|
Эта публикация цитируется в 1 научной статье (всего в 1 статье)
Self-Consistent-Field Method and $\tau$-Functional Method on Group Manifold in Soliton Theory: a Review and New Results
Seiya Nishiyamaa, João da Providênciaa, Constança Providênciaa, Flávio Cordeirob, Takao Komatsuc a Centro de Física Teórica, Departamento de Física, Universidade de Coimbra, P-3004-516 Coimbra, Portugal
b Mathematical Institute, Oxford OX1 3LB, UK
c 3-29-12 Shioya-cho, Tarumi-ku, Kobe 655-0872, Japan
Аннотация:
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.
Ключевые слова:
self-consistent field theory; collective theory; soliton theory; affine KM algebra.
Поступила: 5 сентября 2008 г.; в окончательном варианте 10 января 2009 г.; опубликована 22 января 2009 г.
Образец цитирования:
Seiya Nishiyama, João da Providência, Constança Providência, Flávio Cordeiro, Takao Komatsu, “Self-Consistent-Field Method and $\tau$-Functional Method on Group Manifold in Soliton Theory: a Review and New Results”, SIGMA, 5 (2009), 009, 76 pp.
Образцы ссылок на эту страницу:
https://www.mathnet.ru/rus/sigma355 https://www.mathnet.ru/rus/sigma/v5/p9
|
Статистика просмотров: |
Страница аннотации: | 346 | PDF полного текста: | 51 | Список литературы: | 25 |
|