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。首先是关于Yang-Mills Field和Garnier系统之间的关系，该系统是单一不变的微分方程。第二个是Feynman Path的整体改革。第三个是非孔建立的SIT理论的非标准模型。我们在以下内容中解释了这些。（i）¢^6嵌入在g_2（¢^5）中为坐标之一。 g_2（¢^5）是一个quater nionic kahler歧管，在上面，我们有一个广义的反偶发连接（GASD）的概念。由于Jordan组的4个尺寸对¢^5作用，因此它对G_2（¢^5）作用。这些约旦群体被分类为年轻图。另一方面，对于线性方程式d/d3 u = a（3，s，t）u，其中a为¢^2值，我们有一个异词性变形的概念。众所周知，将奇异性分类也对应于约旦群体。主要结果是（1）我们写下了保存年轻图的胃口方程式列表，（2）两个方程仅等于Garnier系统。 （ii）Feynman将“ Feynman Path的整体”定义为量化经典力学和田间理论。它是一种不可或缺的一体，但在无限的维度空间上。我们定义了双重超产品和双重无限晶格和无穷小的傅立叶变换，并重新制定了Feynman Electronics的Feynman路径积分，遵循Feynman-Hibbs'text。（iii）没有规律性的集合理论称为无孔子发现的集合理论。例如，ACCEL，SCOTT，FINSLER BOFFA SET理论是已知的。我们构建具有任何圆形和非圆形类型的任何顺序长度的路径。此外，对于Boffa集理论，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集理论。