Topics in Noncommutative Geometry, Quantization and Dynamics

非交换几何、量化和动力学主题

• 批准号: 9801612
• 负责人: Slawomir Klimek
• 金额: $ 6.76万
• 依托单位: Indiana University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-06-01 至 2001-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9801612&HistoricalAwards=false
• 关键词: Topics; Noncommutative; Geometry; Quantization; Dynamics

AbstractKlimek The principal investigator proposes to continue and expand in new directions an ongoing program of studying quantized spaces and dynamical systems on such spaces by means of the techniques of operator algebras and non-commutative differential geometry. The ultimate goals of the program are:* To understand the mathematics of quantization of systems (involving both finitely and infinitely many degrees of freedom) with complicated dynamics;* To study the non-commutative geometric principles underlying the physics of the microworld, and in particular possible implications for quantum chaos;* To explore new mathematical phenomena relating operator theory to other fields of mathematics. Non-commutative geometry is a new direction of mathematics created in the eighties. Its prime objective is to study objects, callednon-commutative spaces or manifolds, which are not ordinary geometric objects but still have a geometric flavor. Examples of structuresof this sort arise naturally e.g. in physical theories of subatomic phenomena. Methods of non-commutative geometry became a powerful tool in studying various problems in modern analysis and understanding the fundamentals of the physics of the microworld. It is proposed tostudy examples of non-commutative manifolds and dynamical systems on them with emphasis put on applications to quantum physics and operator theory.

AbstractKlimek 首席研究员建议继续和扩大在新的方向正在进行的计划，研究量子化空间和动力系统的空间，通过技术的算子代数和非交换微分几何。该计划的最终目标是：* 理解具有复杂动力学的系统（涉及无限和无限多个自由度）的量子化数学;* 研究微观世界物理学的非交换几何原理，特别是量子混沌的可能影响;* 探索与算子理论有关的新数学现象。 非对易几何是八十年代创立的一个新的数学方向。它的主要目的是研究对象，称为非交换空间或流形，这不是普通的几何对象，但仍然有几何风味。这种结构的例子自然出现，例如在亚原子现象的物理理论中。非对易几何方法成为研究现代分析中各种问题和理解微观世界物理学基础的有力工具。本文提出研究非对易流形及其上的动力系统的例子，重点放在量子物理和算子理论的应用上。