1.c: IL-J Nuclear Physics Al72 (1971) 145-165; @ North-Holland Publishing Co., Amsterdam l.D.2 Not to be reproduced by photoprint or microfilm without written permission from the publisher BOSON DESCRIPTION OF COLLECTIVE STATES (I). Derivation of the boson transformation for even fermion systems D. JANSSEN, F. DONAU, S. FRAUENDORF Central Institute for Nuclear Research Rossendorf and R. V. JOLOS Joint Institute for Nuclear Research, Dubna Received 25 March 1971 Abstract: Using the algebraic method of Marumori 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 prob- lem of the separation of the “physical” and “unphysical” components has been solved by the introduction of a non-linear boson transformation. 1. Introduction 1.1. ADVANTAGES OF A BOSON TRANSFORMATION AND METHODS KNOWN FOR ITS DERIVATION In order to describe collective excitations in doubly even nuclei one has to solve two tasks: (i) The definition of a collective coordinate as far as possible free of arbitrariness and the derivation of the Hamiltonian depending on this coordinate. (ii) The solution of the Schr6dinger equation. The second task is mainly a numerical problem. For the solution of the first task different methods are proposed. One possibility is given by the transformation of the Hamiltonian into a boson representation and the introduction of normal coordinates using a canonical and unitary transformation in the boson space. Up to now two methods for the derivation of such a boson rep- resentation are treated for the description of the anharmonic effects in spherical doubly even nuclei. The first, which we shall denote the algebraic method, was worked out in the papers of Belyaev and Zelevinski ‘) and Ssrensen “) and by Marumori et zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA al. “). In refs. ‘* ‘) the boson expansion is defined by the algebra of fermion pair and density operators. This is really an expansion in terms of (25+ l)-* and the 145