On the mapping after and before truncation in the boson expansion theory

Using the norm operator method, which extends and corrects the conventional boson expansion theories, we investigate two boson mappings of the boson expansion theory, the so-called mapping after truncation and the mapping before truncation. The difference between them stems from the treatment of the phonon excitation modes; those not adopted as boson excitation modes in the former mapping are first all adopted as boson excitation modes and then truncated later in the latter mapping. If and only if the commutation relations among the phonon operators are closed among the excitation modes adopted as the boson excitation modes in the mapping after truncation, the mapping after and the mapping before truncation coincide, not depending on the types, Hermitian and non-Hermitian. We also investigate the Park operator, which judges whether a boson state vector is physical, and reveal that the conventional claim, which claims that it is applicable only when the mapping is that of the whole fermion space, is incorrect.