Constructive Algebraic Quantum Field Theory

为了构建经典和非交换四维闵可夫斯基空间上的相对论量子场论模型，作者计划发展算子代数和泛函解析技术。未来两年工作的主要重点将是通过D. Buchholz和该提议者最近引入的一种方式对给定模型进行变形来构建这样的模型。除此之外，这将需要进一步发展泛函解析技术，以定义产生量子场算子或相关有界可观测值的适当变形的多重算子积分，证明这些积分的基本自伴随性，并证明自伴随扩展的适当交换性。这将导致相关的冯诺依曼代数的观测，其模块化结构将被研究。此外，我们将开发进一步的技术，以确定这些冯·诺伊曼代数的某些交叉点足够大（或不够大），这将使我们能够确定在构建的模型中足够多的局部可观测值的可用性。除了这里讨论的功能分析之外，这种进展很可能在其他功能分析应用中有用。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成分及其相互作用的规律，但我们对QFT的理解中最严重的差距之一是在四维闵可夫斯基空间的标准模型上缺乏数学上严格构建的非平凡量子场模型。建议的工作将提供这样的模型，以及进一步的数学技术来分析和控制它们。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的量子场论中，如德西特空间和反德西特空间。这样的空间-时间上的非平凡量子场模型也可以被构造。此外，由于本文所研究的变形的特殊性，所提出的工作还将导致在非对易闵可夫斯基空间上进一步严格的量子场模型，这有望与引力的量子化相关。

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.