On the Group Theoretic Approach to the Canonical Anticommutation Relations

Abstract

In [1] Slawny formulated the C^* algebra of the CAR and CCR as twisted group C^* algebras for certain abelian groups. In this note we extend his work, in the case of the CAR algebra, by applying some recent results from the projective representation theory of abelian groups. We show that the Fock representation is monomial and construct pure infinite tensor product states via a group theoretic method. We also sketch some other constructions of irreducible representations including non-product ones.