Mathematical Methods in Physics on Haifei's Home

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.

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.

Weyl and Wigner Representations of Quantum States/Operators

hfwang132@gmail.com (hfwang132) — Tue, 08 Oct 2024 11:36:44 +0800

Preface

In classical mechanics, a physical quantity is a function on phase space, while the state of a system is a point in phase space (or, for an ensemble, a probability distribution on phase space). Upon quantizing this language, one obtains the phase-space representation of quantum states/operators, namely the Wigner representation. The Weyl representation is its Fourier transform.

Most authors define the Wigner representation as:

\[F_W(x,p)=\int \mathrm{d}y \langle x+\frac{y}{2}\mid F \mid x - \frac{y}{2} \rangle e^{\mathrm{i} p y}\]

and then derive its various properties. However, from a physicist’s perspective, the physical meaning of this expression is unclear, and \((x,p)\) do not have equal status, which is somewhat uncomfortable to look at.

What Exactly Is a Pseudovector?

hfwang132@gmail.com (hfwang132) — Fri, 26 May 2023 18:11:03 +0800

In physics books, we often encounter the terms “pseudovector” (pseudo-vector) and “pseudoscalar” (pseudo-scalar).

In fact, on a 3-dimensional manifold, a “pseudovector” is the exterior product of two tangent vectors \(v\in T_pM\wedge T_pM=\bigwedge^2(T_pM)\), while a “pseudoscalar” is the exterior product of three tangent vectors \(s\in T_pM\wedge T_pM\wedge T_pM=\bigwedge^3(T_pM)\).

After equipping the space with an inner product (or a nondegenerate bilinear form), there is a Hodge duality relation between \(\bigwedge^2(T_pM)\) and \(\bigwedge^1(T_pM)\), so we “mistakenly regard” pseudovectors as vectors. Similarly, because there is a Hodge duality relation between \(\bigwedge^3(T_pM)\) and \(\bigwedge^0(T_pM)\) (a scalar field), we “mistakenly regard” pseudoscalars as scalars.

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.

Wave Packets, Group Velocity, and Dispersion Relations

hfwang132@gmail.com (hfwang132) — Wed, 04 May 2022 15:39:54 +0800

Introduction

When I first began studying physics, I was somewhat confused about the concepts of wave packets and group velocity. Looking back now, they are actually very simple; it is just that some textbooks do not explain them clearly. This article reviews the concept of group velocity and several examples of dispersion relations.

Momentum Eigenstates and Fourier Transforms

The wave function of a plane wave

\[|\bm p\rangle=\frac{1}{(2\pi\hbar)^{3/2}}\exp(\mathrm{i}\frac{\bm{p}}{\hbar}\cdot\bm{x})\]

cannot be normalized.

According to the postulates of quantum mechanics, all physically existing wave functions are normalizable. Therefore, momentum eigenstates \(|p\rangle\) do not represent physically existing states. In other words, momentum cannot be measured with complete precision.

A Concise Proof of Vector Product Rules

hfwang132@gmail.com (hfwang132) — Sat, 27 Feb 2021 00:46:19 +0800

I believe that anyone learning vector calculus for the first time finds the product rules for vectors quite daunting:

Easy difficulty:

\[\nabla(fg)=f\nabla g+g\nabla f\]\[\nabla\cdot(f\mathrm{A})=f\nabla\cdot\mathrm{A}+\nabla f\cdot \mathrm{A}\]\[\nabla\times(f\mathrm{A})=f\nabla\times\mathrm{A}+\nabla f\times\mathrm{A}\]

Hard difficulty:

\[\nabla\cdot(\mathrm{A}\times\mathrm{B})=\mathrm{B}\cdot(\nabla\times\mathrm{A})-\mathrm{A}\cdot(\nabla\times\mathrm{B})\]

Hell difficulty:

\[\nabla(\mathrm{A}\cdot \mathrm{B})=\mathrm{A}\times(\nabla\times\mathrm{B})+(\mathrm{A}\cdot\nabla)\mathrm{B}+\mathrm{B}\times(\nabla\times\mathrm{A})+(\mathrm{B}\cdot\nabla)\mathrm{A}\]\[\nabla\times(\mathrm{A}\times\mathrm{B})=\mathrm{A}(\nabla\cdot \mathrm{B})-\mathrm{B}(\nabla\cdot\mathrm{A})+(\mathrm{B}\cdot\nabla)\mathrm{A}-(\mathrm{A}\cdot\nabla)\mathrm{B}\]

DLC: second derivatives

\[\nabla\times(\nabla f)=0\]\[\nabla\cdot(\nabla\times\mathrm{A})=0\]\[\nabla\times(\nabla\times\mathrm{A})=\nabla(\nabla\cdot\mathrm{A})-(\nabla\cdot\nabla)\mathrm{A}\]

A Concise Proof

To make the proof as concise as possible, we would like to use a single expression with subscripts to represent multiple expressions, rather than having to write out x, y, and z every time.

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,

Stop Being Looked Down Upon by Math Majors! What Physicists Need to Know About Uniform Convergence When Interchanging Operations

hfwang132@gmail.com (hfwang132) — Wed, 09 Dec 2020 00:22:44 +0800

Preface

Whenever physicists perform operations such as differentiating under the integral sign or interchanging the order of integration, the mathematicians watching can no longer sit still:

“Does your improper integral/series converge uniformly?”

The physicist replies:

“What is uniform convergence? We have always done it this way.”

Or:

“Assume that this function has sufficiently nice properties.”

Or, even more outrageously:

“Assume that this function looks rather pretty.”

looks good

…