Quantum White Noise and Infinite Dimensional Harmonic Analysis

(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 关于我们 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。通过建立和湮灭算子的类比，研究了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

(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