On questions around reconstruction program

关于重建计划的问题

• 批准号: 1764157
• 负责人: Feng Xu
• 金额: $ 22万
• 依托单位: University of California-Riverside
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-07-01 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1764157&HistoricalAwards=false
• 关键词: questions; around; reconstruction; program

Quantum Mechanics was one of crowning achievements of modern physics and has important applications in daily life such as X-ray, TV, etc. More recently it is the driving principle behind building a quantum computer. The theory of operator algebras was introduced by John von Neumann in order to provide a proper mathematical framework for Quantum Mechanics. The non-commutativity which is a key feature of Quantum Mechanics, is an important aspect of operator algebras. Vaughan Jones's subfactor theory is built on this non-commutative framework. Conformal field theory (CFT) is a theory describing critical phenomena in condensed matter physics, and it also plays an important role in string theory. In recent years there have been remarkable interactions between subfactors and conformal field theory that have led to many interesting mathematical issues. The aim of this project is to find solutions to some of the important mathematical issues that surface in this context which have a wide range of applications in both mathematics and quantum physics.Subfactor theory provides an entry point into a world of mathematics and physics containing large parts of conformal field theory, quantum algebras and low dimensional topology. The research objective of this project is to develop further the connection between these subjects, and to find applications in the other area of mathematics. The project will rely on operator algebraic and subfactor techniques developed in studying CFT, including insights about old and new problems provided by the general framework of subfactors. The project's focus will be on the questions around the reconstruction program which are strongly motivated by recent construction of subfactors. In some cases there are strong indications that the subfactors come from CFT in certain ways that have been well established based on principal investigator's work. Solutions of the problems that are proposed would have important applications in diverse areas of mathematics. Results will be disseminated as research publications and presentations at professional meetings.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

量子力学是现代物理学的最高成就之一，在日常生活中有着重要的应用，如X射线，电视等。算子代数理论是由约翰·冯·诺依曼提出的，目的是为量子力学提供一个合适的数学框架。 非对易性是量子力学的一个重要特征，也是算子代数的一个重要方面。 沃恩·琼斯的子因子理论就是建立在这个非对易框架之上的。共形场论（CFT）是描述凝聚态物理中临界现象的理论，在弦理论中也起着重要的作用。近年来，子因子和共形场论之间的相互作用引起了许多有趣的数学问题。这个项目的目的是找到一些重要的数学问题的解决方案，表面在这方面有广泛的应用在数学和量子物理。子因子理论提供了一个切入点，进入一个世界的数学和物理包含大部分的共形场论，量子代数和低维拓扑。该项目的研究目标是进一步发展这些学科之间的联系，并在数学的其他领域找到应用。该项目将依赖于研究CFT中开发的算子代数和子因子技术，包括子因子一般框架提供的关于新旧问题的见解。该项目的重点将是围绕重建计划的问题，这些问题是由最近的子因素建设强烈推动的。在某些情况下，有强烈的迹象表明，子因素来自CFT在某些方面已经建立了基于主要研究者的工作。所提出的问题的解决方案将在数学的不同领域有重要的应用。研究成果将作为研究出版物和专业会议上的演讲进行传播。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的知识价值和更广泛的影响审查标准进行评估来支持。

项目成果

Some Results On Relative Entropy in Quantum Field Theory

• DOI: 10.1007/s00220-019-03367-x
• 发表时间: 2018-10
• 期刊: Communications in Mathematical Physics
• 影响因子: 2.4
• 作者: Fengjun Xu

Von Neumann Entropy in QFT

QFT 中的冯诺依曼熵

• DOI: 10.1007/s00220-020-03702-7
• 发表时间: 2020
• 期刊: Communications in Mathematical Physics
• 影响因子: 2.4
• 作者: Longo, Roberto;Xu, Feng
• 通讯作者: Xu, Feng

On relative entropy and global index

关于相对熵和全局指数

• DOI: 10.1090/tran/7989
• 发表时间: 2020
• 期刊: Transactions of the American Mathematical Society
• 影响因子: 1.3
• 作者: Xu, Feng

Relative entropy in CFT

CFT 中的相对熵

• DOI: 10.1016/j.aim.2018.08.015
• 发表时间: 2018
• 期刊: Advances in Mathematics
• 影响因子: 1.7
• 作者: Longo, Roberto;Xu, Feng
• 通讯作者: Xu, Feng