The implementation of non-surjective Bogoliubov transformations in Fock states over CAR algebras is investigated. Such a transformation is implementable by a Hilbert space of isometries if and only if the well-known Shale-Stinespring condition is met. In this case, the dimension of the implementing Hilbert space equals the square root of the Watatani index of the associated inclusion of CAR algebras, and both are determined by the Fredholm index of the corresponding one-particle operator. Explicit expressions for the implementing operators are obtained, and the connected components of the semigroup of implementable transformations are described.
We study the semigroup of Bogoliubov endomorphisms of the canonical commutation relations which give rise to representations of the Cuntz algebra O∞ on Fock space and describe the corresponding Cuntz algebra generators in detail.
Abstract. The representations of a group of gauge automorphisms of the canonical commutation or anticommutation relations which appear on the Hilbert spaces of isometries H̺ implementing quasi-free endomorphisms ̺ on Fock space are studied. Such a representation, which characterizes the "charge" of ̺ in local quantum field theory, is determined by the Fock space structure of H̺ itself: Together with a "basic" representation of the group, all higher symmetric or antisymmetric tensor powers thereof also appear. Hence ̺ is 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.
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