物理与工程

对称性是现代物理研究的重要基石,群论则是刻画对称性的核心数学工具。本文旨在探讨如何在“物理学中的群论”课程中以麦克斯韦电磁理论为主线系统讲授群论思想。麦克斯韦理论是一个特别理想的教学范例,其中蕴含了丰富多样的对称性结构:既包括有限群与李群,也涵盖时空对称性与内部对称性,既有传统的0-形式对称性,也有广义的1-形式对称性。我们遵循由浅入深的教学原则,首先回顾麦克斯韦方程组的多种表述形式(微分方程、协变形式、微分形式),以及作用量变分原理。随后从方程和作用量两个层面系统阐述麦克斯韦理论的各类对称性:(1)庞加莱对称性——包括时空平移、洛伦兹变换,对应能量-动量守恒;(2)U(1)规范对称性——作用于电磁势的内部对称性,对应电荷守恒;(3)离散对称性——电荷共轭(C)、宇称(P)、时间反演(T)及其联合变换CPT;(4)共形对称性——尺度变换与特殊共形变换;(5)电磁对偶对称性——交换电场与磁场的离散或连续对称性,引入磁单极子后的广义对偶变换。通过这一系统性教学设计,学生不仅能够深入理解群论的数学结构及其物理意义,还能将抽象的群论概念与具体的电磁现象紧密联系,体会从经典物理到现代规范场论的思想演进。课程内容从基础的洛伦兹群出发,逐步深入到当代物理前沿的广义对称性,既巩固了学生对经典电动力学的理解,又为量子场论、凝聚态物理等后续课程奠定了坚实的群论基础。

Symmetry serves as a fundamental cornerstone of modern physics research, with group theory being the core mathematical tool for characterizing symmetries. This paper aims to explore how to systematically teach group theory concepts using Maxwell's electromagnetic theory as the main thread in the course “Group Theory in Physics”. Maxwell's theory serves as an ideal pedagogical example, containing a rich variety of symmetry structures: it encompasses both finite groups and Lie groups, spacetime symmetries and internal symmetries, as well as traditional 0-form symmetries and generalized 1-form symmetries.Following the pedagogical principle of progressing from simple to complex, we first review the various formulations of Maxwell's equations(differential equations, covariant form, differential forms) and the action principle. Subsequently, we systematically elucidate the various symmetries of Maxwell's theory from both the equation and action perspectives:(1) Poincaré symmetry—including spacetime translations and Lorentz transformations, corresponding to energy-momentum conservation;(2) U(1) gauge symmetry—an internal symmetry acting on the electromagnetic potential, corresponding to charge conservation;(3) discrete symmetries—charge conjugation(C), parity(P), time reversal(T), and their combined CPT transformation;(4) conformal symmetry—scale transformations and special conformal transformations;(5) electromagnetic duality symmetry—discrete or continuous symmetries exchanging electric and magnetic fields, and generalized duality transformations with the introduction of magnetic monopoles.Through this systematic pedagogical design, students can not only gain a deep understanding of the mathematical structure of group theory and its physical significance, but also closely connect abstract group-theoretic concepts with concrete electromagnetic phenomena, experiencing the intellectual evolution from classical physics to modern gauge field theory. The course content starts from the fundamental Lorentz group and gradually delves into generalized symmetries at the forefront of contemporary physics, both reinforcing students' understanding of classical electrodynamics and laying a solid group-theoretic foundation for subsequent courses such as quantum field theory and condensed matter physics.