Constructive Algebraic Quantum Field Theory

构造性代数量子场论

• 批准号: 0901370
• 负责人: Stephen Summers
• 金额: --
• 依托单位: University of Florida
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-08-15 至 2011-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0901370&HistoricalAwards=false
• 关键词: Constructive; Algebraic; Quantum; Field; Theory

The proposer plans to develop operator algebraic and functional-analytic techniques in order to construct models of relativistic quantum field theory both on classical and on noncommutative four-dimensional Minkowski space. The primary focus of the next two years of work will be to construct such models by deforming a given model in a manner recently introduced by D. Buchholz and the proposer. Among other things, this will necessitate the further development of functional-analytic techniques to define multiple operator integrals yielding suitable deformations of quantum field operators or of associated bounded observables, to prove the essential self-adjointness of such integrals, and to prove suitable commutativity properties of the self-adjoint extensions. This would lead to associated von Neumann algebras of observables, whose modular structure would be studied. In addition, will develop further techniques to establish that certain intersections of such von Neumann algebras are sufficiently large (or not), which would allow us to determine the availability of sufficiently many local observables in the constructed models. Such progress is likely to be useful in other applications of functional analysis besides the one discussed here. Relativistic quantum field theory is the most successful theory of elementary particle physics, which itself seeks to describe the fundamental constituents of matter and their laws of interaction, but one of the most serious gaps in our understanding of QFT has been the absence of mathematically rigorously constructed models of nontrivial quantum fields on the standard model of the four-dimensional Minkowski space. The proposed work will provide such models, as well as further mathematical techniques for their analysis and control. Moreover, these methods can also be transferred to quantum field theories on curved spacetimes with a sufficiently large isometry group, such as de Sitter space and anti-de Sitter space. Nontrivial quantum field models on such space--times can then also be constructed. In addition, due to the special nature of the deformation to be studied in this proposal, the proposed work will also result in further rigorous quantum field models on noncommutative Minkowski space, which is expected to be of relevance to the quantization of gravity.

提议者计划发展算子代数和泛函分析技术，以构建经典和非对易四维闵可夫斯基空间上的相对论量子场论模型。未来两年工作的主要重点将是通过以D.布赫霍尔茨和提议者。除此之外，这将需要进一步发展泛函分析技术，以定义多个算子积分，从而产生量子场算子或相关有界可观测量的适当变形，证明这种积分的本质自伴性，并证明自伴扩展的适当交换性。这将导致相关的冯诺依曼代数的观察，其模块结构将被研究。此外，将开发进一步的技术，以建立这样的冯诺依曼代数的某些交叉足够大（或没有），这将使我们能够确定足够多的本地观测的可用性在构建的模型。这样的进展很可能是有用的其他应用程序的功能分析除了这里讨论的。 相对论量子场论是基本粒子物理学中最成功的理论，它本身试图描述物质的基本组成及其相互作用定律，但我们对QFT理解中最严重的差距之一是缺乏在四维闵可夫斯基空间标准模型上数学上严格构建的非平凡量子场模型。拟议的工作将提供这样的模型，以及进一步的数学技术，其分析和控制。 此外，这些方法也可以转移到具有足够大等距群的弯曲时空上的量子场论，如de Sitter空间和反de Sitter空间。 这样的时空上的非平凡量子场模型也可以被构造出来。此外，由于该提案中要研究的变形的特殊性质，所提出的工作也将导致非对易Minkowski空间上更严格的量子场模型，预计这将与引力的量子化有关。