Fock representations of ZF algebras and R-matrices

Abstract

Abstract A variation of the Zamolodchikov–Faddeev algebra over a finite-dimensional Hilbert space $${\mathcal {H}}$$H and an involutive unitary R-Matrix S is studied. This algebra carries a natural vacuum state, and the corresponding Fock representation spaces $${\mathcal {F}}_S({\mathcal {H}})$$FS(H) are shown to satisfy $${\mathcal {F}}_{S\boxplus R}({{\mathcal {H}}}\oplus {{\mathcal {K}}}) \cong {\mathcal {F}}_S({{\mathcal {H}}})\otimes {\mathcal {F}}_R({{\mathcal {K}}})$$FS⊞R(H⊕K)≅FS(H)⊗FR(K), where $$S\boxplus R$$S⊞R is the box-sum of S (on $${{\mathcal {H}}}\otimes {{\mathcal {H}}}$$H⊗H) and R (on $${{\mathcal {K}}}\otimes {{\mathcal {K}}}$$K⊗K). This analysis generalises the well-known structure of Bose/Fermi Fock spaces and a recent result of Pennig. These representations are motivated from quantum field theory (short-distance scaling limits of integrable models).