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Progress of Theoretical Physics

Relation between the Boson Expansion Theory and the Boson-Fermion Expansion Theory

K. Taniguchi, Atsushi Kajiyama, Y. Miyanishi

Published Nov 1, 1994• 4 citations
Publisher

Abstract

The relation between the hermitian type boson expansion theory and the boson-fermion expansion theory recently proposed as the alternative to it is discussed. It is shown that the apparent difference between some of the expansion coefficients in both theories is due to arbitrariness about the definition of the collective state in the fermion space and that one can remove the difference by the appropriate unitary transformation.

DOI
10.1143/ptp.92.975

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The relation between the hermitian type boson expansion theory and the boson-fermion expansion theory recently proposed as the alternative to it is discussed. It is shown that the apparent difference between some of the expansion coefficients in both theories is due to arbitrariness about the definition of the collective state in the fermion space and that one can remove the difference by the appropriate unitary transformation.
We propose a new method that transcribes the fermion system to the boson-fermion system by introducing a unitary operator defined in the whole boson-fermion space. We obtain the boson-fermion expansion of the fermion pair operators such that only the collective pair modes are transformed into boson operators while the non-collective modes are unchanged in the lowest order approximation. The odd- A system is treated in the same formalism as the even-even nuclear system. We briefly discuss the relation between our method and the conventional boson expansion theory.
We propose a new boson expansion theory that does not premise the closed-algebra approximation, indispensable for formulation until now, as an extension of the conventional practical boson expansion methods that have tried to elucidate nuclear collective motion, which is a method that allows the closed-algebra approximation not to be used or to be used appropriately, enables us to obtain the boson expansion easier, and reproduces the fermion subspace onto the boson subspace more faithfully than the conventional practical methods. The two-phonon norm matrix composed of all the phonon excitation modes is investigated in detail, which reveals the mechanism, essential for the boson expansion methods, of how we should construct the fermion subspace to be mapped from the whole fermion space. The conventional practical boson expansion methods have applied the closed-algebra approximation improperly to strengthen the effect of the Pauli principle inappropriately, which should be replaced by those that do not use that approximation or use it properly.
The use of physical boson basis states is stressed for the calculations in the boson space. The explicit form of physical boson basis states in terms of bosons is derived for the nonunitary boson mapping of Dyson. The ambiguity in the normalization introduced due to the use of bi-orthonormal basis states is satisfactorily resolved, resulting in a Hermitian matrix. This Hermitian matrix is found to coincide with the Hamiltonian matrix in the fermion space. The model cases where the use of a boson basis is justified are shown to be consistent with our view of using physical boson basis states.
A comparison between the finite Dyson-type and the infinite Holstein-Primakoff-type boson expansion is presented, considering both particle-hole and pairing type excitations. The significance of truncating to the truly collective branches is emphasized, and the convergence of the Holstein-Primakoff expansion is discussed. It is shown that in both expansions there is no need to construct physical basis states. Complete equivalence between the two types of expansion is exhibited by treating these excitations within models which are schematic but not totally unrealistic.NUCLEAR STRUCTURE Boson expansion theory, convergence, comparison of different methods, finite Dyson expansion, infinite Holstein-Primakoff expansion, pairing and particle-hole collective motions.