Fock representations of Zamolodchikov algebras and R-matrices

Abstract

A variation of the Zamolodchikov-Faddeev algebra over a finite dimensional Hilbert space $\mathcal{H}$ and an involutive unitary $R$-Matrix $S$ is studied. This algebra carries a natural vacuum state, and the corresponding Fock representation spaces $\mathcal{F}_S(\mathcal{H})$ are shown to satisfy $\mathcal{F}_{S\boxplus R}(\mathcal{H}\oplus\mathcal{K}) \cong \mathcal{F}_S(\mathcal{H})\otimes \mathcal{F}_R(\mathcal{K})$, where $S\boxplus R$ is the box-sum of $S$ (on $\mathcal{H}\otimes\mathcal{H}$) and $R$ (on $\mathcal{K}\otimes\mathcal{K}$). This analysis generalises the well-known structure of Bose/Fermi Fock spaces and a recent result of Pennig.\par It is also discussed to which extent the Fock representation depends on the underlying $R$-matrix, and applications to quantum field theory (scaling limits of integrable models) are sketched.