Higher dimensional Yang-Mills field with symmetry and quaternionic structure

具有对称性和四元结构的高维杨-米尔斯场

• 批准号: 12640207
• 负责人: NITTA Takashi
• 金额: $ 0.7万
• 依托单位: Mie University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2000
• 资助国家: 日本
• 起止时间: 2000 至 2001
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640207/
• 关键词: Yang-Mills field; Painlevi equation; Garnier system; nonstandard analysis; Feynman integral; field theory; infinitesimal Fourier transformation; differential equation of functional; Aczel集合論; ディラッフ方程式

We have studied mainly three fields. First is about the relationship between Yang-Mills field and Garnier system that is a differential equation of monodromy invariant. The second is a reformulation of Feynman path integral. The third is a nonstandard model of nonwell founded sit theory. We explain these in the following.(i) ¢^6 is embedded in G_2(¢^5) as one of coordinates. G_2(¢^5) is a quater nionic kahler manifold, over it we have a concept of generalized anti-self-dual connection (GASD). Since 4 dimensional Jordan groups act on ¢^5, it act on G_2 (¢^5). These Jordan groups are classified corresponding to young diagrams. On the other hand for linear equation d/d3 u = A(3, s, t)u, where A is ¢^2-valued, we have a concept of isomonodromic deformation. It is known that the singularities are classified corresponding also to Jordan groups. The main results are (1) we write down a list of GASD equations preserving young diagram, (2) two of the equations are just equal to Garnier systems.(ii) Feynman defined "Feynman path integral" to quantise classical mechanics and field theory. It is some sence an integral but on an infinitesimally dimensional space. We define a double ultra product and a double infinitesimal lattice and an infinitesimal Fourier transformation, and reformulate a Feynman path integral of quantum electrodynamics followed Feynman-Hibbs'text.(iii) A set theory without regularity are called non-well-founded set theory. For example, Acsel, Scott, Finsler Boffa set theory are known. We construct a path with any ordinal length of circular and noncircular types. Furthermore, for Boffa set theory, we study a nonstandard model that includes an original Boffa set theory with an inclusion mapping.

我们主要研究了三个领域：第一是Yang-Mills场与单向不变量微分方程的关系；第二是费曼路径积分的重构；第三是非良好成立的理论模型。我们在下面解释这些。(i) ^6 作为坐标之一嵌入到 G_2(^5) 中，G_2(^5) 是一个四元卡勒流形。我们有一个广义反自对偶连接（GASD）的概念，由于 4 维乔丹群作用于 ^5，所以它作用于 G_2（^5），这些乔丹群对应于年轻图进行分类。另一方面，对于线性方程 d/d3 u = A(3, s, t)u，其中 A 为 ^2 值，我们有一个等单向变形的概念。已知奇点是分类的。主要结果是（1）我们写下了保留年轻图的 GASD 方程列表，（2）其中两个方程正好等于卡尼尔系统。（ii）费曼将“费曼路径积分”定义为它是某种意义上的积分，但在无限小维空间上进行量子化。我们定义了双极乘积和双无穷小晶格以及无穷小傅里叶变换，并重新表述了费曼。量子电动力学的路径积分遵循Feynman-Hibbs的文本。 (iii) 没有正则性的集合论称为非良基集合论，例如Acsel、Scott、Finsler Boffa 集合论我们用任意构造一个路径。此外，对于 Boffa 集合论，我们研究了一个非标准模型，其中包括带有包含映射的原始 Boffa 集合论。