in the broken case, we obtain a general non-perturbative version of the Higgs phenomenon, i.e. the absence of massless Goldstone bosons and of massless vector bosons. The conservation of the (gauge-dependent) current associated with the U(1) axial symmetry in QCD is shown to be compatible with the time dependence of the corresponding charge commutators and a non-vanishing η′ mass, as a consequence of the non-locality of the (conserved) current.