Self-Consistent-Field Method and $\tau$-Functional Method on Group Manifold in Soliton Theory: a Review and New Results

The maximally-decoupled method has been considered as a theory to apply an basic idea of an integrability condition to certain multiple parametrized symmetries. The method is regarded as a mathematical tool to describe a symmetry of a collective submanifold in which a canonicity condition makes the collective variables to be an orthogonal coordinate-system. For this aim we adopt a concept of curvature unfamiliar in the conventional time-dependent (TD) self-consistent field (SCF) theory. Our basic idea lies in the introduction of a sort of Lagrange manner familiar to fluid dynamics to describe a collective coordinate-system. This manner enables us to take a one-form which is linearly composed of a TD SCF Hamiltonian and infinitesimal generators induced by collective variable differentials of a canonical transformation on a group. The integrability condition of the system read the curvature $C= 0$. Our method is constructed manifesting itself the structure of the group under consideration. To go beyond the maximaly-decoupled method, we have aimed to construct an SCF theory, i.e. $\upsilon$ (external parameter)-dependent Hartree–Fock (HF) theory. Toward such an ultimate goal, the $\upsilon$-HF theory has been reconstructed on an affine Kac–Moody algebra along the soliton theory, using infinite-dimensional fermion. An infinite-dimensional fermion operator is introduced through a Laurent expansion of finite-dimensional fermion operators with respect to degrees of freedom of the fermions related to a $\upsilon$-dependent potential with a $\Upsilon$-periodicity. A bilinear equation for the $\upsilon$-HF theory has been transcribed onto the corresponding $\tau$-function using the regular representation for the group and the Schur-polynomials. The $\upsilon$-HF SCF theory on an infinite-dimensional Fock space $F_\infty$ leads to a dynamics on an infinite-dimensional Grassmannian $\mathrm{Gr}_\infty$ and may describe more precisely such a dynamics on the group manifold. A finite-dimensional Grassmannian is identified with a $\mathrm{Gr}_\infty$ which is affiliated with the group manifold obtained by reducting $gl(\infty)$ to $sl(N)$ and $su(N)$. As an illustration we will study an infinite-dimensional matrix model extended from the finite-dimensional $su(2)$ Lipkin–Meshkov–Glick model which is a famous exactly-solvable model.