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A NEW DESCRIPTION OF MOTION OF THE FERMIONIC SO(2N+2) TOP IN THE CLASSICAL LIMIT UNDER THE QUASI-ANTICOMMUTATION RELATION APPROXIMATION
Authors: S Nishiyama, J Da Providencia, C Providencia
Ref.: INTERNATIONAL JOURNAL OF MODERN PHYSICS A 27, 1250054 (2012)
Abstract: The boson images of fermion SO(2N + 1) Lie operators have been given together with those of SO(2N + 2) ones. The SO(2N + 1) Lie operators are generators of rotation in the (2N + 1)-dimensional Euclidean space (N: number of single-particle states of the fermions). The images of fermion annihilation-creation operators must satisfy the canonical anticommutation relations, when they operate on a spinor subspace. In the regular representation space we use a boson Hamiltonian with Lagrange multipliers to select out the spinor subspace. Based on these facts, a new description of a fermionic SO(2N + 2) top is proposed. From the Heisenberg equations of motions for the boson operators, we get the SO(2N + 1) self-consistent field (SCF) Hartree-Bogoliubov (HB) equation for the classical stationary motion of the fermion top. Decomposing an SO(2N + 1) matrix into matrices describing paired and unpaired modes of fermions, we obtain a new form of the SO(2N + 1) SCF equation with respect to the paired-mode amplitudes. To demonstrate the effectiveness of the new description based on the bosonization theory, the extended HB eigenvalue equation is applied to a superconducting toy-model which consists of a particle-hole plus BCS-type interaction. It is solved to reach an interesting and exciting solution which is not found in the traditional HB eigenvalue equation due to the unpaired-made effects. To complete the new description, the Lagrange multipliers must be determined in the classical limit. For this aim a quasi-anticommutation relation approximation is proposed. Only if a certain relation between an SO(2N + 1) parameter z and the N is satisfied, unknown parameters k and l in the Lagrange multipliers can be determined without any inconsistency.
