Electrodynamics on Haifei's Home

Gauge-Field Connections vs. Connections in General Relativity [Higher and More Elegant Electrodynamics · Extra 1]

hfwang132@gmail.com (hfwang132) — Tue, 29 Jul 2025 15:08:10 +0800

In a previous article, we said that the electromagnetic field is a connection. This led many readers to think of the connection in general relativity. What are the differences and commonalities between these two kinds of connections?

The electromagnetic connection \(A\) is a connection on a principal bundle, whereas the general-relativistic connection \(\Gamma\) is a connection on a vector bundle. Their definitions appear to be quite different. Is there a way to relate them?

From Quantum Field Theory to Cavity Quantum Electrodynamics [Higher and More Subtle Electrodynamics · 6]

hfwang132@gmail.com (hfwang132) — Sun, 13 Apr 2025 01:52:37 +0800

In this article, we start from the QED Lagrangian:

\[\begin{aligned} \mathcal{L}= \bar{\psi}(\mathrm{i}\gamma^\mu \partial_\mu -m)\psi - eA_\mu \bar{\psi} \gamma^\mu \psi -\frac{1}{4} F_{\mu\nu} F^{\mu\nu} \end{aligned} \]

This Lagrangian is highly complex: it not only takes into account the electron’s antiparticle—the positron—but the coupling term \(- eA_\mu \bar{\psi} \gamma^\mu \psi\) is also a cubic term, capable of describing various processes such as electron-positron pair creation/annihilation. Due to the presence of the cubic term, this Lagrangian has no analytic solution and can only be solved using perturbation theory in quantum field theory.

Dirac Magnetic Monopoles—Chern Classes and Chern Numbers [Higher and More Wonderful Electrodynamics · 4]

hfwang132@gmail.com (hfwang132) — Sun, 30 Mar 2025 01:13:18 +0800

Previous article in this series:

Godfly: The electromagnetic field is more than just electric and magnetic fields—The AB effect and Berry connection [Higher and More Wonderful Electrodynamics · 3]

In the previous article, we mentioned that if a manifold is topologically trivial, then integrating the Berry curvature over a closed surface \(\Sigma\) should yield zero. This is because, according to Stokes’ theorem:

\[\int_\Sigma F = \int_{\partial \Sigma } A = 0\]

However, on a topologically nontrivial manifold, the integral \(\int_\Sigma F\) need not be zero. This is because a globally single-valued connection \(A\) cannot be defined in this case, so Stokes’ theorem no longer applies. What does this mean?

The Electromagnetic Field Is More Than Just Electric and Magnetic Fields—The AB Effect and Berry Connection [Higher and More Elegant Electrodynamics · 3]

hfwang132@gmail.com (hfwang132) — Sun, 23 Mar 2025 20:20:44 +0800

Previous article in this series:

Godfly: Electrodynamics from the Perspective of Gauge Field Theory [Higher and More Elegant Electrodynamics · 2]

The electromagnetic field is not simply the electric field and the magnetic field.

In other words, the electromagnetic field is more than just the electric field and the magnetic field.

What does this mean?

1. Electromagnetic Potentials Have a Higher Status Than Field Strengths

In a previous article, we left a question open:

Electrodynamics from the Perspective of Hamiltonian Mechanics [Higher and More Elegant Electrodynamics · 5]

hfwang132@gmail.com (hfwang132) — Sat, 22 Mar 2025 23:16:56 +0800

Previous article in this series:

Godfly: Dirac Magnetic Monopoles—Chern Classes and Chern Numbers [Higher and More Elegant Electrodynamics · 4]

In a previous article, we discussed the Hamiltonians of the electromagnetic field and the Dirac field:

\[\begin{aligned} \mathcal{L}= \bar{\psi}(\mathrm{i}\gamma^\mu \partial_\mu -m)\psi - eA_\mu \bar{\psi} \gamma^\mu \psi -\frac{1}{4} F_{\mu\nu} F^{\mu\nu} \end{aligned} \]

So what are their Hamiltonians? Why does quantum field theory not make much use of Hamiltonians? Read on for the answer.

Electrodynamics from the Perspective of Gauge Field Theory [Higher and More Elegant Electrodynamics · 2]

hfwang132@gmail.com (hfwang132) — Mon, 17 Feb 2025 17:15:55 +0800

If you have not yet read the previous article, please see:

Electrodynamics from the Perspective of Differential Geometry [Higher and More Elegant Electrodynamics · 1]

This article uses many concepts from the previous article, so please make sure you have read it.

This article continues to use the metric convention of \((-,+,+,+)\).

Why is electrodynamics a gauge field theory?

To understand what this statement means, we must first understand what electrodynamics is and what gauge field theory is.

Electrodynamics from the Perspective of Differential Geometry [Higher and More Elegant Electrodynamics · 1]

hfwang132@gmail.com (hfwang132) — Fri, 07 Feb 2025 22:55:58 +0800

Preface

You may have heard that Maxwell’s equations have a very simple form:

\[\begin{aligned} \mathrm{d} F&=0 \\ \mathrm{d} \star F &= \mu_0 \star J \end{aligned}\]

Or alternatively,

\[\begin{aligned} \partial_\mu (\star{F})^{\mu \nu}&= 0 \\ \partial_\mu F^{\mu\nu}&= \mu_0 J^\nu \end{aligned}\]

Note: Strictly speaking, \(\mathrm{d}F=0\) (or \(\partial_\mu (\star{F})^{\mu \nu}= 0\)) is not part of the dynamical equations of the electromagnetic field, but rather part of the field’s own structure. This is because \(\mathrm{d}F=0\) follows from the definition of the field strength \(F=\mathrm{d}A\).