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In a slight different way from the previous one, we propose a modifiednon-Euclidian transformation on the SO(2N+2)/U(N+1) Grassmannian which give theprojected SO(2N+1) Tamm-Dancoff equation. We derive a classical time dependent(TD) SO(2N+1) Lagrangian which, through the Euler-Lagrange equation of motionfor SO(2N+2)/U(N+1) coset variables, brings another form of the previousextended-TD Hartree-Bogoliubov (HB) equation. The SO(2N+1) random phaseapproximation (RPA) is derived using Dyson representation for paired andunpaired operators. In the SO(2N) HB case, one boson and two boson excitedstates 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 thegroup theoretical viewpoint, we show the existence of a symplectic two-formomega.

international journal of geometric methods in modern physics

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