Interacting Fock space versus full Fock module

Abstract

We present several examples where moments of creators and an- nihilators on an interacting Fock space may be realized as moments of creators and annihilators on a full Fock module. Motivated by this experience we an- swer the question, whether such a possibility exists for arbitrary interacting Fock spaces, in the armative sense. We treat the problem in full algebraic generality. As a by-product, we find a new notion of positivity for ⁄-algebras which allows to construct tensor products of Hilbert modules over ⁄-algebras. Finally, we consider a subcategory of interacting Fock spaces which are embeddable into a usual full Fock space. We see that a creator a ⁄ (f) on the interacting Fock space is represented by an operator{' ⁄ (f), where ' ⁄ (f) is a usual creator on the full Fock space and { is an operator which does not change the number of particles. In the picture of Hilbert modules the one-particle sector is replaced by a two-sided module over an algebra which contains {. Therefore, { may be absorbed into the creator, so that we are concerned with a usual creator. However, this creator does not act on a Fock space, but rather on a Fock module.