Self-Consistent-Field Method and τ-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...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2009 |
| Автори: | , , , , |
| Формат: | Стаття |
| Мова: | English |
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Інститут математики НАН України
2009
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/149250 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Self-Consistent-Field Method and τ-Functional Method on Group Manifold in Soliton Theory: a Review and New Results / S. Nishiyama, J. da Providência, C. Providência, F. Cordeiro, T. Komatsu // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 89 назв. — англ. |
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nasplib_isofts_kiev_ua-123456789-149250 |
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Nishiyama, S. da Providência, J. Providência, C. Cordeiro, F. Komatsu, T. 2019-02-19T19:22:17Z 2019-02-19T19:22:17Z 2009 Self-Consistent-Field Method and τ-Functional Method on Group Manifold in Soliton Theory: a Review and New Results / S. Nishiyama, J. da Providência, C. Providência, F. Cordeiro, T. Komatsu // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 89 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 37K10; 37K30; 37K40; 37K65 https://nasplib.isofts.kiev.ua/handle/123456789/149250 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. This paper is a contribution to the Special Issue on Kac–Moody Algebras and Applications. One of the authors (S.N.) would like to express his sincere thanks to Professor Alex H. Blin for kind and warm hospitality extended to him at the Centro de F´ısica Te´orica, Universidade de Coimbra. This work was supported by the Portuguese Project POCTI/FIS/451/94. The authors thank YITP, where discussion during the YITP workshop(YITP-W-06-13) on Fundamental Problems and Applications of Quantum Field Theory “Topological Aspects of Quantum Field Theory” – 2006 was useful to complete this work. S.N. also would like to acknowledge partial support from Projects PTDC/FIS/64707/2006 and CERN/FP/83505/2008. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Self-Consistent-Field Method and τ-Functional Method on Group Manifold in Soliton Theory: a Review and New Results Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Self-Consistent-Field Method and τ-Functional Method on Group Manifold in Soliton Theory: a Review and New Results |
| spellingShingle |
Self-Consistent-Field Method and τ-Functional Method on Group Manifold in Soliton Theory: a Review and New Results Nishiyama, S. da Providência, J. Providência, C. Cordeiro, F. Komatsu, T. |
| title_short |
Self-Consistent-Field Method and τ-Functional Method on Group Manifold in Soliton Theory: a Review and New Results |
| title_full |
Self-Consistent-Field Method and τ-Functional Method on Group Manifold in Soliton Theory: a Review and New Results |
| title_fullStr |
Self-Consistent-Field Method and τ-Functional Method on Group Manifold in Soliton Theory: a Review and New Results |
| title_full_unstemmed |
Self-Consistent-Field Method and τ-Functional Method on Group Manifold in Soliton Theory: a Review and New Results |
| title_sort |
self-consistent-field method and τ-functional method on group manifold in soliton theory: a review and new results |
| author |
Nishiyama, S. da Providência, J. Providência, C. Cordeiro, F. Komatsu, T. |
| author_facet |
Nishiyama, S. da Providência, J. Providência, C. Cordeiro, F. Komatsu, T. |
| publishDate |
2009 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
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.
|
| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/149250 |
| citation_txt |
Self-Consistent-Field Method and τ-Functional Method on Group Manifold in Soliton Theory: a Review and New Results / S. Nishiyama, J. da Providência, C. Providência, F. Cordeiro, T. Komatsu // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 89 назв. — англ. |
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