State-Sum Methods in Low-Dimensional Topology and Quantum Gravity - Categorical Underpinnings and Applications

低维拓扑和量子引力中的状态和方法 - 分类基础和应用

• 批准号: 9971510
• 负责人: David Yetter
• 金额: $ 10万
• 依托单位: Kansas State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-08-01 至 2002-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9971510&HistoricalAwards=false
• 关键词: State; Sum; Methods; Low; Dimensional

9971510Yetter This project pursues a number of interrelated and complementarylines of research related to state-sum techniques in low-dimensionaltopology and quantum gravity. Some of these involve the categoricalunderpinnings of state-sum techniques: the investigation of the algebraicstructure of TQFT's, of the deformation theory of categories withstructure, and the categorical underpinnings of ``state-integrals,'' i.e.,measure theoretic analogues of the familiar state-sums. Others aim atapplications of the techniques to topology and quantum gravity: theextension to the Minkowskian signature of the Barrett-Crane state-sumformulation of Euclidean quantum gravity, the investigation andconstruction of topological invariants related to the Barrett-Crane andCrane-Yetter state-sums and their generalizations, and the investigationof the relations between categorical deformation theory and the theoryof Vassiliev knot and link invariants. This project seeks to investigate the structure of 3- and4-dimensional spaces, both the abstract spaces of topology, and thestructure of physical space-times in certain proposed models for aquantum theory of gravity. The problem of quantizing gravity is probablythe deepest unsolved theoretical problem in physics. It points to themost fundamental unknowns in our understanding of the universe, such asthe origin of the universe and black hole physics. Its apparentintractability by using traditional methods suggests that a radically newtype of mathematics may be necessary to solve the problem. The structureof operators on spaces of quantum states, the algebraic structuresdescribing the assembly of spaces or knots and links from simple pieces,and the algebraic data needed to construct state-sums with suitableinvariance properties to investigate both topological structures and toconstruct ``spin-foam'' models of quantum gravity, can all conveniently bedescribed in terms of generalizations and extensions of the theory ofmonoidal categories first introduced by MacLane in the early 1960's.These facts, together with recent progress with the interrelated problemsmentioned above, lead to the hope that the theory of monoidal categoriesmay be the missing tool. Thus, much of the work uses category theory,not as a substitute for set theory as a foundations of mathematics, butas the appropriate sort of algebra for the investigation of spacesabstracted from the one in which we live.***

9971510Yetter这个项目从事了许多与低维拓扑和量子引力中的状态和技术有关的相互关联和互补的研究。其中一些涉及状态和技术的范畴基础：TQFT的代数结构的研究，有结构范畴的形变理论的研究，以及“状态积分”的范畴基础，即测量与常见的状态和的理论相似。其他目的是将这些技术应用于拓扑学和量子引力：欧几里德量子引力的Barrett-Crane态和公式的Minkowskian签名的推广，与Barrett-Crane和Crane-Yetter态和及其推广有关的拓扑不变量的研究和构造，以及范畴形变理论与Vassiliev纽结和链接不变量理论之间的关系的研究。这个项目试图在某些提出的水量子引力理论模型中研究三维和四维空间的结构，既包括抽象的拓扑空间，也包括物理时空的结构。引力的量子化问题可能是物理学中最深刻的未解决的理论问题。它指向了我们对宇宙理解中最基本的未知数，例如宇宙的起源和黑洞物理。它用传统方法显然很难解决，这表明可能需要一种全新的数学来解决这个问题。量子态空间上算符的结构，描述由简单片段组成的空间或纽结和链环的代数结构，以及构造具有适当不变性的状态和以研究拓扑结构和构建量子引力的“自旋泡沫”模型所需的代数数据，都可以很方便地用麦克兰在20世纪60年代初首先引入的么半范畴理论的推广和扩展来描述。这些事实，加上上述相关问题的最新进展，使人们希望一元范畴理论可能是缺失的工具。因此，许多工作使用范畴理论，而不是作为集合理论的替代，作为数学基础，而是作为研究从我们生活的空间抽象出来的空间的适当类型的代数。