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Modified Non-Euclidian Transformation on the SO(2N+2)/U(N+1) Grassmannian and SO(2N+1) Random Phase Approximation for Unified Description of Bose and Fermi Type Collective Excitations

In a slight different way from the previous one, we propose a modified non-Euclidian transformation on the SO(2N+2)/U(N+1) Grassmannian which give the projected SO(2N+1) Tamm-Dancoff equation. We derive a classical time dependent (TD) SO(2N+1) Lagrangian which, through the Euler-Lagrange equation of motion for SO(2N+2)/U(N+1) coset variables, brings another form of the previous extended-TD Hartree-Bogoliubov (HB) equation. The SO(2N+1) random phase approximation (RPA) is derived using Dyson representation for paired and unpaired operators. In the SO(2N) HB case, one boson and two boson excited states are realized. We, however, stress non existence of a higher RPA vacuum. An integrable system is given by a geometrical concept of zero-curvature, i.e., integrability condition of connection on the corresponding Lie group. From the group theoretical viewpoint, we show the existence of a symplectic two-form omega.