规范场论札记(I):电磁学和量子光学中的人造规范势与卡鲁扎–克莱因理论中的衍生电磁规范场

Notes on Gauge Field Theories (I): Synthetic Gauge Potentials in Electromagnetic Optics and Emergent Electromagnetic Gauge Field in the Kaluza-Klein Theory

规范场是驱动物质运动、参与传递相互作用的中介场。在量子电动力学、弱电统一模型和量子色动力学中,规范场是理论原生的基本动力学场。除此之外,规范场还可以“人造”(synthesis)和“呈展”(emergence)两种方式、以非基本或非原生的动力学场的身份展现,前者在一些应用领域(如电磁学、光学、凝聚态物理学)中为不少研究人员所关注,后者为引力–规范统一目的,在微分几何和广义相对论中以高维引力场的身份衍生出来。本文研究电磁学、量子光学系统中的人造规范势(和人造“磁学”)与卡鲁扎–克莱因理论中的衍生电磁规范场。能呈现人造规范势的物理系统包括含时哈密顿量系统、非共面弯曲光纤系统、原子–光场的电或磁偶极矩相互作用系统、横截面非均匀波导体系和各向异性电磁介质。在这些例子中,原子和光场仿佛是在等效的规范势中运动,其波动方程或场方程中的偏导数算符都被携带了等效规范势的协变导数算符取代。对于“呈展的规范场”这个主题而言,卡鲁扎–克莱因理论(五维时空的广义相对论)能统一爱因斯坦引力理论与麦克斯韦电磁场论,电磁相互作用作为高维引力耦合呈现在普通的四维时空,也即电磁力在本质上是引力。推而广之,非阿贝尔版本的卡鲁扎–克莱因理论能将杨–米尔斯规范场解释为高维引力场。本文从引力作用量密度、高维时空线元、短程线方程这些高维引力理论核心构件推导了电动力学基本内容。本文也分析了早期规范理论简史包括魏尔标度变换理论(1918~1919)、卡鲁扎–克莱因理论(1921, 1926)、克莱因矢量规范理论(1938)的研究历史,评述了引力理论和规范场论之间竞争、反哺、互相促进这一曲折发展史及其给后人可能的启迪性意义。

Gauge field is an intermediate field that drives the motion of matter and participates in mediating the interactions or transfers the forces in various physical systems. In quantum electrodynamics, electroweak unified model and quantum chromodynamics, the gauge fields are the fundamental dynamical fields of the theories. In addition, gauge fields can also be synthetic or/and emergent fields, where the former has attracted the intensive attention of many researchers in some applied areas (such as electromagnetism, optics and condensed matter physics) and the latter emerges as a higher-dimensional gravitational field, serving the purpose of gravity-gauge unification, in differen-tial geometry and general relativity. The two topics such as the artificial gauge potentials (and syn-thetic “magnetism”) in electromagnetic optics and the emergent Yang-Mills gauge field are consid-ered in this paper. Such physical systems in electromagnetics and quantum optics that can exhibit the artificial gauge potentials include time-dependent Hamiltonian systems, noncoplanarly curved fiber, atom-light electric- or magnetic-dipole allowed interaction systems, non-uniform cross-section waveguides and anisotropic electromagnetic media. In these illustrative examples, the atoms and optical fields seem to propagate in the presence of some effective gauge potentials and all the spa-tial partial derivatives in their wave or field equations need to be replaced by the covariant derivatives that carry the effective gauge potentials. As far as the emergent gauge field is concerned, the Kaluza-Klein theory (five-dimensional general relativity) can be used to unify the theories of Ein-stein’s gravitation and Maxwell’s electromagnetism, i.e., the electromagnetic interaction emerges as an effect of higher-dimensional gravity and the electromagnetic force is in essence a gravitational interaction. By extension, the Yang-Mills gauge field could also be interpreted as a high-er-dimensional gravitational field in a non-Abelian formalism of the Kaluza-Klein theory. In the present paper, the essential content of classical electrodynamics has been derived by taking full advantage of the fundamental components of higher-dimensional gravity, such as gravitational Lagrangian density, higher-dimensional spacetime line element and geodesic line equation. The brief history of developments in Weyl gauge theory (1918~1919), Kaluza-Klein theory (1921 & 1926) and Klein vectorial gauge theory (1938) will also be analyzed and the history of gravity theory and gauge field theory, where they have competed, fed and promoted each other, will be reviewed in order to reveal the enlightening significance of these theories for us.