Periodicity, type $I{I}_1$ factors and free Poisson laws in interacting Fock spaces

Abstract

We show that the von Neumann algebra generated by position operators in a 2-periodic interacting Fock space is a type $I{I}_1$ factor. On the probabilistic side, we prove that the squared position operators have a Marchenko-Pastur distribution with respect to the vacuum state, yielding a natural realization of free Poisson laws within this framework.