Fermion currents in 1+1 dimensions

Abstract

Let ℱ denote the Fock space for a free massless fermion field in two space-time dimensions. If Q and Q5 denote the charge and chiral charge operators, then ℱ=⊕n1,n2ℱn1,n2, where ℱn1,n2 is the joint eigenspace of Q and Q5 with corresponding eigenvalues n1 and n2. The smeared time-zero free fermion currents formally given by Jμ( f) =∫:ψ̄(x)γμψ(x): f(x) dx [μ=0,1; f∈𝒮(R)] are in fact self-adjoint densely defined operators on ℱ and their exponentials generate a C* algebra which leaves each subspace ℱn1,n2 invariant. If a periodic box cutoff for the fermions is introduced, Uhlenbrock has shown that this C* algebra acts irreducibly in each ‘‘sector’’ ℱn1,n2. We prove this same result without any cutoffs and determine the properties of the representations in each sector. The importance of this result for the problem of constructing fermion fields from ‘‘observables’’ in two-dimensional models, such as those of Thirring and Schwinger, is discussed.