Quantum White Noise and Infinite Dimensional Harmonic Analysis

(1)正态有序白噪声方程。在白噪声演算的基础上，将传统的Ito型量子随机微分方程组推广到正态序白噪声方程，并用白噪声分布描述了其解的性质。利用具有无限维调和分析新性质的单参数变换群解决了白噪声空间上的柯西问题。(2)白噪声的高次方和重整化的Ito公式。通过重整化得到了含有高次方量子白噪声的正态有序白噪声方程的Ito公式，并用白噪声算符理论研究了它的结构。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从标准哈密顿模型(如Anderson…)通过随机极限得到的更多的是固体物理模型和没有偶极近似的量子电动力学模型。这种对非常强的非线性相互作用的直接处理激发了构造相互作用Fock模的理论，从而为新的现象提供了统一的方面。(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微观系统向宏观系统的转变，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。在代数概率的框架下，引入了单态独立性的新概念，并对量子噪声的数学起源导出了相应的中心极限定理和极限过程。通过类比生成算子和湮灭算子，研究了离散图上随机游动与随机游动的关系。(6)其他。量子马尔可夫链，复白噪声和相干态表示，S变换和算符符号的反演公式，量子实在性和局域性，统计不变性。较少

(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