Methods of Mathematical Physics 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?

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 \(F=\mathrm{d}A\).

Geometric Intuition for Lorentz Transformations

hfwang132@gmail.com (hfwang132) — Fri, 26 Feb 2021 23:05:45 +0800

You do not need to know any formulas to gain an intuitive understanding of various phenomena in special relativity (length contraction, time dilation, the twin paradox, etc.).

A Small Modification to the Galilean Transformation

Let us first look at the classical Galilean transformation. In a t-x diagram, the Galilean transformation appears as a shear transformation. Points slide along lines parallel to the x-axis.

Galilean transformation, viewed from Little Red’s and Little Green’s perspectives

Foundations of Axiomatic Set Theory (Part II): Toward Infinity

hfwang132@gmail.com (hfwang132) — Wed, 24 Feb 2021 22:21:02 +0800

Link to the Previous Article

Foundations of Axiomatic Set Theory (Part I): The Empty Set and Russell’s Paradox## Constructing Natural Numbers (ZF3: Axiom of Pairing, ZF4: Axiom of Union)

In the previous article, we successfully defined a unique set called the empty set and resolved Russell’s paradox. Next, we will use the empty set to construct more sets. To do so, we need more constructive axioms.

Axiom of Pairing: \(\forall a \forall b \exists c \forall x((x\in c)\leftrightarrow (x=a )\vee (x=b))\)