Infinite Dimensional Analysis and Quantum Probability Theory

(1)量子理论的随机极限通过随机极限，从正则哈密顿模型推导出量子噪声和量子随机微分方程。将该方法应用于涉及强非线性相互作用的模型，得到了一种新型的量子噪声。例如，一个没有偶极子近似的QED标准模型产生了奇异类型的非交叉图和维格纳半圆定律的非线性变形。为了统一这种现象，引入了相互作用Fock模块的概念。(2)经典和量子随机微积分的非线性推广，在(1)的激励下，我们研究了高次量子白噪声，它被认为是无限层次噪声的第一类。用重整化方法证明了相关的伊藤公式。此外，证明了一类包含高次量子白噪声的正序白噪声方程解的唯一存在性，并建立了与量子随机微分方程的关系。(3)中心极限定理在代数概率论的框架内，统一了已有的独立性概念，引入了单态独立性的新概念。证明了中心极限定理，得到了极限随机过程。这个结果与正交多项式、它们的q-变形和高斯化*概率测度有关。(4)讨论了量子马尔可夫链、量子马尔可夫态的分类以及量子通信信道中的提升问题。研究了量子计算中出现的基本算法与量子马尔可夫链之间的有趣联系。结合量子熵和量子通信详细讨论了一些具体的非线性模型。从代数概率论的角度阐明了量子理论基础中有关贝尔不等式和EPR悖论的问题。

(1) Stochastic Limit of Quantum Theory Through the stochastic limit quantum noises and quantum stochastic differential equations are derived from canonical Hamiltonian models. This *ethod is applied to a model involving strong nonlinear interaction to obtain a new type of quantum noise. For example, a standard model of QED without dipole approximation yields a singular type of noncrossing diagrams and nonlinear deformation of Wigner's semicircle law. The concept of an interacting Fock module is introduced in order to unify this kind of phenomena.(2) Nonlinear Extension of Classical and Quantum Stochastic Calculus Motivated by (1) , we studied higher powers of quantum white noises which are considered as the first class of an infinite hierarchy of noises. Tha associated Ito formula is proved with renormalization. Moreover, unique existence of solutions is proved for a class of normal-ordered white noise equations involving higher powers of quantum white noises and a relation to quantum stochastic differential equations is established.(3) Central Limit Theorems Within the framework of algebraic probability theory, the existing concepts of independence have been unified and the new notion of singleton independence is introduced. The associated central limit theorem is proved and the limit stochastic process is obtained. This result is related to orthogonal polynomials, their q-deformations, and Gaussianization * probability measures.(4) Quantum Markov Chains Classification of quantum Markov states and lifting problems in quantum communication channels are discussed. An interesting connection between basic algorithyms appearing in quantum computing and quantum Markov chains is investigated.(5) Others Some conrete nonlinear models are discussed in detail in connection with quantum entropy and quantum communication. Questions in foundation of quantum theory related to Bell inequality and EPR paradox are clarified from the standpoint of algebraic probability theory.