Mathematical Sciences: Topological Methods in Quantum Field Theory and Methods of Quantum Field Theory in Topology

数学科学：量子场论中的拓扑方法和拓扑学中的量子场论方法

• 批准号: 9201366
• 负责人: Albert Schwarz
• 金额: $ 18.63万
• 依托单位: University of California-Davis
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1992
• 资助国家: 美国
• 起止时间: 1992-07-01 至 1996-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9201366&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Topological; Methods; Quantum

Schwarz proposes to study various problems in quantum field theory. He will try to construct invariants of contact manifolds and of Legendre knots in three-dimensional contact manifolds by means of topological invariants from quantum field theory. There will also be a study of semi-infinite cohomology of infinite- dimensional Lie algebras and the theory of symplectic and contact supermanifolds. These problems are related to the general theory of quantization of constrained systems and to the superstring theory. Modern physics, quantum mechanics and relativity, is a product of the twentieth century. It is founded firmly in the last century's attempt to address the microstructure of matter and to come to grips with the concepts of action-at-a distance, electro-magnetism, and heat radiation. The mathematical foundations for these developments, collectively called mathematical physics, ranges from detailed analysis of Schroedinger operators, which governs the dynamics of particles, to unified field theory, which attempts to unite the four known forces into a single theory.

施瓦茨提出研究量子场论中的各种问题。他将尝试用量子场论中的拓扑不变量来构造接触流形和三维接触流形中的勒让德结的不变量。也将研究无限维李代数的半无限上同调，以及辛和接触超流形的理论。这些问题与约束系统的一般量子化理论和超弦理论有关。现代物理学，量子力学和相对论，是二十世纪的产物。它牢固地建立在上个世纪对物质微观结构的研究和对超距作用、电磁和热辐射等概念的理解上。这些发展的数学基础，统称为数学物理学，范围从控制粒子动力学的薛定谔算子的详细分析，到试图将四种已知力统一为单一理论的统一场论。