Quasi-bosons approximate bosons

In the papers [2–5], operators on the Fermi-Fock space that represent a Bose algebra are constructed. Since the operators fulfilling the CCR are unbounded, the Fermi-Fock space is infinite dimensional, and thus the Fermi one-particle space is infinite dimensional. In this paper we consider the construction in [2] as a limit case of finite dimensional representations of the quasi-bosons. We show that the quasi-Bose operators converge pointwise to the Bose operators. The quasi-Bose operators are naturally divided into two sets, the quasi-annihilation and the quasi-creation operators. The quasi-annihilation operators annihilate the vacuum and commute pairwise. We show that the quasi-creation operators are the adjoints of the quasi-annihilation operators, and hence they commute pairwise as well.