Electromagnetism as a Gauge Theory

In this section, we will explore how electromagnetism can be formulated as a gauge theory. In introductory physics, we learn about electricity and magnetism—two fundamental forces of nature. We begin with electrostatics, which describes the behavior of electric charges at rest, and then see how moving charges create magnetic fields. This typically culminates in Maxwell's equations, which is usually given as a set of four equations.

However, there is a deeper structure to electromagnetism. Specifically, it can be formulated as a gauge theory, which is a powerful framework that describes how fields interact with each other. In this section, we will explore the key concepts of gauge theory and how they apply to electromagnetism. But to do so, we will need to formulate Maxwell's equations in a more general way, using relativistic notation.

At a fundamental level, electromagnetism exists because of the local symmetry present in the Dirac field. From Noether's theorem, we know that this symmetry leads to a local conservation law, which is the conservation of electric charge. As such, electric charge obeys a continuity equation.