I)-5) It is based on the mapping theory which transforms the' (truncated) fermion space into the boson space_ 6 ) Thereby the non collective boson modes are inevitably introduced corresponding to the non-collective degrees of freedom in the original fermion system- The Pauli effect due to fermion statistics is represented in the transformed boson space as the coupling effects between all the boson modes. However we must discard more or less these coupling effects since we can only handle with the limited degrees of boson modes in realistic calculations_ The validity of truncation of the rest of the boson degrees of freedom is not confirmed_ Furthermore it is not so natural to represent the non-collective fermion mode by the boson mode in the lowest order approximation, because no coher~nt properties can be expected. Hence it seems to be plausible to find a new mapping theory in which only the collective branch of modes is transformed to that of boson modes in the lowest order approximation, while the non-collective branch to itself_ This is the main motivation of the present paper. The dynamical nuclear field theory (DNFT) also treats the nuclear system in the boson-fermion system.7),B) In DNFT the self-consistent matching condition is used to eliminate the redundancy originated from introduction of the boson degrees of free dom. By using the perturbation theory for a quasi-degenerate system one can derive the effective operator defined in the limited space such that only the boson excitations exist_ However its relation to BET is not necessarily clear-cut. 9 ) In order to express the above argument concretely, we introduce a unitary operator which maps the fermion space into the boson-fermion space. By using it we can get a new type of bosonization of the correlated fermion pair modes. In the following we give the presentation of our basic idea. Fermion pair operators which we start with are the Tamm-Dancoff type phonon operator and the so-called scattering operator,