Constructive Algebraic Quantum Field Theory

作者计划发展算符代数和泛函分析技术，以便在经典和非对易的四维Minkowski空间上构建相对论量子场论的模型。接下来的两年工作的主要焦点将是通过用D.Buchholz和提出者最近介绍的方式来变形给定的模型来构建这种模型。其中，这将需要泛函分析技术的进一步发展，以定义产生量子场算符或相关有界可观测的适当变形的多重算符积分，以证明此类积分的本质自伴性，并证明自伴扩张的适当的交换性。这将导致相关的von Neumann可观测代数，其模结构将被研究。此外，还将开发进一步的技术来确定此类von Neumann代数的某些交集足够大(或不足够大)，这将允许我们确定所构建的模型中是否有足够多的局部可观测数据。这样的进展很可能对泛函分析的其他应用有用，除了这里讨论的一个。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规律，但我们对量子场论理解中最严重的差距之一是，在四维Minkowski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。拟议的工作将提供这样的模型，以及为其分析和控制提供进一步的数学技术。此外，这些方法也可以转化为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。然后，这样的时空上的非平凡量子场模型也可以构造出来。此外，由于本方案中将要研究的形变的特殊性质，所提出的工作还将导致非对易Minkowski空间上的进一步严格的量子场模型，这有望与引力的量子化有关。

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.