Differential Geometry on Haifei's Home

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:

What Is the Invariance of Differential Forms?

hfwang132@gmail.com (hfwang132) — Fri, 21 Mar 2025 22:49:13 +0800

What Is the Invariance of Differential Forms?

According to textbooks, the invariance of differential forms means that, under a change of variables, the form of a differential equation remains unchanged. That is:

If, when $\mathrm{d}y$ in $y$ denotes the function $y=f(x)$ and $x$ in $\mathrm{d}x$ denotes the function $\phi(x)=x$, $\mathrm{d}y=a\mathrm{d}x$ holds, that is, $\mathrm{d}f=a\mathrm{d}\phi$ holds,

then, when $y$ in $\mathrm{d}y$ denotes the function $y=g(t)$ and $x$ in $\mathrm{d}x$ denotes the function $x=\varphi(t)$, $\mathrm{d}y=a\mathrm{d}x$ likewise holds, that is, $\mathrm{d}g=a\mathrm{d}\varphi$ holds.

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 $F=\mathrm{d}A$.

What is the relationship between Lie derivative and covariant derivative?

hfwang132@gmail.com (hfwang132) — Sat, 15 Jul 2023 16:52:03 +0800

I. Differences and Similarities in Properties

Lie derivative $\mathcal{L}_V$ and covariant derivative $\nabla_V$ share many common points:

Both $\mathcal{L}_V$ and $\nabla_V$ preserve the type of tensors, mapping $\mathcal{T}^p_q(M)$ to $\mathcal{T}^p_q(M)$. $\mathcal{T}^p_q(M)$ represents the set of all smooth tensor fields of type (p, q) on $M$.

Particularly, for (0,0) type tensor fields, i.e., scalar fields $f\in \mathcal{F}(M)$, we have $\mathcal{L}_V f=\nabla_V f=Vf$.

Both satisfy linearity and the Leibniz rule:

$ \begin{aligned} \mathcal{L}_V(\mu A + \lambda B) &= \mu \mathcal{L}_V A + \lambda \mathcal{L}_V B, \\ \mathcal{L}_V (A \otimes B) &= (\mathcal{L}_V A)\otimes B + A \otimes (\mathcal{L}_V B) \end{aligned} $

An Introduction to Differential Geometry (Physics Version)

hfwang132@gmail.com (hfwang132) — Tue, 16 May 2023 21:27:19 +0800

Preface

As a physics student, you have probably heard of one or more of the concepts tensor, differential form, exterior algebra, connection/curvature, and so on. But, like me, you may feel completely lost whenever you hear these concepts. This article therefore aims to help physics students organize these closely interconnected ideas.

The structure of this article is as follows:

a. In Chapter 1, we introduce the stage on which physics takes place: differentiable manifolds, and define scalar fields on manifolds.

Understanding the Invariance of Differential Forms

hfwang132@gmail.com (hfwang132) — Tue, 16 Feb 2021 14:23:58 +0800

21/03/2025: Major revision of this article

What Is the Invariance of Differential Forms?

According to textbooks, the invariance of differential forms means that, under a change of variables, the form of a differential equation remains unchanged. That is:

If, when $\mathrm{d}y$ in $y$ denotes the function $y=f(x)$ and $\mathrm{d}x$ in $x$ denotes the function $\phi(x)=x$, $\mathrm{d}y=a\mathrm{d}x$ holds, that is, $\mathrm{d}f=a\mathrm{d}\phi$ holds,

then, when $\mathrm{d}y$ in $y$ denotes the function $y=g(t)$ and $\mathrm{d}x$ in $x$ denotes the function $x=\varphi(t)$, $\mathrm{d}y=a\mathrm{d}x$ likewise holds, that is, $\mathrm{d}g=a\mathrm{d}\varphi$ holds.

What Is a Differential?

hfwang132@gmail.com (hfwang132) — Mon, 15 Feb 2021 21:44:54 +0800

I. Is a Differential an Infinitesimal?

Physicists like to regard a differential as a very small quantity. This is always convenient in calculations, but it gives one a feeling of imprecision.

In fact, it is indeed imprecise; the second mathematical crisis arose for this reason.

Rigor and accessibility are forever complementary. Regarding a differential as an infinitesimal caters to intuitive sensibilities, yet cannot pass rational scrutiny.

II. A Differential Is a Linear Function

I prefer to think of a differential (at a certain point) as a machine. For example,