The Charge Quantum Numbers of Gauge Invariant Quasi-free Endomorphisms

Abstract

The representations of a group of gauge automorphisms of the canonical commutation or anticommutation relations which appear on the Hilbert spaces of isometries H_ρimplementing quasi-free endomorphisms ρon Fock space are studied. Such a representation, which characterizes the "charge" of ρin local quantum field theory, is determined by the Fock space structure of H_ρitself: Together with a "basic" representation of the group, all higher symmetric or antisymmetric tensor powers thereof also appear. Hence ρis reducible unless it is an automorphism. It is further shown by the example of the massless Dirac field in two dimensions that localization and implementability of quasi-free endomorphisms are compatible with each other.