Approach to a Fermionic SO(2N+2) Rotator Based on the SO(2N+1) Lie Algebra of the

Boson images of fermion SO(2N+1) Lie operators are given together with those of SO(2N+2) ones. The SO(2N+1) Lie operators are generators of rotation in the 2N+1 dimensional Euclidian space. The rotator has coordinate transformations for space fixed and body fixed coordinate frames. Images of the fermion annihilation-creation operators must satifsy canonical anti-commutation 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. From the Heisenberg equations of motions for the boson operators, in the c-number limit of the Lie operators which is obtained from the body fixed transformation of the bosons, we get the SO(2N+1) selfconsistent field (SCF) Hartree-Bogoliubov (HB) equation for the classical stationary motion of the fermion rotators. Decomposing an SO(2N+1) matrix into matrices describing paired and unpaired mode of fermions, we obtain a new form of the SO(2N+1) SCF equation with respect to the paired-mode amplitudes. The new equation is applied to a superconducting toy-model and can be solved to reach an interesting and exciting solution. A determination of the Lagrange multipliers is proposed in the classical limit. Finally we stress to attempt a group theoretical approach to formation of the Lax pair for the SO(2N+2) rotator.