A Solution of an Extended Hartree-Bogoliubov Equation on the Coset Space SO(2N+2)/U(N+1) for Unified Description of Bose and Fermi Type Collective Excitations

The time dependent Hartree-Bogoliubov (TDHB) theory is the first standard approximation in the many-body theoretical description of a superconducting fermion system [1]. The HB wave function for the fermion system represents Bose condensate states of fermion pairs. It is a good approximation for the ground state of the fermion system with a short-range pairing interaction that produces a spontaneous Bose condensation of the fermion pairs. The fermion number-nonconservation of the HB wave function is a consequence of the spontaneous Bose condensation of fermion pairs, which causes coherences in the phases of superconducting (Bose condensed) fermions. The SO(2N) Lie algebra of the fermion pair operators contains the U(N) Lie algebra as a subalgebra. Here SO(2N) and U(N) denote the special orthogonal group of 2N dimensions and the unitary group of N dimensions, respectively (N is the number of single-particle states of the fermions). The canonical transformation of the fermion operators generated by the Lie operators in the SO(2N) Lie algebra induces the well-known generalized Bogoliubov transformation for the fermions. The TDHB equation is derived from the classical Euler-Lagrange equation of motion for the SO(2N)/U(N) coset variables [2]. The TDHB theory is, however, applicable only to even fermion systems. For odd fermion systems we have no TD self-consistent field (SCF) theory with the same level of the mean field approximation as the TDHB theory.