Belyaev-Zelevinsky-Marshalek boson mappings

Let PSI/sub mi/sup +//, PSI/sub mi/(-j≤m≤j , j one-half integral, 2j+1=2Ω,i=1...Λ)be fermion creation and annihilation operators. We define pair and multipole operators not coupled to good angular momentum, which are generators of the unitary symplectic algebra Sp(2Λ). Particular cases studied previously include the LS coupled shell model algebras and the Sp(6) model of Ginocchio. We are interested in a unitary boson mapping from a fermion basis utilizing the subgroup chain Sp(2Λ) > U(Λ) (BZM mapping). This chain is attractive mathematically because the mapping can be done through the intermediary of a Dyson (non-unitary) mapping written down by many authors. It is attractive physically because of its relation to the pseudo-SU(3) model for deformed nuclei and possible relation to the deformed limit of IBM. A general method of unitarizing the Dyson mapping, described previously, yields a result which grows in complexity with Λ. It has now been recognized that the restrictions imposed by the Pauli principle imply that the previous results can be greatly simplified in two limiting cases. In particular, for the smallest values of j and arbitrary Λ, simple, explicit analytic formulas can be given for the mapping. The second limit of small Λ and arbitrary j can also be solved easily. These results pertain to the representation which contains the vacuum state and describes the low-lying (collective) states of even nuclei. The same methods can be applied to the more familiar j-j coupled shell model algebras (orthogonal algebras)