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Progress of Theoretical Physics

The Generalized Bogolyubov-Valatin Transformation of Neutron and Proton System

Y. Takahashi

Published Dec 1, 1971• 8 citations
Publisher

Abstract

The charge independent pairing interaction in the system of neutron and proton with many j-configurations is studied using the five-dimensional quasispin formalism, and the generalized BCS the9ry is obtained as a natural generalization of the BCS theory of identical nucleon system.It is shown that a rotation in the five-dimensional quasispin space can be regarded as the generalized Bogolyubov-Valatin transformation in our generalized BCS theory.The normalized BCS ground state and the solutions of the BCS equations for several models are explicitly given in analogy to the case of identical nucleon system.*> The phase of the modified annihilation operator has been taken to be the same as that of Ichimura.

DOI
10.1143/ptp.46.1749

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Journal of Physics G Nuclear and Particle Physics•Jan 1, 1999•3 citations
The charge independent pairing interaction in the system of neutron and proton with many j-configurations is studied using the five-dimensional quasispin formalism, and the generalized BCS the9ry is obtained as a natural generalization of the BCS theory of identical nucleon system.It is shown that a rotation in the five-dimensional quasispin space can be regarded as the generalized Bogolyubov-Valatin transformation in our generalized BCS theory.The normalized BCS ground state and the solutions of the BCS equations for several models are explicitly given in analogy to the case of identical nucleon system.*> The phase of the modified annihilation operator has been taken to be the same as that of Ichimura.
The new method presented by B. Bremond and J. G. Valatin is generalized which at-tache ~ four body correlation to the Bogoliubov transformation. The generalization carried in this paper is similar to that from the Bogoliubov transformation to the generalized Bogoliubov one. We find a criterion of whether a trial ground state vector has an essential feature of four body correlation or it is like that of the generalized Bogoliubov transformation. This method is applied to a system with a charge independent pairing interaction. There appear two. different kinds of single quasiparticle excitation energy due to proton-neutron interaction.
The homogeneous canonical transformation of boson operators is studied from the standpoint of the SU(1, 1) quasi-spin formalism. It is shown that this transformation can be regarded as a rotation in the SU(1, 1) quasi-spin space. This situation is just analogous to the SU(2) quasi-spin formalism of many-fermion problem, in which fermion Bogolyubov transformation can be regarded as a rotation in the SU(2) quasi-spin space. The transformation law of state vectors under this boson Bogolyubov transformation is then written down explicitly in terms of the SU(1, 1) d-functions obtained in the preceding paper. For clarity, we treat the simplest nontrivial example of one-dimensional harmonic oscillator or, equivalently, a system consisting of one kind of bosons. Generalization to many-boson problem and its possible applications to Bose fluid and nuclear structure are briefly discussed.
Abstract The structure of the Bogoljubov -Valatin transformation and the HFB state is investigated by group-theoretical methods. A direct proof of the Bloch-Messiah factorization theorem shows that the second factor which contains the BCS part of the BV transformation can be chosen always real. An unitary representation of the Bogoljubov-Valatin group in the Fock space is constructed which allows to write down various equivalent expressions for the HFB state vector. The relation between the "unitary form" and the usual form of | HFB〉 is clarified using the "quasispin" concept in connection with a theorem about the factorization of the D(3) Lie group.