Twisted tensor products of field algebras

Abstract

Let $\mathcal{A}$ be a C*-algebra, $\mathbf{h}$ a Hilbert space, and ${\mathcal{C}}_{\mathbf{h}}$ the CAR algebra over $\mathbf{h}$. We construct a twisted tensor product of $\mathcal{A}$ by ${\mathcal{C}}_{\mathbf{h}}$ such that the two factors are not necessarily one in the relative commutant of the other. The resulting C*-algebra may be regarded as a generalized CAR algebra constructed over a suitable Hilbert $\mathcal{A}$-bimodule. As an application, we exhibit a class of fixed-time models where a free Dirac field (giving rise to the ${\mathcal{C}}_{\mathbf{h}}$ factor) in general is not relatively local to a free scalar field (which yields the $\mathcal{A}$ factor). In some of the models, gauge-invariant combinations of the two (not relatively local) fields form a local observable net.