1999 Fiscal Year Final Research Report Summary
Quantum White Noise and Infinite Dimensional Harmonic Analysis
| Project/Area Number |
09440057
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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 | Nagoya University |
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Principal Investigator |
OBATA Nobuaki Nagoya University, Graduate School of Mathematics, Associate Professor, 大学院・多元数理科学研究所, 助教授 (10169360)
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| Co-Investigator(Kenkyū-buntansha) |
ARIMATSU Toshihiko Tsukuba University, Department of Physics, Professor, 物理学系, 教授 (50134200)
MINAMI Kazuhiko Graduate School of Mathematics, Associate Professor, 大学院・多元数理科学研究所, 助教授 (40271530)
AOMOTO Kazuhiko Graduate School of Mathematics, Professor, 大学院・多元数理科学研究所, 教授 (00011495)
HORA Akihito Okayama University, Department of Environmental and Mathematical Sciences, Associate Professor, 環境理工学部, 助教授 (10212200)
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| Project Period (FY) |
1997 – 1999
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| Keywords | white noise / quantum noise / central limit theorem / Ito formula / quantum stochastic differential equation / dissipative quantum system / stochastic limit of quantum theory / quantum Markov chain |
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| Research Abstract |
(1) Normal-Ordered White Noise Equations. On the basis of white noise calculus traditional quantum stochastic differential equations of Ito type are extended to normal-ordered white noise equations and properties of their solutions are described by means of white noise distributions. Cauchy problems on white noise space are solved through one-parameter transformation groups with new aspect to infinite dimensional harmonic analysis.
(2) Higher Powers of White Noises and Renormalized Ito Formula. The Ito formula associated with normal-ordered white noise equations involving higher powers of quantum white noises emerges with renormalization and its structure is investigated by means of white noise operator theory. It is applied to (quantum) stochastic differential equations with very singular coefficients.
(3) Stochastic Limit of Quantum Theory. Quantum noises and quantum stochastic differential equations are derived through stochastic limit from standard Hamiltonian models such as Anderson
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model in solid state physics and models of quantum electrodynamics without dipole approximation. Such direct treatment of very strong nonlinear interaction motivate to construct theory of interaction Fock module to give a unified aspect to new phenomenon.
(4) Mathematical Analysis of Dissipative Quantum Systems. In order to describe the transition from micro-systems to macro -systems within the framework of canonical operator formalism, the theory of Non-Equilibrium Thermo Field Dynamics is developed. Relationship among equations describing dissipative quantum systems is clarified with a unified point of view.
(5) Central Limit Theorems. Within the framework of algebraic probability the new concept of singleton independence is introduced and the associated central limit theorems and limit processes are derived towards mathematical origin of quantum noises. Connection with random walks on discrete graphs is investigated through analogues of the creation and annihilation operators.
(6) Others. Quantum Markov chains, complex white noise and coherent state representation, inversion formulas of S-transform and operator symbols, quantum reality and locality, statistical invariance. Less
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