规范场论札记(II)：局域电磁对偶变换理论、广义卡鲁扎–克莱因衍生杨–米尔斯规范场理论与高维引力规范场论

Notes on Gauge Field Theories (II): Local Electromagnetic Dual Transformation The-ory, Generalized Kaluza-Klein Emergent Yang-Mills Gauge Field Theory and Higher-Dimensional Gravitational Gauge Field Theory

摘要: 本文继续研究、评述与本专题上篇“规范场论札记(I)”中的“人造”(synthetic)规范场和“呈展”(emergent)规范场有关的两个主题。本文“规范场论札记(II)”主要研究三个具体问题：i) 在前人的磁荷和对偶变换理论基础之上，提出了局域电磁对偶变换对称性和对偶规范场概念，指出对于电磁波而言，对偶规范势的效果使得真空中的电磁波好似在介电系数和磁导率均为张量的各向异性介质中传播；ii) 给出了一种非阿贝尔版本或广义的卡鲁扎–克莱因理论的详细推导过程，其目的是为了具体介绍一个统一爱因斯坦广义相对论引力和杨–米尔斯规范相互作用的理论。在此模型中，杨–米尔斯规范场以高维引力度规场分量的身份在普通四维时空中呈现出来，或者说，杨–米尔斯规范场在本质上是一种非阿贝尔卡鲁扎–克莱因高维引力场；iii) 介绍了笔者提出的一种自旋联络高维引力规范理论，其引力拉格朗日量用黎曼曲率平方项构造，高维自旋仿射联络(洛伦兹联络)具有杨–米尔斯规范势的特征、矢量场和旋量物质场的高维自旋流在四维时空内表现为杨–米尔斯规范荷流，从而该高维引力规范理论可以统一引力相互作用和杨–米尔斯规范相互作用。在文献中，引力理论有很多家，虽然本文仅对非阿贝尔版本的卡鲁扎–克莱因理论和引力规范理论的长处和缺陷作了评述，但希望本文对此二者的析评有助于读者举一反三、触类旁通理解文献中各类引力理论的风格特征和优缺点。

Abstract: This paper continues to explore and review the two topics of “synthetic” gauge field and “emergent” gauge field, which have already been considered in the last paper entitled “Notes on Gauge Field Theories (I)”. The present paper on “Notes on Gauge Field Theories (II)” includes three topics: i) A theory of local electromagnetic dual transformation symmetry and dual gauge field is suggested based on the previous theories of magnetic charge and dual transformation, and it is pointed out that the effect of dual gauge potential makes an electromagnetic wave in vacuum seem to propagate in an anisotropic medium whose permittivity and permeability are both tensors; ii) A non-Abelian version or generalized Kaluza-Klein theory is given in detail. The purpose is to introduce a theory of fundamental interaction that unifies Einstein’s general-relativity gravity and Yang-Mills gauge interaction. In this model, the Yang-Mills gauge potential is a higher-dimensional gravitational metricfield off-diagonal component emerging in the ordinary four-dimensional spacetime, or in other words, the Yang-Mills gauge field is essentially a non-Abelian Kaluza-Klein higher-dimensional gravitational field; iii) A theory of higher-dimensional spin-connection gravitational gauge field theory, of which the gravitational Lagrangian density is quadratic in the Riemannian curvature, is reviewed. The higher-dimensional spin-affine connection (the Lorentz connection) can serve as a Yang-Mills gauge potential and the spin currents of vectorial and spinorial matter fields play a role of Yang-Mills gauge charge currents in the four-dimensional spacetime, and so the gravitational interaction and the Yang-Mills gauge interaction can be unified into the present higher-dimensional spin-connection gravitational gauge theory, which was suggested by us. There have been many theories of gravitation in the literature. Although the merits and weaknesses of only the non-Abelian version of Kaluza-Klein theory and the gauge theory of gravitation are reviewed in this paper, we expect that the analysis of these two theories would still help readers to draw parallels among the relevant gravity theories and to understand the stylistic characteristics, advantages and disad-vantages of various gravitation theories in the literature.