Boson-Fermion Expansions

Abstract

Boson expansion theory is extended so as to allow for the coupling of several fermion-like valence particles or holes to the collective vibrations of a closed shell, described in terms of boson degrees of freedom. The result is a faithful mapping of the original many-fermion problem into a certain boson-fermion subspace without redundant variables. Generalizations of the Dyson and Holstein-Primakoff representations are derived with the aid of al­ gebraic techniques. § l. Introduction The boson expansion method exploits the remarkable fact that fermion behavior can be perfectly replicated within a subspace of a suitable boson Hilbert space.ll It provides an ideal microscopic avatar for phenomenological theories of collective motion, from the venerable Bohr-Mottelson model to that current cynosure, the IBA, all of which are formulated in terms of bosons.2> Following the pioneering work of Beliaev and Zelevinsky 3> and of Marumori, Yamamura and coworkers, 4> the subject of boson expansions has had a slow and, at times, precarious evolution/> but is now in the midst of a renaissance. It has even been applied recently to particle physics.5> The renewed popularity of boson expansion theory is certainly due in part to its utility and the interest in the IBA; but the opportunities for rederivation by the most fashion­ able methods, such as path-integration, 6> generalized coherent-state representa­ tions,n etc., have probably also played a role. In the usual boson expansion method, the entire fermion space is mapped injectively into the physical subspace of the so-called ideal space. The ortho­ gonal complement of the physical subspace is called the unphysical subspace, since it has nothing to do with the fermion problem. If the number of particles is even, the physical subspace is purely bosonic with a distinct boson for each fermion-pair excitation.8>,9> If the number of particles is odd, the physical subspace lies in the tensor product of the boson space with the space of an ideal odd fermion. Therefore, all fermion pairs are bosonized, and at most one nucleon may persist as fermion-like.*>,lo>.m