Quantum White Noise and Infinite Dimensional Harmonic Analysis

量子白噪声和无限维谐波分析

• 批准号: 09440057
• 负责人: OBATA Nobuaki
• 金额: $ 8.51万
• 依托单位: Nagoya University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 1997
• 资助国家: 日本
• 起止时间: 1997 至 1999
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09440057/
• 关键词: white noise; quantum noise; central limit theorem; Ito formula; quantum stochastic differential equation; dissipative quantum system; stochastic limit of quantum theory; quantum Markov chain; 量子ホワイトノイズ; ホワイトノイズ解析; 無限次元解析; 統計的独立性; ランダムウォーク; 直交

(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 … More 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

(1)正态有序白色噪声方程在白色噪声演算的基础上，将传统的Ito型量子随机微分方程推广到正规序白色噪声方程，并利用白色噪声分布描述了其解的性质.利用单参数变换群求解了白色噪声空间上的柯西问题，为无穷维调和分析提供了新的思路。(2)白色噪声的高次幂和重整化伊藤公式。利用白色噪声算符理论研究了含有高次量子白色噪声的正态有序白色噪声方程的重整化Ito公式及其结构.它适用于（量子）随机微分方程的非常奇异的系数。(3)量子理论的随机极限。从安德森等标准哈密顿模型出发，通过随机极限推导出量子噪声和量子随机微分方程 ...更多信息 固体物理学中的量子力学模型和没有偶极近似的量子电动力学模型。这种对强非线性相互作用的直接处理，促使建立相互作用理论Fock模型，以便对新现象给予统一的认识。(4)耗散量子系统的数学分析。为了在正则算符的框架内描述从微观系统到宏观系统的转变，发展了非平衡热场动力学理论。用统一的观点阐明了描述耗散量子系统的方程之间的关系。(5)中心极限定理在代数概率的框架内，引入了单点独立性的新概念，并导出了相应的中心极限定理和极限过程，探讨了量子噪声的数学根源。通过创建和消灭算子的类似物来研究离散图上随机游动的联系。(6)他人量子马尔可夫链，复白色噪声和相干态表象，S变换的反演公式和算符符号，量子实在性和定域性，统计不变性。少

项目成果

L.Accardi, Y.Hashimoto and N.Obata: "Notions of independence related to the free group" Infinite Dimen.Anal.Quantum Prob.1. 201-220 (1998)

L.Accardi、Y.Hashimoto 和 N.Obata：“与自由群相关的独立概念”Infinite Dimen.Anal.Quantum Prob.1。

Luigi Accardi,Y.G.Lu,I.Volovich: "The QED Hilbert module and interacting Fock spaces"International Institute for Advanced Studies. 120 (1997)

Luigi Accardi、Y.G.Lu、I.Volovich：“QED 希尔伯特模块和相互作用的福克空间”国际高等研究院。

洞 彰人: "Central limit theorems and asymptotic spectral analysis of large graphs" Infinite Dimen.Anal.Quantum Prob.1. 221-246 (1998)

Akito Dora：“大图的中心极限定理和渐近谱分析”Infinite Dimen.Anal.Quantum Prob.1 (1998)。

有光敏彦: "Quantum stochastic systems in terms of non-equilibrium thermo field dynamics" Quantum Communications,Computing,and Measurement(O.Hirota et al.Eds.)Plenum. 371-380 (1997)

Toshihiko Arimitsu：“非平衡热场动力学的量子随机系统”量子通信、计算和测量（O.Hirota 等人编辑）Plenum 371-380 (1997)。

Toshihiko Arimitsu,et al: "A comment on the Shibata-Hashitsume Langevin equation--Is it of the Stratonovich type or of the Ito type?"Physica A. 256. 129-148 (1997)

Toshihiko Arimitsu 等：“对 Shibata-Hashitsume Langevin 方程的评论——它是 Stratonovich 型还是 Ito 型？”Physica A. 256. 129-148 (1997)