规范场与纤维丛:它的内容,方法和意义  

Gauge Field and Fiber Bundle:Its Contents,Methods,and Meanings

作　　者：赵松年[1] 路博 陈肯 黄旭 ZHAO Song-nian;LU Bo;CHEN Ken;HUANG Xu(Institute of Atmospheric Physics,Chinese Academy of Sciences,Beijing 100029,China;College of science,Beijing University of Posts and Telecommunications,Beijing 100029,China)

机构地区：[1]中国科学院大气物理研究所,北京100029 [2]北京邮电大学理学院,北京100029

出　　处：《物理学进展》2023年第1期10-24,共15页Progress In Physics

基　　金：国家自然科学基金(No:62071488)的支持。

摘　　要：从本质上讲,规范场是物理学中(量子场论,基本粒子理论)重要的研究领域(1979年、1999年和2004年共有六位物理学家获得诺贝尔奖,他们的研究工作直接或间接与规范场有关),而纤维丛则是数学中(微分几何、群论、李代数)的热门课题(1986年唐纳森因研究纤维丛获得菲尔茨奖)。近年来,对杨–米尔斯方程、纤维丛和规范场的研究正在深入开展,因此,本文着重从物理概念出发,分别论述规范场在量子场论中,纤维丛在微分几何中相关概念的形成、发展,以及与杨–米尔斯方程之间的关系,特别是从电磁场、弱力和强力的统一方面,显示了规范场的重要性和深远意义,为了使更多的相关专业读者能在这一重要的领域中迅速获得必要的专业知识,产生探索和创新的热情,在综述中对阿贝尔规范场到非阿贝尔规范场,以及对称性自发破缺的基本概念和处理方法,进行了详细的论述,特别是初步探讨了杨–米尔斯方程的空间属性,目的是希望能更好地将规范场、纤维丛这二者与杨–米尔斯方程联系起来,加深对纤维丛的联络在更深层次的了解,由于这类问题是一个有意义的研究方向,值得有志者去深入探索。Essentially,gauge fields are an important area of research in physics(quantum field theory,elementary particle theory)(six physicists won Nobel prizes in 1979,1999 and 2004for their work directly or indirectly related to gauge fields),and fiber bundles are an important area of research in mathematics(differential geometry,group theory,Lie algebras)(Donaldson won the Fields Medal in 1986 for his work on fiber bundle).In recent years,the research of fiber bundle and gauge field based on Yang-Mills equation is being carried out deeply.Therefore,this paper focuses on the formation and development of the related concepts of gauge field in quantum field theory and fiber bundle in differential geometry,and the relationship between gauge field and Yang-Mills equation,especially from the unification of electromagnetic force,weak force and strong force.It shows the importance and far-reaching significance of gauge field.In order to enable more relevant professional readers to quickly acquire necessary professional knowledge and generate enthusiasm for exploration and innovation in this important field,the basic concepts and processing methods from Abelian gauge field to non-Abelian gauge field and spontaneous symmetry breaking are discussed in detail in this review.In particular,the spatial properties of the Yang-Mills equation are discussed in order to better connect the gauge field and fiber bundle with the Yang-Mills equation,so as to deepen the understanding of the connection of the fiber bundles in a deeper level.Since this kind of problem is a meaningful research direction,it is worth to be explored by interested people.

关 键 词：纤维丛 规范场 联络 转动群 对称性 不变性 

分 类 号：O412[理学—理论物理]