Boson description of collective states

Abstract

Using the algebraic method of Mammon et al. an extension of the boson representations of Holstein and Primakoff and Dyson for spin operators is given for the case of fermion pair and density operators. Furthermore we demonstrate that an equivalent formulation arises when the generator-coordinate method is applied. Using Dyson's concept in an exact way a finite boson expansion of the fermion pair and density operators is derived. As a consequence the resulting Hamiltonian contains the boson operators at most in sixth order. However this Dyson transformation is not unitary and therefore the Hamiltonian is not hermitean. Thus the diagonalization of the Hamiltonian leads to a bi-orthogonal set of eigenstates. Similar to the Dyson theory these states contain components which violate the Pauli principle. The problem of the separation of the “physical” and “unphysical” components has been solved by the introduction of a non-linear boson transformation.