Self consistent field method and τ-function : Construction of SCF theory on an infinite Fock space

孝男 小松

Alvin K. Benson

Seiya Nishiyama

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., υ (external parameter)-dependent Hartree-Fock (HF) theory.Toward such an ultimate goal, the υ-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 υ-dependent potential with a Υ-periodicity.A bilinear equation for the υ-HF theory has been transcribed onto the corresponding τ -function using the regular representation for the group and the Schur-polynomials.The υ-HF SCF theory on an infinite-dimensional Fock space F ∞ leads to a dynamics on an infinite-dimensional Grassmannian Gr ∞ and may describe more precisely such a dynamics on the group manifold.A finite-dimensional Grassmannian is identified with a Gr ∞ which is affiliated with the group manifold obtained by reducting gl(∞) 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.

Takao Komatsu, Seiya Nishiyama

A unified aspect of the self-consistent field (SCF) method and the τ-functional method is presented. SCF theory in the τ-functional space F ∞ manifestly results in a gauge theory of fermions and then a collective motion appears as a motion of fermion gauges with a common factor. This provides a new algebraic tool for the microscopic understanding of fermion many-body systems.

Seiya Nishiyama, João da Providência

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.