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Remarks on the Mean-Field Theory Based on the SO(2N+1) Lie Algebra of the Fermion Operators

Toward a unified algebraic theory for mean-field Hamiltonian (MFH) describing paired- and unpaired-mode effects, in this paper, we propose a generalized HB (GHB) MFH in terms of the SO(2N+1)Lie algebra of fermion pair and creation-annihilation operators. We diagonalize the GHB-MFH and throughout the diagonalization of which, we can first obtain the unpaired mode amplitudes which are given by the SCF parameters appeared in the HBT together with the additional SCF parameter in the GHB-MFH and by the parameter specifying the property of the SO(2N +1) 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 the results. We construct the Killing potential in the coset space SO(2N)/U(N) on the Kaehler symmetric space which is equivalent with the generalized density matrix (GDM). We show another approach to the fermion MFH based on such a GDM. We derive an SO(2N +1) GHB MF operator and a modified HB eigenvalue equation. We discuss on the MF theory related to the algebraic MF theory based on the GDM and the coadjoint orbit leading to the non-degenerate symplectic form.