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The representations of a group of gauge automorphisms of the canonicalcommutation or anticommutation relations which appear on the Hilbert spaces ofisometries H_\rho implementing quasi-free endomorphisms \rho on Fock space arestudied. Such a representation, which characterizes the "charge" of \rho inlocal quantum field theory, is determined by the Fock space structure of H_\rhoitself: Together with a "basic" representation of the group, all highersymmetric or antisymmetric tensor powers thereof also appear. Hence \rho isreducible unless it is an automorphism. It is further shown by the example ofthe massless Dirac field in two dimensions that localization andimplementability of quasi-free endomorphisms are compatible with each other.Comment: 15 pages, no figure

THE CHARGE QUANTUM NUMBERS OF GAUGE INVARIANT QUASI-FREE ENDOMORPHISMS