Quantum White Noise and Infinite Dimensional Harmonic Analysis

(1)正序白噪声方程。在白噪声演算的基础上，将传统的伊藤型量子随机微分方程推广到正序白噪声方程，并用白噪声分布描述了其解的性质。利用单参数变换群解决了白噪声空间上的柯西问题，为无穷维谐波分析提供了新的思路。(2)高次白噪声和重归一化伊藤公式。通过对高次量子白噪声进行重整，得到了与正序白噪声方程相关的伊藤公式，并用白噪声算子理论研究了伊藤公式的结构。它适用于具有非常奇异系数的（量子）随机微分方程。(3)量子理论的随机极限。量子噪声和量子随机微分方程是由固体物理中的标准哈密顿模型（如Anderson…More模型）和非偶极子近似的量子电动力学模型通过随机极限导出的。这种对强非线性相互作用的直接处理，促使我们构建相互作用的Fock模理论，从而对新现象给出一个统一的认识。(4)耗散量子系统的数学分析。为了在正则算符形式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。在代数概率论的框架内，引入了单态无关的新概念，推导了量子噪声数学起源的中心极限定理和极限过程。通过对产生算子和湮灭算子的类比，研究了离散图上随机游走的联系。(6)其他。量子马尔可夫链，复杂白噪声和相干态表示，s变换和算子符号的反演公式，量子实在性和局域性，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