Boson Expansions for Fermion Pair Operators: The SinglejLevel

A new version of a method due originally to Marumori for the representation of fermion pair operators as expansions in boson variables is developed and applied to nucleons confined to a single $j$ shell. In the present formulation, a seniority basis of good angular momentum states is imaged in a subspace of the boson space — the physical boson space — characterized by the same quantum numbers. This transformation has the following properties, only the first of which is shared with previous work: (i) Pauli-principle restrictions are satisfied. (ii) The fermion pair operators approach the leading terms of their expansions as $j\to \infty$. (iii) For fixed $j$ one can identify maximal subspaces of the physical boson space in which the boson expansions converge and which define the maximum extent to which the model physical system can exhibit vibrational behavior. An extension appropriate to systems with an odd number of nucleons is also described. By consideration of the direct product, the method described here should be useful for shell-model calculations of a restricted set of states for large particle number.