Mathematical Sciences: Mathematical Problems of Quantum Field Theory

[9500704]施瓦茨该提案由几个项目组成，致力于量子场论的数学问题。特别地，我们计划使用超几何技术来证明新的定理，给出半经典近似的精确条件。量子场论的方法将被用来构造一些几何结构的不变量。几乎所有的项目都与弦理论有关，弦理论现在被认为有机会描述自然界中存在的所有相互作用。我们将尝试构建假设的“通用弦论”，并利用公理量子场论的方法对弦论中空间的维度有更深的理解。量子场论是一门与数学和物理学研究的中心领域重叠的学科。在物理学中，它是试图理解构成所有物质的基本粒子的核心。在数学中，它是一个广泛应用于几何、分析和概率论等领域的技术体。这项研究背后的潜在目标是朝着揭示统一自然力量的基本理论的目标取得进展。＊＊＊

9500704 Schwarz The proposal consists of several projects, devoted to mathematical problems of quantum field theory. In particular, we are planning to use the technique of supergeometry to prove new theorems giving conditions of exactness of semiclassical approximation. The methods of quantum field theory will be used to construct invariants of some geometric structures. Almost all projects are related to string theory, which is now considered as having a chance to describe all interactions existing in Nature. We will try to construct the hypothetical "universal string theory" and reach a deeper understanding of the dimensionality of the space in string theory, using methods of axiomatic quantum field theory. Quantum field theory is a subject which overlaps central areas of research in both mathematics and physics. In physics it is at the heart of the attempt to understand the fundamental particles from which all matter is formed. In mathematics it is a body of techniques which have wide applications in areas such as geometry, analysis and probability. The underlying goal behind this research is to make progress towards the aim of uncovering the fundamental theory which unifies the forces of nature. ***