Algebro-geometric approach to a fermion self-consistent field theory on coset space SU(m+n)/S(U(m) x U(n))
The integrability-condition method is regarded as a mathematical tool to describe the symmetry of collective sub-manifold. We here adopt the particle-hole representation. In the conventional time-dependent (TD) self-consistent field (SCF) theory, we take the one-form linearly composed of the TD SCF Hamiltonian and the infinitesimal generator induced by the collective-variable differential of canonical transformation on a group. Standing on the differential geometrical viewpoint, we introduce a Lagrange-like manner familiar to fluid dynamics to describe collective coordinate systems. We construct a geometric equation, noticing the structure of coset space SU(m+n)/S(U(m) x U(n)). To develop a perturbative method with the use of the collective variables, we aim at constructing a new fermion SCF theory, i.e., renewal of TD Hartree-Fock (TDHF) theory by using the canonicity condition under the existence of invariant subspace in the whole HF space. This is due to a natural consequence of the maximally decoupled theory because there exists an invariant subspace, if the invariance principle of Schredinger equation is realized. The integrability condition of the TDHF equation determining a collective sub-manifold is studied, standing again on the differential geometric viewpoint. A geometric equation works well over a wide range of physics beyond the random phase approximation.