1997 Fiscal Year Final Research Report Summary
Mathematical Problems in Quantum Field Theory and Infinite-Dimensional Analysis
| Project/Area Number |
08454021
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
解析学
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| Research Institution | HOKKAIDO UNIVERSITY |
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Principal Investigator |
ARAI Asao Hokkaido University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (80134807)
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| Co-Investigator(Kenkyū-buntansha) |
MIKAMI Toshio Hokkaido University, Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (70229657)
INOUE Akihiko Hokkaido University Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (50168431)
TSUDA Ichiro Hokkaido University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (10207384)
KISHIMOTO Akitaka Hokkaido University Graduate School of Science, Professor, 大学院・理学研究科, 教授 (00128597)
AGEMI Rentaro Hokkaido University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (10000845)
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| Project Period (FY) |
1996 – 1997
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| Keywords | canonical commutation relations (CCR) / gauge theory / qauntum field / Dirac operator / supersymmetric qauntum field theory / Fock space / Schrodinger operator |
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| Research Abstract |
(1) Representations of canonical commutation relations (CCR) appearing in a (non-commutative) gauge theory on a non-simply connected region of R^3 have been analyzed indetail. These representations are realized by the set of the position and the physical momentum operators. Properties of the strongly continuous 1-parameter unitary groups generated by the physical momentum operators (commutation relations, irreducibility etc.) were clarified as well as connections to the Aharonov-Bohm effect, representations of quantum groups and quantum lattice gauge theory. Moreover, the coupling of the quantum system to a quantized radiation field was considered. As a result, new classes of representations of CCR were discovered on the tensor product Hibert space of L^2 (R^3) and the Fock space of the quantized radiation field. These are original discoveries.
(2) A necessary and sufficient condition for two Dirac operators on the boson-fermion Fock space to strongly anticommute was characterized in terms of the strong anticommutativity of Dirac operators on the one-particle base Hilbert space.
(3) A class of representations of CCR with infinite degrees of freedom (or indexed by an infinite dimensional Hilbert space) was constructed in connection with perturbation problem of embedded eigenvalues in quantum field models.
(4) A new estimate for the groundstate energy of the Schrodinger operator was derived.
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