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Progress of Theoretical Physics

A Possible Quantization of Time-Dependent Hartree-Bogoliubov Theory Based on the SO(2N+1) Algebra

M. Yamamura

Published Feb 1, 1980• 2 citations
Publisher

Abstract

A possible quantization is proposed with respect to time-dependent Hartree-Bogoliubov theory based on the SO(2N+1) algebra which was presented by Fukutome, Nishiyama and the present author. The adopted quantization procedure is an extension of Marshalek and Holzwarth's idea used in the case of the conventional time-dependent Hartree-Bogoliubov theory based on the SO(2N) algebra. As a natural consequent of the quantization, Holstein-Primakoff type boson expansion theory can be derived in a unified form of even and odd particle system.

DOI
10.1143/ptp.63.486

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Remarks on the mean-field theory based on the SO(2N + 1) Lie algebra of the fermion operators

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Toward a unified algebraic theory for mean-field Hamiltonian describing paired- and unpaired-mode effects, in this paper, we propose a generalized Hartree–Bogoliubov mean-field Hamiltonian in terms of fermion pair and creation-annihilation operators of the [Formula: see text] Lie algebra. We diagonalize the generalized Hartree–Bogoliubov mean-field Hamiltonian and throughout its diagonalization we can first obtain the unpaired mode amplitudes which are given by the self-consistent field parameters appeared in the Hartree–Bogoliubov theory together with the additional self-consistent field parameter in the generalized Hartree–Bogoliubov mean-field Hamiltonian and by the parameter specifying the property of the [Formula: see text] group. Consequently, it turns out that the magnitudes of these amplitudes are governed by such parameters. Thus, it becomes possible to make clear a new aspect of such results. We construct the Killing potential in the coset space [Formula: see text] on the Kähler symmetric space which is equivalent to the generalized density matrix. We show another approach to the fermion mean-field Hamiltonian based on such a generalized density matrix. We derive an [Formula: see text] generalized Hartree–Bogoliubov mean-field Hamiltonian operator and a modified Hartree–Bogoliubov eigenvalue equation. We discuss on the mean-field theory related to the algebraic mean-field theory based on the generalized density matrix and the coadjoint orbit leading to the nondegenerate symplectic form.

arXiv:1810.11264•International Journal of Geometric Methods in Modern Physics•Oct 7, 2019•3 citations
A possible quantization is proposed with respect to time-dependent Hartree-Bogoliubov theory based on the SO(2N+1) algebra which was presented by Fukutome, Nishiyama and the present author. The adopted quantization procedure is an extension of Marshalek and Holzwarth's idea used in the case of the conventional time-dependent Hartree-Bogoliubov theory based on the SO(2N) algebra. As a natural consequent of the quantization, Holstein-Primakoff type boson expansion theory can be derived in a unified form of even and odd particle system.
A time dependent theory is developed for the SO(2N+1) wave function which yields both paired and unpaired modes of Fermion. The time dependent SO(2N+1) equation is derived and shown to have two kinds of solution, the Hartree-Bogoliubov solution which describes even Fermion systems and the improper Hartree-Bogoliubov solution which describes odd Fermion systems. The time dependent improper Hargree-Bogoliubov equation leads to a new random phase approximation which gives a self-consistent unified description of Bose and Fermi type collective excitations in an odd Fermion system.
An induced representation of an SO(2N + 1) group has been obtained from a group extension of the SO(2N) Bogoliubov transformation for fermions to a new canonical transformation group. Embedding the SO(2N + 1) group into an SO(2N + 2) group and using the SO(2N + 2)/U(N + 1) coset variables, we develop an extended time dependent Hartree–Bogoliubov (TDHB) theory in which paired and unpaired modes are treated in an equal manner. The extended TDHB theory applicable to both even and odd fermion systems is a time dependent self-consistent field (TDSCF) theory with the same level of the mean field approximation as the usual TDHB theory for even fermion systems. We start from the Hamiltonian of the fermion system which includes, however, the Lagrange multiplier terms to select the physical spinor subspace. The extended TDHB equation is derived from the classical Euler–Lagrange equation of motion for the SO(2N + 2)/U(N + 1) coset variables in the TDSCF. The final form of the extended TDHB equation can be expressed through the variables as the representatives of the paired mode and the unpaired mode. We introduce the quasi anti-commutation relation approximation for the fermion. The parameters included in the Lagrangian multiplier terms are determined under the quasi anti-commutation relation approximation.
Toward a unified algebraic theory for mean-field Hamiltonian describing paired- and unpaired-mode effects, in this paper, we propose a generalized Hartree–Bogoliubov mean-field Hamiltonian in terms of fermion pair and creation-annihilation operators of the [Formula: see text] Lie algebra. We diagonalize the generalized Hartree–Bogoliubov mean-field Hamiltonian and throughout its diagonalization we can first obtain the unpaired mode amplitudes which are given by the self-consistent field parameters appeared in the Hartree–Bogoliubov theory together with the additional self-consistent field parameter in the generalized Hartree–Bogoliubov mean-field Hamiltonian and by the parameter specifying the property of the [Formula: see text] group. Consequently, it turns out that the magnitudes of these amplitudes are governed by such parameters. Thus, it becomes possible to make clear a new aspect of such results. We construct the Killing potential in the coset space [Formula: see text] on the Kähler symmetric space which is equivalent to the generalized density matrix. We show another approach to the fermion mean-field Hamiltonian based on such a generalized density matrix. We derive an [Formula: see text] generalized Hartree–Bogoliubov mean-field Hamiltonian operator and a modified Hartree–Bogoliubov eigenvalue equation. We discuss on the mean-field theory related to the algebraic mean-field theory based on the generalized density matrix and the coadjoint orbit leading to the nondegenerate symplectic form.