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Title: | Self-Consistent-Field Method and $τ$-Functional Method on Group Manifold in Soliton Theory: a Review and New Results |

Authors: | Nishiyama, Seiya Providencia, João da Providência, Constança Cordeiro, Flávio Komatsu, Takao |

Keywords: | self-consistent field theory; collective theory; soliton theory; affine KM algebra |

Issue Date: | 22-Jan-2009 |

Publisher: | Department of Applied Research, Institute of Mathematics of National Academy of Science of Ukraine |

Project: | POCTI/FIS/451/94 PTDC/FIS/64707/2006 CERN/FP/83505/2008 |

Serial title, monograph or event: | Symmetry, Integrability and Geometry: Methods and Applications (SIGMA) |

Volume: | 5 |

Abstract: | 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., (external parameter)-dependent Hartree–Fock (HF) theory. Toward such an ultimate goal, the -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 -dependent potential with a -periodicity. A bilinear equation for the -HF theory has been transcribed onto the corresponding -function using the regular representation for the group and the Schur-polynomials. The -HF SCF theory on an infinite-dimensional Fock space F1 leads to a dynamics on an infinite-dimensional Grassmannian Gr1 and may describe more precisely such a dynamics on the group manifold. A finite-dimensional Grassmannian is identified with a Gr1 which is affiliated with the group manifold obtained by reducting gl(1) 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. |

URI: | https://hdl.handle.net/10316/110420 |

ISSN: | 18150659 |

DOI: | 10.3842/SIGMA.2009.009 |

Rights: | openAccess |

Appears in Collections: | FCTUC Física - Artigos em Revistas Internacionais |

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