| Project/Area Number |
09640178
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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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 | Nagoya University |
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Principal Investigator |
ACCARDI Luigi Graduate School of Mathematics, Nagoya University Professor, 大学院・多元数理科学研究科, 教授 (80283464)
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| Co-Investigator(Kenkyū-buntansha) |
OHYA Masamori Faculty of Engineering Science Tokyo Science University Professor, 理工学部, 教授 (90112896)
MINAMI Kazuhiko Graduate School of Mathematics, Associate Professor, 大学院・多元数理科学研究科, 助教授 (40271530)
AOMOTO Kazuhiko Graduate School of Mathematics, Professor, 大学院・多元数理科学研究科, 教授 (00011495)
OBATA Nobuaki Graduate School of Mathematics, Associate Professor, 大学院・多元数理科学研究科, 助教授 (10169360)
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| Project Period (FY) |
1997 – 1998
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| Project Status |
Completed (Fiscal Year 1998)
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| Budget Amount *help |
¥3,000,000 (Direct Cost: ¥3,000,000)
Fiscal Year 1998: ¥1,300,000 (Direct Cost: ¥1,300,000)
Fiscal Year 1997: ¥1,700,000 (Direct Cost: ¥1,700,000)
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| Keywords | Quantum noise / Stochastic limit / Quantum Ito formula / Quantum Markov chain / Quantum communication / Quantum entropy / Nonlinear interaction / Interacting Fock space / レヴィ・ラプラシアン / ヤンミルズ方程式 / 量子論の確率極限 / 相互作用フォック加群 / ウィグナー半円則 / シングルトン独立性 / 量子ランダムウォーク |
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| Research Abstract |
(1) Stochastic Limit of Quantum Theory Through the stochastic limit quantum noises and quantum stochastic differential equations are derived from canonical Hamiltonian models. This *ethod is applied to a model involving strong nonlinear interaction to obtain a new type of quantum noise. For example, a standard model of QED without dipole approximation yields a singular type of noncrossing diagrams and nonlinear deformation of Wigner's semicircle law. The concept of an interacting Fock module is introduced in order to unify this kind of phenomena.
(2) Nonlinear Extension of Classical and Quantum Stochastic Calculus Motivated by (1) , we studied higher powers of quantum white noises which are considered as the first class of an infinite hierarchy of noises. Tha associated Ito formula is proved with renormalization. Moreover, unique existence of solutions is proved for a class of normal-ordered white noise equations involving higher powers of quantum white noises and a relation to quantum stochastic differential equations is established.
(3) Central Limit Theorems Within the framework of algebraic probability theory, the existing concepts of independence have been unified and the new notion of singleton independence is introduced. The associated central limit theorem is proved and the limit stochastic process is obtained. This result is related to orthogonal polynomials, their q-deformations, and Gaussianization * probability measures.
(4) Quantum Markov Chains Classification of quantum Markov states and lifting problems in quantum communication channels are discussed. An interesting connection between basic algorithyms appearing in quantum computing and quantum Markov chains is investigated.
(5) Others Some conrete nonlinear models are discussed in detail in connection with quantum entropy and quantum communication. Questions in foundation of quantum theory related to Bell inequality and EPR paradox are clarified from the standpoint of algebraic probability theory.
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