# Complex solutions to Maxwell’s equations

Sachin Munshi∗ and Rongwei Yang  Department of Mathematics and Statistics,
SUNY at Albany, Albany, NY 12222, U.S.A<EMAIL_ADDRESS><EMAIL_ADDRESS>Department of Mathematics and Statistics, SUNY at Albany,
Albany, NY 12222, U.S.A<EMAIL_ADDRESS>
###### Abstract.

This paper provides a view of Maxwell’s equations from the perspective of
complex variables. The study is made through complex differential forms and
the Hodge star operator in $\mathbb{C}^{2}$ with respect to the Euclidean and
the Minkowski metrics. It shows that holomorphic functions give rise to
nontrivial solutions, and the inner product between the electric and the
magnetic fields is considered in this case. Further, it obtains a simple
necessary and sufficient condition regarding harmonic solutions to the
equations. In the end, the paper gives an interpretation of the Lorenz gauge
condition in terms of the codifferential operator.

###### Key words and phrases:

Maxwell’s equations, differential forms, Hodge star operator, harmonic
functions, holomorphic functions, Lorenz gauge

###### 2010 Mathematics Subject Classification:

Primary 35Q61, 78A25; Secondary 32A10

∗ Corresponding author

##  1\. Introduction

Named after the physicist and mathematician James C. Maxwell, Maxwell’s
equations form the foundation of classical electromagnetism, optics, and
electrodynamics (see for instance [1, 7, 13]). They are a set of partial
differential equations that describe the interactions between the electric and
magnetic fields that emerge from distributions of electric charges and
currents, and of course, how these fields change in time. Maxwell’s equations
are the differential equations

$\displaystyle\nabla\cdotp\mathbf{B}$ $\displaystyle=$ $\displaystyle 0,$
(1.1a) $\displaystyle\nabla\times\mathbf{E}+\frac{\partial\mathbf{B}}{\partial
t}$ $\displaystyle=$ $\displaystyle 0,$ (1.1b)
$\displaystyle\nabla\cdotp\mathbf{E}$ $\displaystyle=$
$\displaystyle\frac{\rho}{\epsilon_{0}},$ (1.1c)
$\displaystyle\nabla\times\mathbf{B}-\frac{1}{c^{2}}\frac{\partial\mathbf{E}}{\partial
t}$ $\displaystyle=$ $\displaystyle\mu_{0}\mathbf{J},$ (1.1d)

where $\mathbf{E}=(E_{1},E_{2},E_{3})$ is the electric field,
$\mathbf{B}=(B_{1},B_{2},B_{3})$ is the magnetic field, the scalar $\rho$ is
the electric charge density, and the vector $\mathbf{J}$ is the electric
current density vector. Moreover, $\epsilon_{0}$ is the vacuum permittivity,
$\mu_{0}$ is the vacuum permeability, and $c:=1/\sqrt{\epsilon_{0}\mu_{0}}$ is
the speed of light in vacuum. For ease of use, most modern physicists and
mathematicians simply set $c=1$, as shall we throughout this paper. For
convenience, we fix the notation $x_{0}=ct=t$. Then in a more compact form,
equations (1.1a)-(1.1d) can be written in differential forms over Minkowski
space-time as

$\displaystyle dF$ $\displaystyle=$ $\displaystyle 0,$ (1.2)
$\displaystyle\star\text{ }d\star F$ $\displaystyle=$ $\displaystyle J,$ (1.3)

where

$\displaystyle F$
$\displaystyle=-dx_{0}\wedge\left(E_{1}dx_{1}+E_{2}dx_{2}+E_{3}dx_{3}\right)$
(1.4) $\displaystyle-B_{1}dx_{2}\wedge dx_{3}+B_{2}dx_{1}\wedge
dx_{3}-B_{3}dx_{1}\wedge dx_{2},$

and it is referred to as the Faraday $2$-form. Here, $J$ is the current
$1$-form, $d$ is the exterior differential operator, and $\star$ is the Hodge
star operator ([1]). These will be described in detail in this paper. By the
Poincaré lemma, Equation (1.2) implies that locally $F=d\omega$ for some
differentiable $1$-form
$\omega=\eta_{0}dx_{0}+\eta_{1}dx_{1}+\eta_{2}dx_{2}+\eta_{3}dx_{3}.$ This
paper investigates the case when $\eta_{j},0\leq j\leq 3$, are all harmonic
functions in $(x_{0},x_{1},x_{2},x_{3})$ with respect to the Euclidean or
Minkowski metric. If one takes advantage of the identification
$\mathbb{R}^{4}\simeq\mathbb{C}^{2}$, i.e.

$\mathbb{R}^{4}\ni\left(x_{0},x_{1},x_{2},x_{3}\right)\mapsto\left(x_{0}+ix_{1},x_{2}+ix_{3}\right):=\left(z_{1},z_{2}\right)\in\mathbb{C}^{2},$

then $\omega$ can be written as
$\omega(z)=f_{1}dz_{1}+f_{2}dz_{2}+f_{\bar{1}}d\bar{z}_{1}+f_{\bar{2}}d\bar{z}_{2}$.
The following is the main result.

######  Theorem 1.1.

Let $f_{j},f_{\bar{j}},\text{ }j=1,2$ be harmonic functions on
$\mathbb{C}^{2}$. Then the complex differential form $F_{\omega}:=d\omega$ is
a solution to the source-free Maxwell’s equations in the Euclidean metric if
and only if
$\bar{\partial}_{1}f_{1}+\bar{\partial}_{2}f_{2}+\partial_{1}f_{\bar{1}}+\partial_{2}f_{\bar{2}}$
is constant.

A parallel theorem holds with respect to the Minkowski metric on
$\mathbb{R}^{4}\simeq\mathbb{C}^{2}$. A solution $F_{\omega}$ to the Maxwell’s
equations with respect to the Minkowski metric is said to be wavelike if each
of the functions $f_{j},f_{\bar{j}},\text{ }j=1,2$, is a solution to the
d’Alembertian equation (or wave equation)

$\left(\frac{\partial^{2}}{\partial x_{0}^{2}}-\frac{\partial^{2}}{\partial
x_{1}^{2}}-\frac{\partial^{2}}{\partial
x_{2}^{2}}-\frac{\partial^{2}}{\partial x_{3}^{2}}\right)u=0.$

Known solutions to the Maxwell’s equations are all wavelike, so the next
result seems surprising.

######  Theorem 1.2.

There exist non-wavelike solutions to the source-free Maxwell’s equations with
respect to the Minkowski metric.

The paper is organized as follows.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Complex Differential Forms
    2. 2.2 Hodge Star Operator $\star$
    3. 2.3 Self-dual and Anti-self-dual Forms
    4. 2.4 The Hodge Laplacian
  3. 3 Maxwell’s Equations
    1. 3.1 Classical Version
    2. 3.2 Differential Forms Version
  4. 4 Harmonic Solutions To Maxwell’s Equations
    1. 4.1 Euclidean Metric Case
    2. 4.2 Minkowski Metric Case
  5. 5 On the Lorenz gauge
  6. 6 Concluding Remarks

##  2\. Preliminaries

Recall that $\mathbb{R}^{4}$ is just Euclidean space of real dimension 4\. As
a vector space, a point in $\mathbb{R}^{4}$ may be considered as a row vector
$\mathbf{x}=\left(x_{0},x_{1},x_{2},x_{3}\right),x_{i}\in\mathbb{R}$. Note
that in the Lorentzian signature, $\mathbb{R}^{4}$ is denoted as
$\mathbb{R}^{1,3}$, and is referred to as Minkowski space-time, with $x_{0}$
as the time variable (denoted $ct$), where $c$ is the speed of light, and
$x_{1},x_{2},x_{3}$ are the space variables. Going back to viewing
$\mathbb{R}^{4}$ in the Euclidean metric, we have the identification
$\mathbb{R}^{4}\simeq\mathbb{C}^{2}$, i.e.

$\mathbb{R}^{4}\ni\left(x_{0},x_{1},x_{2},x_{3}\right)\mapsto\left(x_{0}+ix_{1},x_{2}+ix_{3}\right):=\left(z_{1},z_{2}\right)\in\mathbb{C}^{2}.$

###  2.1. Complex Differential Forms

We first introduce some preliminaries on complex differential forms. Consider
$\mathbb{C}^{n}$ with points given by coordinates
$z=\left(z_{1},z_{2},\dots,z_{n}\right)$. The tangent space of
$\mathbb{C}^{n}$ is

$T\left(\mathbb{C}^{n}\right)=\text{span}\left\\{\frac{\partial}{\partial
z_{k}},\frac{\partial}{\partial\bar{z}_{k}}:1\leq k\leq n\right\\},$

and the cotangent space is given by

$T^{\ast}\left(\mathbb{C}^{n}\right)=\text{span}\left\\{dz_{k},d\bar{z}_{k}:1\leq
k\leq n\right\\}.$

######  Definition 2.1.

Let $f$ be a smooth function on a domain $\mathcal{M}\subset\mathbb{C}^{n}$.
Consider the linear operators $\partial,\bar{\partial},d$ defined to act on
$f$ as follows:

$\displaystyle\partial f=\sum_{k=1}^{n}\frac{\partial f}{\partial
z_{k}}dz_{k},\hskip 14.22636pt\bar{\partial}f=\sum_{k=1}^{n}\frac{\partial
f}{\partial\bar{z}_{k}}d\bar{z}_{k},\hskip
14.22636ptdf=\left(\partial+\bar{\partial}\right)f.$

Here, $d$ is called the complex exterior differential operator, while
$\partial,\bar{\partial}$ are called the Dolbeault operators.

Recall that a smooth function $f$ on a domain $\mathcal{M}$ is said to be
holomorphic if $\bar{\partial}f=0$ everywhere on $\mathcal{M}$. Clearly, this
means that $f$ is analytic in each variable ([16]). Note that if $f$ is
holomorphic, then $df=\partial f$. The following fact is well-known.

######  Fact 2.2.

$\partial^{2}=\bar{\partial}^{2}=d^{2}=0$.

For a multi-index $I=i_{1}i_{2}\cdots i_{p}$ we assume
$i_{1}<i_{2}<\cdots<i_{p}$ and define its length $|I|=p$.

######  Definition 2.3.

The space of complex $\left(p,q\right)$-forms on $\mathcal{M}$ is defined as

$\Omega^{p,q}\left(\mathcal{M}\right):=\left\\{\sum_{|I|=p,|J|=q}f_{I,J}dz_{I}\wedge
d\bar{z}_{J}:p+q\leq 2n,f_{I,J}\in
C^{\infty}\left(\mathcal{M}\right)\right\\}.$

We may drop the $\mathcal{M}$ from the definition for convenience. Clearly,
$\Omega^{1,0}$ is the space of complex differential forms containing only the
$dz_{k}$ terms, and $\Omega^{0,1}$ is the space of forms containing only the
$d\bar{z}_{k}$ terms. Then in terms of the exterior product on differential
forms we have

$\Omega^{p,q}=\underbrace{\Omega^{1,0}\wedge\cdots\wedge\Omega^{1,0}}_{p}\wedge\underbrace{\Omega^{0,1}\wedge\cdots\wedge\Omega^{0,1}}_{q}.$

Slightly abusing notation, we shall adhere to the following definition
throughout this paper.

######  Definition 2.4.

$\Omega^{k}:=\bigoplus_{p+q=k}\Omega^{p,q}$ is the space of all complex
differential forms of total degree $k=p+q$.

######  Definition 2.5.

For each $p>0$, the forms given by $\sum_{|I|=p}f_{I}dz_{I}$, where each
$f_{I}$ is holomorphic, are called holomorphic p-forms, and they form a
holomorphic section of $\Omega^{p,0}$.

Note that if $\eta=\sum_{|I|=p}f_{I}d\bar{z}_{I}$, with $f_{I}$ holomorphic,
then $\bar{\partial}\eta=0$.

###  2.2. Hodge Star Operator $\star$

We briefly go over some basics of Hodge theory withholding any discussion on
topology or manifold theory. We identify $\mathbb{C}^{n}$ with
${\mathbb{R}}^{2n}$ with the representation $z_{k}=x_{2k-2}+ix_{2k-1},\
k=1,2,...,n$. Given a nondegenerate self-adjoint $2n\times 2n$ matrix
$g=\left(g_{st}\right)$, it induces a sesquilinear form
$\langle\cdot,\cdot\rangle$ on the tangent space
$T\left(\mathbb{R}^{2n}\right)$ with complex coefficients by the evaluations

$\left\langle\alpha\frac{\partial}{\partial
x_{s}},\beta\frac{\partial}{\partial
x_{t}}\right\rangle_{g}:=\alpha\overline{\beta}g_{st},$

where $\alpha$ and $\beta$ are complex numbers and $0\leq s,t\leq 2n-1$. Then
on the cotangent space $T^{*}\left(\mathbb{R}^{2n}\right)$ one has the
corresponding sesquilinear form given by

$\langle\alpha dx_{s},\beta dx_{t}\rangle_{g}:=\alpha\overline{\beta}g^{st},$

where $(g^{st})=g^{-1}$. Using the fact $z_{k}=x_{2k-2}+ix_{2k-1}$ above and
the following representation

$\frac{\partial}{\partial z_{k}}=\frac{1}{2}\left(\frac{\partial}{\partial
x_{2k-2}}-i\frac{\partial}{\partial x_{2k-1}}\right),k=1,2,...,n,$

one may regard the aforementioned sesquilinear form
$\langle\cdot,\cdot\rangle_{g}$ as a sesquilinear form on
$T^{*}\left(\mathbb{C}^{n}\right)$. Further, it can be extended to a
sesquilinear form on $\Omega^{p}$ such that for
$\eta=\eta_{1}\wedge\cdots\wedge\eta_{p},\
\xi=\xi_{1}\wedge\cdots\wedge\xi_{p}$ one has

$\langle\eta,\xi\rangle_{g}=\det[\langle\eta_{s},\xi_{t}\rangle_{g}]_{s,t=1}^{p},\text{
}\eta,\xi\in\Omega^{p}.$

One observes that if the matrix $g$ is positive definite then
$\langle\cdot,\cdot\rangle_{g}$ is an inner product on the set of constant
$p$-forms for each $1\leq p\leq 2n$.

######  Definition 2.6.

The Hodge star operator
$\star:\Omega^{p}\left(\mathbb{C}^{n}\right)\rightarrow\Omega^{2n-p}\left(\mathbb{C}^{n}\right)$
with respect to the bilinear form $\langle\cdot,\cdot\rangle_{g}$ is a linear
operator such that for $\eta,\xi\in\Omega^{p}\left(\mathbb{C}^{n}\right)$ one
has

$\eta\wedge\star\bar{\xi}=\langle\eta,\xi\rangle_{g}\textup{vol}_{g},$

where $\textup{vol}_{g}=\left(\frac{i}{2}\right)^{n}\sqrt{|\det
g|}dz_{1}\wedge d\bar{z}_{1}\wedge\cdots\wedge dz_{n}\wedge d\bar{z}_{n}$ is
the volume form ([11]).

######  Example 2.7.

For the Euclidean metric on ${\mathbb{R}}^{2n}$ we have $g=I_{2n}$. Let
$\Omega^{p}\left(\mathbb{R}^{2n}\right)$ denote the space of real differential
$p$-forms over $\mathbb{R}^{2n}$. Then it is well-known that

$\star\left(dx_{0}\wedge dx_{2}\wedge\cdots\wedge dx_{p-1}\right)=dx_{p}\wedge
dx_{p+2}\wedge\cdots\wedge dx_{2n-1}.$ (2.1)

And for a permutation $\sigma=\left(i_{0},i_{1},\dots,i_{2n-1}\right)$, we
have

$\star\left(dx_{i_{0}}\wedge dx_{i_{1}}\wedge\cdots\wedge
dx_{i_{p-1}}\right)=\left(-1\right)^{\sigma}dx_{i_{p}}\wedge
dx_{i_{p+1}}\wedge\cdots\wedge dx_{i_{2n-1}}$

depending on the parity of $\sigma$. In particular, $\star\left(dx_{0}\wedge
dx_{1}\wedge\cdots\wedge dx_{2n-1}\right)=1$ 111$dx_{0}\wedge
dx_{1}\wedge\cdots\wedge dx_{2n-1}$ is the volume form.. Moreover, for a
$p$-form $\omega$,

$\star^{2}\omega=\left(-1\right)^{p\left(2n-p\right)}\omega.$

This is only true in the Euclidean metric. Generally,
$\star^{2}\omega=\left(-1\right)^{p\left(2n-p\right)}s\omega$, where $s$ is
the parity of the signature of the inner product defined by the metric.

The next two examples give the Hodge dual on $\mathbb{C}^{2}$ with respect to
the Euclidean metric and the Minkowski metric, respectively, and they will be
used later. Using the earlier identification
$z_{1}=x_{0}+ix_{1},z_{2}=x_{2}+ix_{3}$, we have

$dz_{1}=dx_{0}+idx_{1},dz_{2}=dx_{2}+idx_{3},d\bar{z}_{1}=dx_{0}-idx_{1},d\bar{z}_{2}=dx_{2}-idx_{3},$

and these complex $\left(1,0\right)$\- and $\left(0,1\right)$-forms span the
complex cotangent space $T^{\ast}\left(\mathbb{C}^{2}\right)$. Clearly,
$T^{\ast}\left(\mathbb{C}^{2}\right)$ is also spanned by the real forms
$dx_{k},0\leq k\leq 3$ with complex coefficients.

######  Example 2.8.

One may directly use Definition 2.6 to compute the Hodge dual of complex
differential forms, or one may first write them as real forms, apply Example
2.7 and then convert back to complex forms. We shall use the latter approach.

Let’s start with $\star dz_{1}$. Since $dz_{1}=dx_{0}+idx_{1}$, it follows
that

$\displaystyle\star dz_{1}$ $\displaystyle=\star dx_{0}+i\star dx_{1}$
$\displaystyle=\left(dx_{1}\wedge dx_{2}\wedge
dx_{3}\right)-i\left(dx_{0}\wedge dx_{2}\wedge dx_{3}\right)$
$\displaystyle=\left(dx_{1}-idx_{0}\right)\wedge\left(dx_{2}\wedge
dx_{3}\right)$
$\displaystyle=\left(dx_{1}-idx_{0}\right)\wedge\frac{i}{2}\left(dz_{2}\wedge
d\bar{z}_{2}\right)$
$\displaystyle=\left(dx_{0}+idx_{1}\right)\wedge\frac{1}{2}\left(dz_{2}\wedge
d\bar{z}_{2}\right)$ $\displaystyle=\frac{1}{2}\left(dz_{1}\wedge dz_{2}\wedge
d\bar{z}_{2}\right),$

where the fourth line follows from the fact that

$\displaystyle dz_{2}\wedge d\bar{z}_{2}$
$\displaystyle=\left(dx_{2}+idx_{3}\right)\wedge\left(dx_{2}-idx_{3}\right)=-2idx_{2}\wedge
dx_{3}.$

Likewise one verifies that

$\displaystyle\star dz_{2}=-\frac{1}{2}\left(dz_{1}\wedge dz_{2}\wedge
d\bar{z}_{1}\right),\star d\bar{z}_{1}=\frac{1}{2}\left(dz_{2}\wedge
d\bar{z}_{1}\wedge d\bar{z}_{2}\right),\star
d\bar{z}_{2}=-\frac{1}{2}\left(dz_{1}\wedge d\bar{z}_{1}\wedge
d\bar{z}_{2}\right),$

and

$\displaystyle\star\left(dz_{1}\wedge d\bar{z}_{1}\right)=dz_{2}\wedge
d\bar{z}_{2},$ $\displaystyle\star\left(dz_{2}\wedge
d\bar{z}_{2}\right)=dz_{1}\wedge d\bar{z}_{1},\hskip 14.22636pt$
$\displaystyle\star\left(dz_{1}\wedge d\bar{z}_{2}\right)=-dz_{1}\wedge
d\bar{z}_{2},$ $\displaystyle\star\left(dz_{2}\wedge
d\bar{z}_{1}\right)=-dz_{2}\wedge d\bar{z}_{1},$
$\displaystyle\star\left(dz_{1}\wedge dz_{2}\right)=dz_{1}\wedge dz_{2},\hskip
14.22636pt$ $\displaystyle\star\left(d\bar{z}_{1}\wedge
d\bar{z}_{2}\right)=d\bar{z}_{1}\wedge d\bar{z}_{2}.$

The Hodge star of $3$-forms can be computed using the fact that
$\star^{2}\omega=(-1)^{p(n-p)}\omega$, and one has

$\displaystyle\star\left(dz_{1}\wedge dz_{2}\wedge
d\bar{z}_{1}\right)=2dz_{2},\hskip 14.22636pt\star\left(dz_{1}\wedge
dz_{2}\wedge d\bar{z}_{2}\right)=-2dz_{1},$
$\displaystyle\star\left(dz_{1}\wedge d\bar{z}_{1}\wedge
d\bar{z}_{2}\right)=2d\bar{z}_{2},\hskip 14.22636pt\star\left(dz_{2}\wedge
d\bar{z}_{1}\wedge d\bar{z}_{2}\right)=-2d\bar{z}_{1}.$

Finally, for the unique $\left(2,2\right)$-form, we have

$\displaystyle\star\left(dz_{1}\wedge dz_{2}\wedge d\bar{z}_{1}\wedge
d\bar{z}_{2}\right)=4.$

######  Example 2.9.

The Minkowski metric on ${\mathbb{R}}^{1,3}$ has the signature
$\left(+---\right)$ represented by the metric matrix

$g_{\textbf{mink}}=\begin{pmatrix}1&0&0&0\\\ 0&-1&0&0\\\ 0&0&-1&0\\\
0&0&0&-1\end{pmatrix}.$ (2.2)

In this case, the inner product of complex $1$-forms is determined by the
facts:

$\langle dz_{1},dz_{1}\rangle=\langle dz_{1},dz_{2}\rangle=\langle
dz_{1},d\bar{z}_{2}\rangle=\langle dz_{2},d\bar{z}_{2}\rangle=0$

and

$\langle dz_{1},d\bar{z}_{1}\rangle=\langle dz_{2},dz_{2}\rangle=2.$

Calculation of the Hodge star operator on $\mathbb{C}^{2}$ with respect to the
Minkowski metric is similar to that in Example 2.8. Here we only list its
action on 2-forms and 3-forms for later use:

$\displaystyle\star\left(dz_{1}\wedge dz_{2}\right)=-dz_{2}\wedge
d\bar{z}_{1},$ $\displaystyle\star\left(dz_{1}\wedge
d\bar{z}_{2}\right)=-d\bar{z}_{1}\wedge d\bar{z}_{2},$
$\displaystyle\star\left(dz_{2}\wedge d\bar{z}_{1}\right)=dz_{1}\wedge
dz_{2},$ $\displaystyle\star\left(d\bar{z}_{1}\wedge
d\bar{z}_{2}\right)=dz_{1}\wedge d\bar{z}_{2},$
$\displaystyle\star\left(dz_{1}\wedge d\bar{z}_{1}\right)=-dz_{2}\wedge
d\bar{z}_{2},$ $\displaystyle\star\left(dz_{2}\wedge
d\bar{z}_{2}\right)=dz_{1}\wedge d\bar{z}_{1};$

and

$\displaystyle\star\left(dz_{1}\wedge dz_{2}\wedge
d\bar{z}_{1}\right)=2dz_{2},\hskip 28.45274pt\star\left(dz_{1}\wedge
dz_{2}\wedge d\bar{z}_{2}\right)=2dz_{\bar{1}},$
$\displaystyle\star\left(dz_{1}\wedge d\bar{z}_{1}\wedge
d\bar{z}_{2}\right)=2d\bar{z}_{2},\hskip 28.45274pt\star\left(dz_{2}\wedge
d\bar{z}_{1}\wedge d\bar{z}_{2}\right)=2dz_{1}.$

###  2.3. Self-dual and Anti-self-dual Forms

We now focus on the Hodge star operator on the complex differential forms on
$\mathbb{C}^{2}$ with respect to the Euclidean metric and the Minkowski
metric.

1. Under the Euclidean metric, we have $\star^{2}\omega=\omega$ for every $\omega\in\Omega^{2}(\mathbb{C}^{2})$. Self-dual and anti-self-dual forms are the eigenvectors corresponding to the eigenvalues $1$ and $-1$, respectively, of the Hodge star operator on $\mathbb{C}^{2}$.

######  Definition 2.10.

A differential form $\omega$ is said to be self-dual if it is equal to its
Hodge dual, i.e. $\star\omega=\omega$. If $\star\omega=-\omega$, then $\omega$
is said to be anti-self-dual.

Out of the complex differential forms we considered in the previous
subsection, six of them correspond to the pair $\left(p,q\right)$ such that
$p+q=2$. Let $\Omega_{+}^{2},\Omega_{-}^{2}$ denote the bases of self-dual and
anti-self-dual forms, respectively, in $\Omega^{2}$. Then by the computations
in Example 2.8 we have that

$\displaystyle\Omega_{+}^{2}$ $\displaystyle=\text{span}\left\\{dz_{1}\wedge
dz_{2},d\bar{z}_{1}\wedge d\bar{z}_{2},dz_{1}\wedge d\bar{z}_{1}+dz_{2}\wedge
d\bar{z}_{2}\right\\},$ (2.3) $\displaystyle\Omega_{-}^{2}$
$\displaystyle=\text{span}\left\\{dz_{1}\wedge d\bar{z}_{2},dz_{2}\wedge
d\bar{z}_{1},dz_{1}\wedge d\bar{z}_{1}-dz_{2}\wedge d\bar{z}_{2}\right\\}.$

2. Under the Minkowski metric, we have $\star^{2}\omega=-\omega$ for any $\omega\in\Omega^{2}(\mathbb{C}^{2})$. Hence $\star$ has eigenvalues $i,-i$. With a bit of abuse of terminology, the eigenspaces corresponding to them are often also called self-dual and anti-self-dual forms, respectively. For consistency we shall also denote them by $\Omega^{2}_{+}$ and $\Omega^{2}_{-}$, respectively. Then by the computations in Example 2.9, we have

$\displaystyle\Omega_{+}^{2}$ $\displaystyle=\text{span}\left\\{dz_{1}\wedge
dz_{2}+idz_{2}\wedge d\bar{z}_{1},d{z}_{1}\wedge d\bar{z}_{1}+idz_{2}\wedge
d\bar{z}_{2},d{z}_{1}\wedge d\bar{z}_{2}+id\bar{z}_{1}\wedge
d\bar{z}_{2}\right\\},$ (2.4) $\displaystyle\Omega_{-}^{2}$
$\displaystyle=\text{span}\left\\{dz_{1}\wedge dz_{2}-idz_{2}\wedge
d\bar{z}_{1},d{z}_{1}\wedge d\bar{z}_{1}-idz_{2}\wedge
d\bar{z}_{2},d{z}_{1}\wedge d\bar{z}_{2}-id\bar{z}_{1}\wedge
d\bar{z}_{2}\right\\}.$

###  2.4. The Hodge Laplacian

The Hodge star operator gives rise to a Hermitian bilinear form on compactly
supported differentiable $p$-forms defined by

$(\eta,\xi)_{g}:=\int_{\mathbb{C}^{n}}\eta\wedge\star\bar{\xi}=\int_{\mathbb{C}^{n}}\langle\eta,{\xi}\rangle_{g}\textup{vol}_{g}.$

The differential operators
$d:\Omega^{p-1}\left(\mathbb{C}^{n}\right)\rightarrow\Omega^{p}\left(\mathbb{C}^{n}\right)$,
where $1\leq p\leq 2n$, have the following natural adjoint with respect to the
bilinear form $(\cdot,\cdot)_{g}$.

######  Definition 2.11.

The co-differential operator
$d^{*}:\Omega^{p}\left(\mathbb{C}^{n}\right)\rightarrow\Omega^{p-1}\left(\mathbb{C}^{n}\right)$
is defined by

$d^{*}\omega:=\left(-1\right)^{n\left(p+1\right)+1}\star d\star\omega,$ (2.5)

where $\omega$ is any differential $p$-form. In particular, for
$\mathbb{C}^{2}$ we have $d^{*}\omega=-\star d\star\omega$.

######  Definition 2.12.

A differential $p$-form $\omega$ is said to be Hodge-Laplace harmonic (HL-
harmonic for short) if

$\Delta\omega:=\left(dd^{*}+d^{*}d\right)\omega=0,$ (2.6)

where $\Delta$ is referred to as the Hodge Laplacian (or the Laplace-de Rham
operator).

For more information on complex differential forms we refer readers to [1, 2,
9, 19].

##  3\. Maxwell’s Equations

Maxwell equations have several equivalent formulations ([1, 6, 13, 18]). From
the viewpoint of physics, there are versions of Maxwell’s equations based on
electric and magnetic potentials that allow one to solve the equations as a
boundary value problem within the realms of classical physics and quantum
mechanics. Quantum mechanics is not in the scope of this paper, so we shall
only consider Maxwell’s equations in the classical sense. Moreover, the space-
time formulations of Maxwell’s equations are primarily used in high-energy
physics and gravitational physics, in conjunction with Einstein’s theories of
special relativity and general relativity. From the viewpoint of mathematics,
Maxwell’s equations are important in vector calculus, potential field theory,
gauge theory, differential geometry, topology, and many other studies ([1, 7,
9]).

###  3.1. Classical Version

Let $\mathbf{E}=\left(E_{1},E_{2},E_{3}\right)$ and
$\mathbf{B}=\left(B_{1},B_{2},B_{3}\right)$ be electric and magnetic fields,
respectively, in a convex region of $\mathbb{R}^{4}$ with Lorentzian signature
$+---$222This is essentially the Minkowski space-time $\mathbb{R}^{1,3}$.,
where the $E_{j},B_{j}$ are scalar-valued functions of time and space. In this
vector space, we represent points as coordinate column vectors
$\left(x_{0},x_{1},x_{2},x_{3}\right)$, where $x_{0}=ct$ ($c$ being the speed
of light) is the time variable and $x_{1},x_{2},x_{3}$ are the space variables
333For simplicity, we often set $c=1$.. Maxwell’s equations are given by the
following partial differential equations:

$\displaystyle\nabla\cdotp\mathbf{B}$ $\displaystyle=$ $\displaystyle 0,$
(3.1a) $\displaystyle\nabla\times\mathbf{E}+\frac{\partial\mathbf{B}}{\partial
t}$ $\displaystyle=$ $\displaystyle 0,$ (3.1b)
$\displaystyle\nabla\cdotp\mathbf{E}$ $\displaystyle=$ $\displaystyle\rho,$
(3.1c) $\displaystyle\nabla\times\mathbf{B}-\frac{\partial\mathbf{E}}{\partial
t}$ $\displaystyle=$ $\displaystyle\mathbf{J},$ (3.1d)

where the scalar $\rho$ is the electric charge density and the vector
$\mathbf{J}$ is the electric current density vector. The equations (3.1a) and
(3.1b) are homogeneous, while the equations (3.1c) and (3.1d) are
inhomogeneous. From vector calculus,
$\nabla\cdotp\left(\nabla\times\mathbf{A}\right)=0$ for any smooth vector
field $\mathbf{A}=(A_{1},A_{2},A_{3})$ and
$\nabla\times\left(\nabla\phi\right)=0$ for any scalar function $\phi$ 444From
physics viewpoint, a scalar field.. Now by Poincaré’s lemma, in
$\mathbb{R}^{3}$, if $\nabla\cdotp\mathbf{B}=0$, then
$\mathbf{B}=\nabla\times\mathbf{A}$ for some vector field $\mathbf{A}$. This
leads us to the potential field theory aspect of electromagnetism. So in this
context, one considers the magnetic vector potential $\mathbf{A}$ and the
electric scalar potential $\phi$ such that

$\displaystyle\mathbf{B}$ $\displaystyle=\nabla\times\mathbf{A},$ (3.2)
$\displaystyle\mathbf{E}$ $\displaystyle=-\frac{\partial\mathbf{A}}{\partial
t}-\nabla\phi.$ (3.3)

Putting (3.2) and (3.3) in equations (3.1c) and (3.1d), we obtain

$\rho=-\frac{\partial}{\partial
t}\left(\nabla\cdotp\mathbf{A}\right)-\Delta\phi,$ (3.4)

where $\Delta$ in (3.4) is the Laplacian in $\mathbb{R}^{3}$.

So if we let
$\mathbf{A}^{{}^{\prime}}=\mathbf{A}-\nabla\psi,\phi^{{}^{\prime}}=\phi+\frac{\partial\psi}{\partial
t}$, for any scalar field $\psi$, then the 4-vector
$\left(\phi^{{}^{\prime}},\mathbf{A}^{{}^{\prime}}\right)$ solves (3.2) and
(3.3), but not uniquely. In particular, this gives one the gauge freedom to
choose $\phi$ such that $\left(\phi,\mathbf{A}\right)$ satisfies the
Lorenz555Not to be confused with Lorentz! gauge:

$\frac{\partial\phi}{\partial t}+\nabla\cdotp\mathbf{A}=0,$ (3.5)

which implies that

$\displaystyle\rho$ $\displaystyle=\left(\frac{\partial^{2}}{\partial
t^{2}}-\Delta\right)\phi,$ (3.6) $\displaystyle\mathbf{J}$
$\displaystyle=\left(\frac{\partial^{2}}{\partial
t^{2}}-\Delta\right)\mathbf{A},$ (3.7)

where $\frac{\partial^{2}}{\partial t^{2}}-\Delta$ is the d’Alembertian, or
wave operator. Since Lorentz transformations keep the Minkowski metric
invariant, the d’Alembertian gives a Lorentz scalar. Further, Maxwell’s
equations are Lorentz invariant and gauge invariant.

###  3.2. Differential Forms Version

Set $\omega=\phi dx_{0}-A_{1}dx_{1}-A_{2}dx_{2}-A_{3}dx_{3},$ where ${\bf
A}=(A_{1},A_{2},A_{3})$ is a time-dependent smooth vector field in
${\mathbb{R}}^{3}$. This $1$-form $\omega$ is often referred to as the
magnetic potential 1-form. With the Lorenz gauge from (3.5), we assume
$\omega$ satisfies the following normalization:

$\frac{\partial\phi}{\partial x_{0}}+\frac{\partial A_{1}}{\partial
x_{1}}+\frac{\partial A_{2}}{\partial x_{2}}+\frac{\partial A_{3}}{\partial
x_{3}}=0.$ (3.8)

Now set

$J=\rho dx_{0}+J_{1}dx_{1}+J_{2}dx_{2}+J_{3}dx_{3}.$ (3.9)

Define the $2$-form $F_{\omega}=d\omega$. This is called the Faraday field
strength or simply the Faraday $2$-form. Using (3.2) and (3.3), we have

$\displaystyle F_{\omega}$
$\displaystyle=-dx_{0}\wedge\left(E_{1}dx_{1}+E_{2}dx_{2}+E_{3}dx_{3}\right)$
(3.10) $\displaystyle-B_{1}dx_{2}\wedge dx_{3}+B_{2}dx_{1}\wedge
dx_{3}-B_{3}dx_{1}\wedge dx_{2}.$

With the exterior derivative $d$ and the Hodge star operator $\star$, the
Maxwell’s equations take the concise form

$\displaystyle dF_{\omega}$ $\displaystyle=0,$ (3.11)
$\displaystyle\star\text{ }d\star F_{\omega}$ $\displaystyle=J,$ (3.12)

where (3.11) is referred to as the Bianchi identity. Equation (3.11) is
equivalent to the homogeneous Maxwell’s equations (3.1a) and (3.1b); while
(3.12) is equivalent to the inhomogeneous Maxwell’s equations (3.1c) and
(3.1d). We may also refer to (3.11) and (3.12) as the exterior differential
form of Maxwell’s equations. And we say $\omega$, or alternatively
$F_{\omega}$, is a solution to the Maxwell’s equations if (3.11) and (3.12)
are satisfied.

It is now apparent that solutions to the Maxwell’s equations are not unique,
since if $\omega$ is a solution then ${\omega^{\prime}}$ is also a solution
for $\omega^{\prime}=\omega+d\psi$, where $\psi$ is any smooth function. This
fact gives the gauge freedom of choosing $\phi$ that satisfies the Lorenz
gauge condition (3.5). This differential form formulation of Maxwell’s
equations also makes the following remark apparent.

######  Remark 3.1.

In vacuum, that is, when $\rho=0,J=0$, it follows from (3.11) and (3.12) that
every self-dual or anti-self-dual 2-form $F_{\omega}$ is a solution to the
source-free Maxwell’s equations.

##  4\. Harmonic Solutions To Maxwell’s Equations

In this section we would like to consider Maxwell’s equations in vacuum, and
study some complex solutions and, in particular, harmonic solutions to those
equations. But before doing this, we shall first construct a complex
differential form solution to Maxwell’s equations initially not in vacuum. So
let’s begin by considering smooth functions
$f_{1},f_{2},f_{\bar{1}},f_{\bar{2}}$ in the two variables $z_{1},z_{2}$.
Define the complex differential form

$\omega\left(z\right)=f_{1}dz_{1}+f_{2}dz_{2}+f_{\bar{1}}d\bar{z}_{1}+f_{\bar{2}}d\bar{z}_{2}.$
(4.1)

The form $\omega$ acts as a potential, much like the magnetic potential
discussed in Section 3.2. In a more general context, it is often referred to
as a connection form. The associated curvature form or curvature field is
defined as $F_{\omega}:=d\omega+\omega\wedge\omega$. Since $\omega$ is scalar-
valued, we have $\omega\wedge\omega=0$ and therefore $F_{\omega}=d\omega$.
Direct computation shall verify that

$\displaystyle F_{\omega}$
$\displaystyle=\left(\partial_{1}f_{2}-\partial_{2}f_{1}\right)dz_{1}\wedge
dz_{2}+\left(\bar{\partial}_{1}f_{\bar{2}}-\bar{\partial}_{2}f_{\bar{1}}\right)d\bar{z}_{1}\wedge
d\bar{z}_{2}$
$\displaystyle+\left(\partial_{1}f_{\bar{2}}-\bar{\partial}_{2}f_{1}\right)dz_{1}\wedge
d\bar{z}_{2}+\left(\partial_{2}f_{\bar{1}}-\bar{\partial}_{1}f_{2}\right)dz_{2}\wedge
d\bar{z}_{1}$ (4.2)
$\displaystyle+\left(\partial_{1}f_{\bar{1}}-\bar{\partial}_{1}f_{1}\right)dz_{1}\wedge
d\bar{z}_{1}+\left(\partial_{2}f_{\bar{2}}-\bar{\partial}_{2}f_{2}\right)dz_{2}\wedge
d\bar{z}_{2}.$

Now in terms of real 2-forms, we can write (4.2) in the following way:

$\displaystyle F_{\omega}$
$\displaystyle=-2i\left(\partial_{1}f_{\bar{1}}-\bar{\partial}_{1}f_{1}\right)dx_{0}\wedge
dx_{1}$
$\displaystyle+(\left(\partial_{1}f_{2}-\partial_{2}f_{1}\right)+\left(\bar{\partial}_{1}f_{\bar{2}}-\bar{\partial}_{2}f_{\bar{1}}\right)$
$\displaystyle+\left(\partial_{1}f_{\bar{2}}-\bar{\partial}_{2}f_{1}\right)-\left(\partial_{2}f_{\bar{1}}-\bar{\partial}_{1}f_{2}\right))dx_{0}\wedge
dx_{2}$
$\displaystyle+i(\left(\partial_{1}f_{2}-\partial_{2}f_{1}\right)-\left(\bar{\partial}_{1}f_{\bar{2}}-\bar{\partial}_{2}f_{\bar{1}}\right)$
$\displaystyle-\left(\partial_{1}f_{\bar{2}}-\bar{\partial}_{2}f_{1}\right)-\left(\partial_{2}f_{\bar{1}}-\bar{\partial}_{1}f_{2}\right))dx_{0}\wedge
dx_{3}$ (4.3)
$\displaystyle-2i\left(\partial_{2}f_{\bar{2}}-\bar{\partial}_{2}f_{2}\right)dx_{2}\wedge
dx_{3}$
$\displaystyle+(-\left(\partial_{1}f_{2}-\partial_{2}f_{1}\right)-\left(\bar{\partial}_{1}f_{\bar{2}}-\bar{\partial}_{2}f_{\bar{1}}\right)$
$\displaystyle+\left(\partial_{1}f_{\bar{2}}-\bar{\partial}_{2}f_{1}\right)-\left(\partial_{2}f_{\bar{1}}-\bar{\partial}_{1}f_{2}\right))dx_{1}\wedge
dx_{3}$
$\displaystyle+i(\left(\partial_{1}f_{2}-\partial_{2}f_{1}\right)-\left(\bar{\partial}_{1}f_{\bar{2}}-\bar{\partial}_{2}f_{\bar{1}}\right)$
$\displaystyle+\left(\partial_{1}f_{\bar{2}}-\bar{\partial}_{2}f_{1}\right)+\left(\partial_{2}f_{\bar{1}}-\bar{\partial}_{1}f_{2}\right))dx_{1}\wedge
dx_{2}.$

One can easily determine the electric and magnetic components in view of
(3.10):

$\displaystyle E_{1}$
$\displaystyle=2i\left(\partial_{1}f_{\bar{1}}-\bar{\partial}_{1}f_{1}\right),$
$\displaystyle E_{2}$
$\displaystyle=-\left(\left(\partial_{1}f_{2}-\partial_{2}f_{1}\right)+\left(\bar{\partial}_{1}f_{\bar{2}}-\bar{\partial}_{2}f_{\bar{1}}\right)+\left(\partial_{1}f_{\bar{2}}-\bar{\partial}_{2}f_{1}\right)-\left(\partial_{2}f_{\bar{1}}-\bar{\partial}_{1}f_{2}\right)\right),$
$\displaystyle E_{3}$
$\displaystyle=-i\left(\left(\partial_{1}f_{2}-\partial_{2}f_{1}\right)-\left(\bar{\partial}_{1}f_{\bar{2}}-\bar{\partial}_{2}f_{\bar{1}}\right)-\left(\partial_{1}\bar{f}_{2}-\bar{\partial}_{2}f_{1}\right)-\left(\partial_{2}\bar{f}_{1}-\bar{\partial}_{1}f_{2}\right)\right),$
$\displaystyle B_{1}$
$\displaystyle=2i\left(\partial_{2}f_{\bar{2}}-\bar{\partial}_{2}f_{2}\right),$
$\displaystyle B_{2}$
$\displaystyle=-\left(\partial_{1}f_{2}-\partial_{2}f_{1}\right)-\left(\bar{\partial}_{1}f_{\bar{2}}-\bar{\partial}_{2}f_{\bar{1}}\right)+\left(\partial_{1}f_{\bar{2}}-\bar{\partial}_{2}f_{1}\right)-\left(\partial_{2}f_{\bar{1}}-\bar{\partial}_{1}f_{2}\right),$
$\displaystyle B_{3}$
$\displaystyle=-i\left(\left(\partial_{1}f_{2}-\partial_{2}f_{1}\right)-\left(\bar{\partial}_{1}f_{\bar{2}}-\bar{\partial}_{2}f_{\bar{1}}\right)+\left(\partial_{1}f_{\bar{2}}-\bar{\partial}_{2}f_{1}\right)+\left(\partial_{2}f_{\bar{1}}-\bar{\partial}_{1}f_{2}\right)\right).$

In general, since ${\bf E}$ and ${\bf B}$ are complex vectors in
$\mathbb{C}^{3}$, the “electromagnetic dynamics” occurs in $7$-dimensional
space ($\dim\mathbb{C}^{3}$ plus one dimension for time). The spatial inner
product is defined by

$\langle{\bf E},{\bf
B}\rangle=E_{1}\bar{B_{1}}+E_{2}\bar{B_{2}}+E_{3}\bar{B_{3}}.$

To facilitate the computation of this inner product and the energy density
$\frac{1}{2}(|{\bf E}|^{2}+|{\bf B}|^{2})$, we write (4.2) as

$F_{\omega}=F_{12}dz_{1}\wedge dz_{2}+F_{\bar{1}\bar{2}}d\bar{z}_{1}\wedge
d\bar{z}_{2}+\sum_{j,k=1,2}F_{j\bar{k}}dz_{j}\wedge d{\overline{z_{k}}}.$
(4.4)

Then the following can be verified by direct computation:

$\displaystyle\langle{\bf E},{\bf
B}\rangle=4F_{1\bar{1}}\overline{F_{2\bar{2}}}+2\left[(F_{12}-F_{2\bar{1}})(\overline{F_{12}+F_{2\bar{1}}})+(F_{\bar{1}\bar{2}}+F_{1\bar{2}})(\overline{F_{\bar{1}\bar{2}}-F_{1\bar{2}}})\right].$
(4.5)

The energy density of the electromagnetic dynamics can be computed as

$\displaystyle\frac{1}{2}(|{\bf E}|^{2}+|{\bf
B}|^{2})=2(|F_{12}|^{2}+|F_{\bar{1}\bar{2}}|^{2}+\sum_{j,k=1,2}|F_{j\bar{k}}|^{2}).$
(4.6)

If $g$ is the Euclidean metric on $\mathbb{C}^{2}$, then one can also verify
that

$|{\bf E}|^{2}+|{\bf B}|^{2}=\langle F_{\omega},F_{\omega}\rangle.$

In particular, one observes that if $f_{\bar{i}}=\bar{f}_{i},i=1,2$ then both
the electric field ${\bf E}$ and the magnetic field ${\bf B}$ are real vector-
fields in ${\mathbb{R}}^{3}$. For example, one can write

$E_{1}=2iF_{1\bar{1}}=2i\left(\overline{\bar{\partial}_{1}f_{1}}-\bar{\partial}_{1}f_{1}\right)$

and see that $E_{1}$ is $4$ times the imaginary part of
$\bar{\partial}_{1}f_{1}$. Other components can be checked similarly.
Alternatively, one may observe that since $\omega$ is real in this case, the
curvature field $F_{\omega}=d\omega$ must be real. In this case, one has
$F_{\bar{1}\bar{2}}=\overline{F_{12}}$ and
$F_{1\bar{2}}=-\overline{F_{2\bar{1}}}$, and it follows that

$\displaystyle\langle{\bf E},{\bf
B}\rangle=4\left(F_{1\bar{1}}\overline{F_{2\bar{2}}}+|F_{12}|^{2}-|F_{2\bar{1}}|^{2}\right).$
(4.7)

If ${\bf E}$ and ${\bf B}$ are real vectors, then $\cos^{-1}\frac{\langle{\bf
E},{\bf B}\rangle}{|{\bf E}||{\bf B}|}$ is the angle between the electric
field and the magnetic field. But when ${\bf E}$ and ${\bf B}$ are complex
vectors, the physical meaning of $\langle{\bf E},{\bf B}\rangle$ is less
clear.

Although the original Maxwell’s equations were formulated under the Minkowski
metric, the differential form formulation (3.11) and (3.12) makes good sense
under other metrics. In the sequel we shall consider the Maxwell’s equations
in this formulation under two different metrics.

###  4.1. Euclidean Metric Case

Under the Euclidean metric, applying the Hodge star operator to (4.2) (cf.
Example 2.8), one has

$\displaystyle\star F_{\omega}$
$\displaystyle=\left(\partial_{1}f_{2}-\partial_{2}f_{1}\right)dz_{1}\wedge
dz_{2}+\left(\bar{\partial}_{1}f_{\bar{2}}-\bar{\partial}_{2}f_{\bar{1}}\right)d\bar{z}_{1}\wedge
d\bar{z}_{2}$
$\displaystyle-\left(\partial_{1}f_{\bar{2}}-\bar{\partial}_{2}f_{1}\right)dz_{1}\wedge
d\bar{z}_{2}-\left(\partial_{2}f_{\bar{1}}-\bar{\partial}_{1}f_{2}\right)dz_{2}\wedge
d\bar{z}_{1}$ (4.8)
$\displaystyle+\left(\partial_{2}f_{\bar{2}}-\bar{\partial}_{2}f_{2}\right)dz_{1}\wedge
d\bar{z}_{1}+\left(\partial_{1}f_{\bar{1}}-\bar{\partial}_{1}f_{1}\right)dz_{2}\wedge
d\bar{z}_{2}.$

It is immediate that $F_{\omega}$ is self-dual if and only if

$\displaystyle\partial_{1}f_{\bar{1}}-\bar{\partial}_{1}f_{1}$
$\displaystyle=\partial_{2}f_{\bar{2}}-\bar{\partial}_{2}f_{2},$ (4.9)
$\displaystyle\partial_{1}f_{\bar{2}}-\bar{\partial}_{2}f_{1}$
$\displaystyle=\partial_{2}f_{\bar{1}}-\bar{\partial}_{1}f_{2}=0,$

and it is anti-self-dual if and only if

$\displaystyle\partial_{1}f_{\bar{1}}-\bar{\partial}_{1}f_{1}$
$\displaystyle=-\left(\partial_{2}f_{\bar{2}}-\bar{\partial}_{2}f_{2}\right),$
(4.10) $\displaystyle\partial_{1}f_{2}-\partial_{2}f_{1}$
$\displaystyle=\bar{\partial}_{1}f_{\bar{2}}-\bar{\partial}_{2}f_{\bar{1}}=0.$

The following fact follows immediately from (4.5).

######  Corollary 4.1.

Let $F_{\omega}$ be a self-dual solution to the Maxwell equations in vaccum
with respect to the Euclidean metric on $\mathbb{C}^{2}$. Then $\langle{\bf
E},{\bf B}\rangle\geq 0$ holds with equality only if $\omega$ is a trivial
solution.

###### Proof.

In fact, Equations (4.9) shows that $F_{1\bar{1}}=F_{2\bar{2}}$ and
$F_{1\bar{2}}=F_{2\bar{1}}=0$. Hence by (4.5) one has

$\langle{\bf E},{\bf
B}\rangle=2(2|F_{1\bar{1}}|^{2}+|F_{12}|^{2}+|F_{\bar{1}\bar{2}}|^{2})\geq 0.$

If $\langle{\bf E},{\bf B}\rangle=0$ then all six coefficients of the
$2$-forms in $F_{\omega}$ are $0$ and hence $\omega$ is a trivial solution.∎

Since $F_{\omega}=d\omega$, $F_{\omega}$ is exact. Hence if $F_{\omega}$ is
self-dual or anti-self-dual then

$d\star F_{\omega}=\pm dF_{\omega}=\pm d^{2}\omega=0.$

Now if $f_{1},f_{2}$ are holomorphic and $f_{\bar{1}},f_{\bar{2}}$ are
conjugate holomorphic, then the equations in (4.9) are automatically
satisfied. We thus have the following fact.

######  Proposition 4.2.

The form $F_{\omega}$, determined by $\omega$ in (4.1) with $f_{1},f_{2}$
holomorphic and $f_{\bar{1}},f_{\bar{2}}$ conjugate holomorphic, is a self-
dual solution for the source-free Maxwell’s equations (4.2) and (4.8) under
the Euclidean metric. Further, in this case

$\displaystyle F_{\omega}$
$\displaystyle=\left(\partial_{1}f_{2}-\partial_{2}f_{1}\right)dz_{1}\wedge
dz_{2}+\left(\bar{\partial}_{1}f_{\bar{2}}-\bar{\partial}_{2}f_{\bar{1}}\right)d\bar{z}_{1}\wedge
d\bar{z}_{2}.$ (4.11)

Consequently, we have

######  Corollary 4.3.

If $\omega$ is of the form (4.1) and $f_{1},f_{2}$ are holomorphic and
$f_{\bar{1}},f_{\bar{2}}$ are conjugate holomorphic, then

$\displaystyle E_{1}$ $\displaystyle=B_{1}=0,$ $\displaystyle E_{2}$
$\displaystyle=B_{2}=-\left((\partial_{1}f_{2}-\partial_{2}f_{1})+(\bar{\partial}_{1}f_{\bar{2}}-\bar{\partial}_{2}f_{\bar{1}})\right),$
$\displaystyle E_{3}$
$\displaystyle=B_{3}=-i\left((\partial_{1}f_{2}-\partial_{2}f_{1})-(\bar{\partial}_{1}f_{\bar{2}}-\bar{\partial}_{2}f_{\bar{1}})\right).$

It is surprising that in this case the electric field and the magnetic field
coincide, or in other words they are mathematically indistinguishable. The
same phenomenon occurs later in Example 4.6 on Dirac monople.

Of course self-duality (resp. anti-self-duality) of the differential form
$F_{\omega}$ does not necessarily require $f_{j},f_{\bar{j}},\text{ }j=1,2$
being holomorphic (resp. conjugate holomorphic). For instance,
$f_{1}=z_{1}+\bar{z}_{1},f_{2}=z_{2}+\bar{z}_{2},f_{\bar{1}}=\bar{z}_{1}-z_{1},f_{\bar{2}}=\bar{z}_{2}-z_{2}$
satisfy (4.9), while
$f_{1}=z_{1}-\bar{z}_{2},f_{2}=z_{2}-\bar{z}_{1},f_{\bar{1}}=\bar{z}_{1}-z_{1},f_{\bar{2}}=z_{2}-\bar{z}_{2}$
satisfy (4.10). The next example exihibits a $1$-dimensional electromagnetic
dynamics.

######  Example 4.4.

Let
$\tau\left(z\right)=f_{1}\left(dz_{1}+d\bar{z}_{1}\right)+f_{2}\left(dz_{2}+d\bar{z}_{2}\right)$,
where $f_{1}$ and $f_{2}$ are holomorphic in $z_{1}$ and $z_{2}$. We claim
that ${F}:=d\tau$ is self-dual if only if ${F}=m\left(dz_{1}\wedge
d\bar{z}_{1}+dz_{2}\wedge d\bar{z}_{2}\right)$ for some constant $m$. First,
it is clear that if ${F}=m\left(dz_{1}\wedge d\bar{z}_{1}+dz_{2}\wedge
d\bar{z}_{2}\right)$ then it is self-dual. Conversely, since $f_{1},f_{2}$ are
holomorphic, using the definition of ${F}$, after some simplification we have

$\displaystyle{F}$ $\displaystyle=\partial_{1}f_{1}dz_{1}\wedge
d\bar{z}_{1}+\left(\partial_{1}f_{2}-\partial_{2}f_{1}\right)dz_{1}\wedge
dz_{2}+\partial_{1}f_{2}dz_{1}\wedge d\bar{z}_{2}$ (4.12)
$\displaystyle+\partial_{2}f_{1}dz_{2}\wedge
d\bar{z}_{1}+\partial_{2}f_{2}dz_{2}\wedge d\bar{z}_{2}.$

After setting ${F}=\star{F}$ and comparing coefficients, it follows that

$\partial_{1}f_{1}=\partial_{2}f_{2},\text{
}\partial_{1}f_{2}=0=\partial_{2}f_{1}.$ (4.13)

The second relation in (4.13) implies $f_{1}$ is independent of $z_{2}$, and
likewise $f_{2}$ is independent of $z_{1}$. From the first relation, since
$\partial_{1}f_{1}=\partial_{2}f_{2}$, $f_{1}$ and $f_{2}$ must be constant.
Therefore, we can write $f_{1}=mz_{1}+c_{1},f_{2}=mz_{2}+c_{2}$ for constants
$m,c_{1},c_{2}\in\mathbb{C}$. Substituting the conditions in (4.13) into
(4.12), we have

$\displaystyle{F}$ $\displaystyle=m\left(dz_{1}\wedge
d\bar{z}_{1}+dz_{2}\wedge d\bar{z}_{2}\right),$
$\displaystyle=-2im\left(dx_{0}\wedge dx_{1}+dx_{2}\wedge dx_{3}\right).$
(4.14)

One can read off the electric and magnetic components in (4.14) by comparing
with (3.10):

$\displaystyle E_{1}$ $\displaystyle=B_{1}=-2im,$ $\displaystyle E_{2}$
$\displaystyle=B_{2}=E_{3}=B_{3}=0,$

which shows that the electromagnetic dynamics in this case is $1$-dimensional.

We now continue with the discussion on the general case that
$f_{i},f_{\bar{i}},i=1,2$ are smooth functions on $\mathbb{C}^{2}$. After
applying the exterior derivative to (4.8) and some simplification we obtain

$\displaystyle d\star F_{\omega}$
$\displaystyle=\left(-\partial_{2}\bar{\partial}_{1}f_{1}+\left(2\partial_{1}\bar{\partial}_{1}+\partial_{2}\bar{\partial}_{2}\right)f_{2}-\partial_{1}\partial_{2}f_{\bar{1}}-\partial_{2}^{2}f_{\bar{2}}\right)dz_{1}\wedge
dz_{2}\wedge d\bar{z}_{1}$
$\displaystyle+\left(-\left(\partial_{1}\bar{\partial}_{1}+2\partial_{2}\bar{\partial}_{2}\right)f_{1}+\partial_{1}\bar{\partial}_{2}f_{2}+\partial_{1}^{2}f_{\bar{1}}+\partial_{1}\partial_{2}f_{\bar{2}}\right)dz_{1}\wedge
dz_{2}\wedge d\bar{z}_{2}$ (4.15)
$\displaystyle+\left(-\bar{\partial}_{1}\bar{\partial}_{2}f_{1}-\bar{\partial}_{2}^{2}f_{2}-\bar{\partial}_{2}\partial_{1}f_{\bar{1}}+\left(2\bar{\partial}_{1}\partial_{1}+\bar{\partial}_{2}\partial_{2}\right)f_{\bar{2}}\right)dz_{1}\wedge
d\bar{z}_{1}\wedge d\bar{z}_{2}$
$\displaystyle+\left(\bar{\partial}_{1}^{2}f_{1}+\bar{\partial}_{1}\bar{\partial}_{2}f_{2}-\left(\bar{\partial}_{1}\partial_{1}+2\bar{\partial}_{2}\partial_{2}\right)f_{\bar{1}}+\bar{\partial}_{1}\partial_{2}f_{\bar{2}}\right)dz_{2}\wedge
d\bar{z}_{1}\wedge d\bar{z}_{2}.$

Applying the Hodge star operator to (4.15) and rearranging terms, we have

$\displaystyle\star\text{ }d\star F_{\omega}$
$\displaystyle=2\left(\left(\partial_{1}\bar{\partial}_{1}+2\partial_{2}\bar{\partial}_{2}\right)f_{1}-\partial_{1}\bar{\partial}_{2}f_{2}-\partial_{1}^{2}f_{\bar{1}}-\partial_{1}\partial_{2}f_{\bar{2}}\right)dz_{1}$
$\displaystyle+2\left(-\bar{\partial}_{1}^{2}f_{1}-\bar{\partial}_{1}\bar{\partial}_{2}f_{2}+\left(\bar{\partial}_{1}\partial_{1}+2\bar{\partial}_{2}\partial_{2}\right)f_{\bar{1}}-\bar{\partial}_{1}\partial_{2}f_{\bar{2}}\right)d\bar{z}_{1}$
(4.16)
$\displaystyle+2\left(-\partial_{2}\bar{\partial}_{1}f_{1}+\left(2\partial_{1}\bar{\partial}_{1}+\partial_{2}\bar{\partial}_{2}\right)f_{2}-\partial_{1}\partial_{2}f_{\bar{1}}-\partial_{2}^{2}f_{\bar{2}}\right)dz_{2}$
$\displaystyle+2\left(-\bar{\partial}_{1}\bar{\partial}_{2}f_{1}-\bar{\partial}_{2}^{2}f_{2}-\bar{\partial}_{2}\partial_{1}f_{\bar{1}}+\left(2\bar{\partial}_{1}\partial_{1}+\bar{\partial}_{2}\partial_{2}\right)f_{\bar{2}}\right)d\bar{z}_{2}.$

The RHS of (4.16) can be viewed as a complex current form that we may denote
by $J$. For the sake of simplicity, we can rewrite (4.16) as

$\star\text{ }d\star
F_{\omega}=J:=P_{1}dz_{1}+P_{\bar{1}}d\bar{z}_{1}+P_{2}dz_{2}+P_{\bar{2}}d\bar{z}_{2},$
(4.17)

where the coefficients $P_{j},P_{\bar{j}},j=1,2$ can be easily read off from
(4.16). Likewise, we can write the RHS of (4.16) in terms of real 1-forms in
view of (3.9):

$J=\left(P_{1}+P_{\bar{1}}\right)dx_{0}+i\left(P_{1}-P_{\bar{1}}\right)dx_{1}+\left(P_{2}+P_{\bar{2}}\right)dx_{2}+i\left(P_{2}-P_{\bar{2}}\right)dx_{3}.$

Again, if $f_{\bar{i}}=\bar{f}_{i},i=1,2$ then $J$ is real. Here
$P_{1}+P_{\bar{1}}$ can be considered as a scalar electric charge density
$\rho$, while the last three coefficients for the above current 1-form
correspond to the last three components of the electric current density vector
$\mathbf{J}=\left(\rho,J_{1},J_{2},J_{3}\right)$ in Section 3.1. So to
summarize, with smooth complex-valued functions
$f_{1},f_{2},f_{\bar{1}},f_{\bar{2}}$ in two variables $z_{1},z_{2}$, we
determined from the complex differential form $\omega$ in (4.1), a solution
$F_{\omega}$ (in (4.2)) to a complex analogue of Maxwell’s equations
$dF_{\omega}=0$ and $\star d\star F_{\omega}=J$.

Let
$\nabla^{2}:=4\left(\partial_{1}\bar{\partial}_{1}+\partial_{2}\bar{\partial}_{2}\right)$
be the Laplacian on $\mathbb{C}^{2}$ in the Euclidean metric. Then a complex
function $f$ is said to be harmonic if $\nabla^{2}f=0$ on $\mathbb{C}^{2}$.
The following is the main result of this subsection.

######  Theorem 4.5.

Let $f_{j},f_{\bar{j}},\text{ }j=1,2$ be harmonic functions. Then the complex
differential form $\omega$ is a solution to the source-free Maxwell’s
equations in the Euclidean metric if and only if
$\bar{\partial}_{1}f_{1}+\bar{\partial}_{2}f_{2}+\partial_{1}f_{\bar{1}}+\partial_{2}f_{\bar{2}}$
is constant.

###### Proof.

First, we can rewrite (4.15) as

$\displaystyle d\star F_{\omega}$
$\displaystyle=\left(\nabla^{2}f_{2}-\partial_{2}\bar{\partial}_{1}f_{1}-\partial_{2}^{2}f_{\bar{2}}-\partial_{1}\partial_{2}f_{\bar{1}}+\partial_{1}\bar{\partial}_{1}f_{2}\right)dz_{1}\wedge
dz_{2}\wedge d\bar{z}_{1}$
$\displaystyle+\left(-\nabla^{2}f_{1}+\partial_{1}\bar{\partial}_{2}f_{2}+\partial_{1}^{2}f_{\bar{1}}+\partial_{1}\partial_{2}f_{\bar{2}}-\partial_{2}\bar{\partial}_{2}f_{1}\right)dz_{1}\wedge
dz_{2}\wedge d\bar{z}_{2}$ (4.18)
$\displaystyle+\left(\nabla^{2}f_{\bar{2}}-\partial_{1}\bar{\partial}_{2}f_{\bar{1}}-\bar{\partial}_{2}^{2}f_{2}+\partial_{1}\bar{\partial}_{1}f_{\bar{2}}-\bar{\partial}_{1}\bar{\partial}_{2}f_{1}\right)dz_{1}\wedge
d\bar{z}_{1}\wedge d\bar{z}_{2}$
$\displaystyle+\left(-\nabla^{2}f_{\bar{1}}+\partial_{2}\bar{\partial}_{1}f_{\bar{2}}+\bar{\partial}_{1}^{2}f_{1}-\partial_{2}\bar{\partial_{2}}f_{\bar{1}}+\bar{\partial}_{1}\bar{\partial}_{2}f_{2}\right)dz_{2}\wedge
d\bar{z}_{1}\wedge d\bar{z}_{2}.$

Since $f_{j},f_{\bar{j}},\text{ }j=1,2$ are harmonic, we have
$\nabla^{2}f_{j}=0$ and $\nabla^{2}f_{\bar{j}}=0$, which implies

$\displaystyle d\star F_{\omega}$
$\displaystyle=\left(-\partial_{2}\bar{\partial}_{1}f_{1}-\partial_{2}^{2}f_{\bar{2}}-\partial_{1}\partial_{2}f_{\bar{1}}-\partial_{2}\bar{\partial}_{2}f_{2}\right)dz_{1}\wedge
dz_{2}\wedge d\bar{z}_{1}$
$\displaystyle+\left(\partial_{1}\bar{\partial}_{2}f_{2}+\partial_{1}^{2}f_{\bar{1}}+\partial_{1}\partial_{2}f_{\bar{2}}+\partial_{1}\bar{\partial}_{1}f_{1}\right)dz_{1}\wedge
dz_{2}\wedge d\bar{z}_{2}$
$\displaystyle+\left(-\partial_{1}\bar{\partial}_{2}f_{\bar{1}}-\bar{\partial}_{2}^{2}f_{2}-\partial_{2}\bar{\partial}_{2}f_{\bar{2}}-\bar{\partial}_{1}\bar{\partial_{2}}f_{1}\right)dz_{1}\wedge
d\bar{z}_{1}\wedge d\bar{z}_{2}$
$\displaystyle+\left(\partial_{2}\bar{\partial}_{1}f_{\bar{2}}+\bar{\partial}_{1}^{2}f_{1}+\partial_{1}\bar{\partial}_{1}f_{\bar{1}}+\bar{\partial}_{1}\bar{\partial}_{2}f_{2}\right)dz_{2}\wedge
d\bar{z}_{1}\wedge d\bar{z}_{2}.$

For $F_{\omega}$ to satisfy the source-free Maxwell’s equations, we require
$d\star F_{\omega}=0$, which we may now write in matrix form as

$\text{diag}\\{-\partial_{2},\partial_{1},-\bar{\partial}_{2},\bar{\partial}_{1}\\}\begin{pmatrix}\bar{\partial_{1}}&\bar{\partial_{2}}&\partial_{1}&\partial_{2}\\\
\bar{\partial_{1}}&\bar{\partial_{2}}&\partial_{1}&\partial_{2}\\\
\bar{\partial_{1}}&\bar{\partial_{2}}&\partial_{1}&\partial_{2}\\\
\bar{\partial_{1}}&\bar{\partial_{2}}&\partial_{1}&\partial_{2}\end{pmatrix}\begin{pmatrix}f_{1}\\\
f_{2}\\\ f_{\bar{1}}\\\ f_{\bar{2}}\end{pmatrix}=\mathbf{0},$

where “diag” stands for a diagonal matrix and $\mathbf{0}$ denotes the column
4-vector of zeroes. Clearly, the above is true if and only if
$\bar{\partial}_{1}f_{1}+\bar{\partial}_{2}f_{2}+\partial_{1}f_{\bar{1}}+\partial_{2}f_{\bar{2}}$
is a constant.∎

######  Example 4.6.

With $z_{1}=x_{0}+ix_{1},z_{2}=x_{2}+ix_{3}$, consider the form

$\displaystyle\omega\left(z\right)$
$\displaystyle=i\left(x_{0}dx_{1}-x_{1}dx_{0}+x_{2}dx_{3}-x_{3}dx_{2}\right)$
$\displaystyle=\frac{1}{2}\left(\eta\left(z\right)-\overline{\eta\left(z\right)}\right),$
(4.19)

where $\eta\left(z\right)=\bar{z}_{1}dz_{1}+\bar{z}_{2}dz_{2}$. In this case,
we have
$f_{1}=\frac{1}{2}\bar{z}_{1},f_{2}=\frac{1}{2}\bar{z}_{2},f_{\bar{1}}=-\frac{1}{2}z_{1},f_{\bar{2}}=-\frac{1}{2}z_{2}$,
where $f_{j},f_{\bar{j}},\text{ }j=1,2$ are not holomorphic (resp. conjugate
holomorphic) but are all clearly harmonic. Moreover, we have the associated
curvature form

$F_{\omega}=d\omega=-\left(dz_{1}\wedge d\bar{z}_{1}+dz_{2}\wedge
d\bar{z}_{2}\right),$

which is self-dual in the Euclidean metric. Now observe that

$\displaystyle\partial_{1}f_{\bar{1}}-\bar{\partial}_{1}f_{1}$
$\displaystyle=-1=\partial_{2}f_{\bar{2}}-\bar{\partial}_{2}f_{2},$
$\displaystyle\partial_{1}f_{\bar{2}}$
$\displaystyle=0=\bar{\partial}_{1}f_{2},$
$\displaystyle\partial_{2}f_{\bar{1}}$
$\displaystyle=0=\bar{\partial}_{2}f_{1}.$

The above conditions clearly satisfy the self-duality of $F_{\omega}$.
Moreover,
$\bar{\partial}_{1}f_{1}+\bar{\partial}_{2}f_{2}+\partial_{1}f_{\bar{1}}+\partial_{2}f_{\bar{2}}=0$.

The above example is related to the Dirac monopole ([17]), which is a
hypothetical magnetic charge. The original idea was proposed in a 1931 paper
by Paul Dirac ([3]). Evidently, the above example shows that the existence of
Dirac monopoles does not conflict with Maxwell’s equations in vacuum (away
from the magnetic monople). See [3, 15] and the references therein for more
background on this rather intriguing subject. A notable fact here is that
${\bf E}={\bf B}$. It is easy to compute that ${\bf E}={\bf B}=(-2i,0,0)$, and
therefore

$\langle{\bf E},{\bf B}\rangle=\frac{1}{2}(|{\bf E}|^{2}+|{\bf B}|^{2})=4.$

Theorem 4.5 leads to easy constructions of non-self-dual solutions to
Equations (3.11) and (3.12) in vaccum.

######  Example 4.7.

In fact, a simple working example that satisfies the conditions in Theorem
4.5, but fails the conditions for $F_{\omega}$ to be self-dual nor anti-self-
dual, is
$f_{1}=2\bar{z}_{1}-z_{2},f_{2}=z_{1}+2\bar{z}_{2},f_{\bar{1}}=z_{1}+\bar{z}_{1},f_{\bar{2}}=z_{2}+\bar{z}_{2}$.

###  4.2. Minkowski Metric Case

We believe that much of the work in Section 4.1 can be done in a parallel
manner with respect to other bilinear forms $\langle\cdot,\cdot\rangle_{g}$ on
${\mathbb{R}}^{4}$, where $g$ is a nondegenerate constant $4\times 4$ self-
adjoint matrix. But since the Minkowski metric on ${\mathbb{R}}^{1,3}$ is more
conforming with our reality, and it is indeed where the Maxwell’s equations
were initially studied, we shall work it out in details in this subsection.

Recall that the d’Alembertian in the Minkowski metric is given by

$\Box=\frac{\partial^{2}}{\partial x_{0}^{2}}-\frac{\partial^{2}}{\partial
x_{1}^{2}}-\frac{\partial^{2}}{\partial
x_{2}^{2}}-\frac{\partial^{2}}{\partial
x_{3}^{2}}=\frac{\partial^{2}}{\partial x_{0}^{2}}-\nabla^{2},$ (4.20)

where again $\nabla^{2}$ is the Laplacian in $\mathbb{R}^{3}$. Using the
identities

$\displaystyle\partial_{1}=\frac{1}{2}\left(\frac{\partial}{\partial
x_{0}}-i\frac{\partial}{\partial x_{1}}\right),\hskip
28.45274pt\bar{\partial}_{1}=\frac{1}{2}\left(\frac{\partial}{\partial
x_{0}}+i\frac{\partial}{\partial x_{1}}\right)$
$\displaystyle\partial_{2}=\frac{1}{2}\left(\frac{\partial}{\partial
x_{2}}-i\frac{\partial}{\partial x_{3}}\right),\hskip
28.45274pt\bar{\partial}_{2}=\frac{1}{2}\left(\frac{\partial}{\partial
x_{2}}+i\frac{\partial}{\partial x_{3}}\right),$ (4.21)

we have the following.

######  Lemma 4.8.

In terms of complex variables, the d’Alembertian in the Minkowski metric is
given by

$\Box=2\left(\partial_{1}^{2}+\bar{\partial}_{1}^{2}-2\partial_{2}\bar{\partial}_{2}\right).$
(4.22)

###### Proof.

Using the first two identities in (4.21), note that

$\displaystyle\partial_{1}+\bar{\partial}_{1}$
$\displaystyle=\frac{\partial}{\partial x_{0}},\hskip
28.45274pti\left(\partial_{1}-\bar{\partial}_{1}\right)=\frac{\partial}{\partial
x_{1}}.$

Then the first two terms of the d’Alembertian in (4.20) are

$\displaystyle\left(\frac{\partial}{\partial x_{0}}+\frac{\partial}{\partial
x_{1}}\right)\left(\frac{\partial}{\partial x_{0}}-\frac{\partial}{\partial
x_{1}}\right)$
$\displaystyle=\left(\partial_{1}+\bar{\partial}_{1}+i\left(\partial_{1}-\bar{\partial}_{1}\right)\right)\left(\partial_{1}+\bar{\partial}_{1}-i\left(\partial_{1}-\bar{\partial}_{1}\right)\right)$
$\displaystyle=\left(\partial_{1}+\bar{\partial}_{1}\right)^{2}+\left(\partial_{1}-\bar{\partial}_{1}\right)^{2}$
$\displaystyle=2\left(\partial_{1}^{2}+\bar{\partial}_{1}^{2}\right).$

Similarly, using the last two identities in (4.21), it follows that the last
two terms of the d’Alembertian in (4.20) are
$-4\partial_{2}\bar{\partial}_{2}$. Therefore,
$\Box=2\left(\partial_{1}^{2}+\bar{\partial}_{1}^{2}-2\partial_{2}\bar{\partial}_{2}\right)$.
∎

######  Definition 4.9.

A function $f$ is said to be $M$-harmonic if $\Box f=0$.

Here the “$M$” refers to the Minkowski metric. It is well-known that
$M$-harmonic functions $\psi$ describe waves propagating in
${\mathbb{R}}^{1,3}$. A smooth $1$-form as defined in (4.1) is said to be
wavelike if the functions $f_{j},f_{\bar{j}},j=1,2$ are all $M$-harmonic.
Likewise, the curvature field $F_{\omega}$ is said to be wavelike if its
coefficient functions $F_{12},F_{\bar{1}\bar{2}}$, and $F_{j\bar{k}},j,k=1,2$
in (4.4) are all $M$-harmonic. It is easy to check that if $\omega$ is
wavelike then $F_{\omega}$ is also wavelike, for instance,

$\Box
F_{12}=\Box\left(\partial_{1}f_{2}-\partial_{2}f_{1}\right)=\partial_{1}\Box
f_{2}-\partial_{2}\Box f_{1}=0,$

and other coefficients are checked similarly. In particular, this implies that
$\Box{E_{j}}=\Box{B_{j}}=0,1\leq j\leq 3$, i.e., ${\bf E}$ and ${\bf B}$ are
waves in the space $\mathbb{C}^{3}$. However, as we will see a bit later,
there exists non-wavelike $\omega$ for which $F_{\omega}$ is wavelike.

Now let $f_{j},f_{\bar{j}},\text{ }j=1,2$ be complex smooth functions as
before. Then $F_{\omega}$ is still the same as in (4.2). However in the
Minkowski metric, by Example 2.9 we have

$\displaystyle\star F_{\omega}$
$\displaystyle=\left(\bar{\partial}_{1}f_{\bar{2}}-\bar{\partial}_{2}f_{\bar{1}}\right)dz_{1}\wedge
d\bar{z}_{2}+\left(\partial_{2}f_{1}-\partial_{1}f_{2}\right)dz_{2}\wedge
d\bar{z}_{1}$
$\displaystyle+\left(\partial_{2}f_{\bar{1}}-\bar{\partial}_{1}f_{2}\right)dz_{1}\wedge
dz_{2}+\left(\bar{\partial}_{2}f_{1}-\partial_{1}f_{\bar{2}}\right)d\bar{z}_{1}\wedge
d\bar{z}_{2}$ (4.23)
$\displaystyle+\left(\partial_{2}f_{\bar{2}}-\bar{\partial}_{2}f_{2}\right)dz_{1}\wedge
d\bar{z}_{1}+\left(\bar{\partial}_{1}f_{1}-\partial_{1}f_{\bar{1}}\right)dz_{2}\wedge
d\bar{z}_{2}.$

In the Minkowski metric on $\mathbb{C}^{2}$, the curvature form $F_{\omega}$
is self-dual (resp. anti-self-dual) provided $\star F_{\omega}=\pm
iF_{\omega}$ ([1, 5]). So self-duality of $F_{\omega}$ requires

$\displaystyle\partial_{2}f_{\bar{1}}-\bar{\partial}_{1}f_{2}$
$\displaystyle=i\left(\partial_{1}f_{2}-\partial_{2}f_{1}\right),$
$\displaystyle\bar{\partial}_{1}f_{\bar{2}}-\bar{\partial}_{2}f_{\bar{1}}$
$\displaystyle=i\left(\partial_{1}f_{\bar{2}}-\bar{\partial}_{2}f_{1}\right),$
(4.24) $\displaystyle\partial_{2}f_{\bar{2}}-\bar{\partial}_{2}f_{2}$
$\displaystyle=i\left(\partial_{1}f_{\bar{1}}-\bar{\partial}_{1}f_{1}\right).$

On the other hand, anti-self-duality of $F_{\omega}$ requires

$\displaystyle\partial_{2}f_{\bar{1}}-\bar{\partial}_{1}f_{2}$
$\displaystyle=-i\left(\partial_{1}f_{2}-\partial_{2}f_{1}\right),$
$\displaystyle\bar{\partial}_{1}f_{\bar{2}}-\bar{\partial}_{2}f_{\bar{1}}$
$\displaystyle=-i\left(\partial_{1}f_{\bar{2}}-\bar{\partial}_{2}f_{1}\right),$
(4.25) $\displaystyle\partial_{2}f_{\bar{2}}-\bar{\partial}_{2}f_{2}$
$\displaystyle=-i\left(\partial_{1}f_{\bar{1}}-\bar{\partial}_{1}f_{1}\right).$

The following fact is immediate.

######  Corollary 4.10.

If $\omega$ is a self-dual or anti-self-dual solution to the Maxwell’s
equations in vaccum with respect to the Minkowski metric, then $\langle{\bf
E},{\bf B}\rangle$ is either $0$ or purely imaginary. In particular, if
$\omega$ is a real self-dual or anti-self-dual solution then $\langle{\bf
E},{\bf B}\rangle=0$.

###### Proof.

If $\omega$ is self-dual, then (4.25) indicates that

$F_{2\bar{1}}=iF_{12},\ \ F_{\bar{1}\bar{2}}=iF_{1\bar{2}},\ \
F_{2\bar{2}}=iF_{1\bar{1}}.$

Applying these relations to (4.5), one has

$\displaystyle\langle{\bf E},{\bf B}\rangle=$
$\displaystyle-4i|F_{1\bar{1}}|^{2}+2(1-i)^{2}|F_{12}|^{2}+2(1+i)^{2}|F_{1\bar{2}}|^{2}$
$\displaystyle=-4i\left(|F_{1\bar{1}}|^{2}+|F_{12}|^{2}-|F_{1\bar{2}}|^{2}\right).$

In the case $\omega$ is anti-self-dual, parallel computations yield

$\langle{\bf E},{\bf
B}\rangle=4i\left(|F_{1\bar{1}}|^{2}+|F_{12}|^{2}-|F_{1\bar{2}}|^{2}\right).$

If $\omega$ is real then $\langle{\bf E},{\bf B}\rangle$ is real and therefore
it must be equal to $0$.∎

To proceed, as in the previous subsection we assume $F_{\omega}$ is neither
self-dual nor anti-self-dual in the Minkowski metric. So for the source-free
Maxwell’s equations to be satisfied, we require $d\star F_{\omega}=0$. In this
case, we have

$\displaystyle d\star F_{\omega}$
$\displaystyle=\left(\partial_{1}\partial_{2}f_{1}-\left(\partial_{1}^{2}+\bar{\partial}_{1}^{2}-\partial_{2}\bar{\partial}_{2}\right)f_{2}+\partial_{2}\bar{\partial}_{1}f_{\bar{1}}-\partial_{2}^{2}f_{\bar{2}}\right)dz_{1}\wedge
dz_{2}\wedge d\bar{z}_{1}$
$\displaystyle+\left(\partial_{1}\bar{\partial}_{1}f_{1}-\bar{\partial}_{1}\bar{\partial}_{2}f_{2}-\left(\partial_{1}^{2}-2\partial_{2}\bar{\partial}_{2}\right)f_{\bar{1}}-\partial_{2}\bar{\partial}_{1}f_{\bar{2}}\right)dz_{1}\wedge
dz_{2}\wedge d\bar{z}_{2}$ (4.26)
$\displaystyle+\left(\partial_{1}\bar{\partial}_{2}f_{1}-\bar{\partial}_{2}^{2}f_{2}+\bar{\partial}_{1}\bar{\partial}_{2}f_{\bar{1}}-\left(\partial_{1}^{2}+\bar{\partial}_{1}^{2}-\partial_{2}\bar{\partial}_{2}\right)f_{\bar{2}}\right)dz_{1}\wedge
d\bar{z}_{1}\wedge d\bar{z}_{2}$
$\displaystyle+\left(-\left(\bar{\partial}_{1}^{2}-2\partial_{2}\bar{\partial}_{2}\right)f_{1}-\partial_{1}\bar{\partial}_{2}f_{2}+\partial_{1}\bar{\partial}_{1}f_{\bar{1}}-\partial_{1}\partial_{2}f_{\bar{2}}\right)dz_{2}\wedge
d\bar{z}_{1}\wedge d\bar{z}_{2}.$

Observe that for $f_{j},f_{\bar{j}},\text{ }j=1,2$ holomorphic and
respectively conjugate holomorphic, equations (4.24) and (4.25) imply that all
the coefficients of the $2$-forms in $F_{\omega}$ are $0$. Hence there is no
nontrivial self-dual or anti-self-dual solution to the source-free Maxwell
equations in this case. However, it follows from the above computation that

$d\star
F_{\omega}=\left(\partial_{1}\partial_{2}f_{1}-\partial_{1}^{2}f_{2}\right)dz_{1}\wedge
dz_{2}\wedge
d\bar{z}_{1}+\left(\bar{\partial}_{1}\bar{\partial}_{2}f_{\bar{1}}-\bar{\partial}_{1}^{2}f_{\bar{2}}\right)dz_{1}\wedge
d\bar{z}_{1}\wedge d\bar{z}_{2}.$

Hence $d\star F_{\omega}=0$ if and only if

$\partial_{1}\partial_{2}f_{1}-\partial_{1}^{2}f_{2}=\bar{\partial}_{1}\bar{\partial}_{2}f_{\bar{1}}-\bar{\partial}_{1}^{2}f_{\bar{2}}=0.$
(4.27)

Further, if $f_{\bar{j}}=\overline{f_{j}},\text{ }j=1,2$ then the above two
equations are the same. One thus obtains the following fact.

######  Proposition 4.11.

Let $f_{1}$ and $f_{2}$ be holomorphic functions and
$f_{\bar{j}}=\overline{f_{j}},\text{ }j=1,2$. Then $\omega$ is a solution to
the Maxwell’s equations in vacuum with respect to the Minkowski metric if and
only if $\partial_{2}f_{1}-\partial_{1}f_{2}$ is independent of the variable
$z_{1}$.

Similar to Proposition 4.2 and Corollary 4.3, in this case we have

$\displaystyle F_{\omega}$
$\displaystyle=\left(\partial_{1}f_{2}-\partial_{2}f_{1}\right)dz_{1}\wedge
dz_{2}+\overline{\left(\partial_{1}f_{2}-\partial_{2}f_{1}\right)}d\bar{z}_{1}\wedge
d\bar{z}_{2},$
$\displaystyle=2\Re\left((\partial_{1}f_{2}-\partial_{2}f_{1})dz_{1}\wedge
dz_{2}\right),$

and consequently,

$\displaystyle E_{1}$ $\displaystyle=B_{1}=0,$ $\displaystyle E_{2}$
$\displaystyle=B_{2}=2\Re(\partial_{1}f_{2}-\partial_{2}f_{1}),$
$\displaystyle E_{3}$
$\displaystyle=B_{3}=-2\Im(\partial_{2}f_{1}-\partial_{1}f_{2}),$

where $\Re(a)$ and $\Im(a)$ stand for the real, and respectively, imaginary
part of a complex number $a$. Observe that ${\bf E}={\bf B}$ in this case,
which resembles the Dirac monople example we examined earlier.

######  Example 4.12.

There are plenty of holomorphic functions $f_{1}$ and $f_{2}$ that satisfy the
condition in the above proposition. For instance, let

$f_{1}=z_{1}^{2}h(z_{2})+g(z_{2}),\
f_{2}=\frac{z_{1}^{3}}{3}\partial_{2}h(z_{2}),$

where $g$ and $h$ are arbitrary one-variable entire functions. Then
$\partial_{2}f_{1}-\partial_{1}f_{2}=\partial_{2}g(z_{2})$, which is
independent of $z_{1}$. Further, since in this case

$\Box f_{1}=2h(z_{2}),\ \ \ \Box f_{2}=2z_{1}\partial_{2}h(z_{2}),$

which can be nonzero, the $1$-form $\omega$ may not be wavelike. Further, it
is easy to see that $\partial_{2}g(z_{2})$ is M-harmonic and hence
$F_{\omega}$ is wavelike.

We state this observation as follows.

######  Corollary 4.13.

There are real analytic non-wavelike solutions to the Maxwell’s equations in
vacuum.

######  Remark 4.14.

Corollary 4.3 and the above observations also indicate that, under both the
Euclidean metric and the Minkowski metric, the Maxwell’s equations in vacuum
have solutions in which the electric field and the magnetic field are
mathematically indistinguishable. However, it is not clear if such solutions
exist in nature.

Now coming back to our familiar wavelike solutions we have the following fact.
Its proof is similar to that of Theorem 4.5.

######  Theorem 4.15.

Assume $\omega$ as in (4.1) is wavelike. Then it is a solution to the
Maxwell’s equations in vacuum under the Minkowski metric if and only if
$\partial_{1}f_{1}-\bar{\partial}_{2}f_{2}+\bar{\partial}_{1}f_{\bar{1}}-\partial_{2}f_{\bar{2}}$
is constant.

###### Proof.

Let $\mathbf{0}$ denote the column 4-vector of zeroes. Since $\omega$ is
wavelike, we have $\Box f_{j}=\Box f_{\bar{j}}=0,j=1,2$, which means

$\left(\partial_{1}^{2}+\bar{\partial}_{1}^{2}-2\partial_{2}\bar{\partial}_{2}\right)f_{j}=0,\left(\partial_{1}^{2}+\bar{\partial}_{1}^{2}-2\partial_{2}\bar{\partial}_{2}\right)f_{\bar{j}}=0.$
(4.28)

Plugging (4.28) into (4.26) and rearranging terms, we have

$\displaystyle d\star F_{\omega}$
$\displaystyle=\left(\partial_{1}\partial_{2}f_{1}-\partial_{2}\bar{\partial}_{2}f_{2}+\partial_{2}\bar{\partial}_{1}f_{\bar{1}}-\partial_{2}^{2}f_{\bar{2}}\right)dz_{1}\wedge
dz_{2}\wedge d\bar{z}_{1}$
$\displaystyle+\left(\partial_{1}\bar{\partial}_{1}f_{1}-\bar{\partial}_{1}\bar{\partial}_{2}f_{2}+\bar{\partial}_{1}^{2}f_{\bar{1}}-\partial_{2}\bar{\partial}_{1}f_{\bar{2}}\right)dz_{1}\wedge
dz_{2}\wedge d\bar{z}_{2}$
$\displaystyle+\left(\partial_{1}\bar{\partial}_{2}f_{1}-\bar{\partial}_{2}^{2}f_{2}+\bar{\partial}_{1}\bar{\partial}_{2}f_{\bar{1}}-\partial_{2}\bar{\partial}_{2}f_{\bar{2}}\right)dz_{1}\wedge
d\bar{z}_{1}\wedge d\bar{z}_{2}$
$\displaystyle+\left(\partial_{1}^{2}f_{1}-\partial_{1}\bar{\partial}_{2}f_{2}+\partial_{1}\bar{\partial}_{1}f_{\bar{1}}-\partial_{1}\partial_{2}f_{\bar{2}}\right)dz_{2}\wedge
d\bar{z}_{1}\wedge d\bar{z}_{2}.$

To satisfy the source-free Maxwell’s equations, we need $d\star F_{\omega}=0$,
which in matrix form is

$\text{diag}\\{\partial_{2},\bar{\partial}_{1},\bar{\partial}_{2},\partial_{1}\\}\begin{pmatrix}\partial_{1}&-\bar{\partial}_{2}&\bar{\partial}_{1}&-\partial_{2}\\\
\partial_{1}&-\bar{\partial}_{2}&\bar{\partial}_{1}&-\partial_{2}\\\
\partial_{1}&-\bar{\partial}_{2}&\bar{\partial}_{1}&-\partial_{2}\\\
\partial_{1}&-\bar{\partial}_{2}&\bar{\partial}_{1}&-\partial_{2}\end{pmatrix}\begin{pmatrix}f_{1}\\\
f_{2}\\\ f_{\bar{1}}\\\ f_{\bar{2}}\end{pmatrix}=\mathbf{0}.$

Clearly, this is true if and only if
$\partial_{1}f_{1}-\bar{\partial}_{2}f_{2}+\bar{\partial}_{1}f_{\bar{1}}-\partial_{2}f_{\bar{2}}$
is constant and this completes the proof.∎

##  5\. On the Lorenz gauge

It was indicated in Section 3.2 that given a smooth $4$-vector
$(\phi,A_{1},A_{2},A_{3})$ one can associate with it the magnetic potential
$1$-form $\omega=\phi dx_{0}-A_{1}dx_{1}-A_{2}dx_{2}-A_{3}dx_{3}.$ The Lorenz
gauge condition (3.5) stipulates the normalization (3.8) regarding the sum of
partial derivatives, namely,

$\frac{\partial\phi}{\partial x_{0}}+\frac{\partial A_{1}}{\partial
x_{1}}+\frac{\partial A_{2}}{\partial x_{2}}+\frac{\partial A_{3}}{\partial
x_{3}}=0.$

Theorems 4.5 and 4.15 indeed give a mathematical explanation as to why the
Lorenz gauge matters. Here we give a unified treatment.

######  Corollary 5.1.

Let $\omega$ be a smooth $1$-form as defined in (4.1). Then the sum of partial
derivatives appearing in Theorem 4.5 is $-\frac{1}{2}d^{*}\omega$, and that in
Theorem 4.15 is $\frac{1}{2}d^{*}\omega$.

###### Proof.

First, recall that in the Euclidean metric over $\mathbb{C}^{2}$ we have that
$d^{*}\omega=-\star d\star\omega$. Then using the calculations in Example 2.8
one easily verifies that

$\star\omega=\frac{1}{2}\left(f_{1}dz_{1}\wedge dz_{2}\wedge
d\bar{z}_{2}+f_{\bar{1}}dz_{2}\wedge d\bar{z}_{1}\wedge
d\bar{z}_{2}-f_{2}dz_{1}\wedge dz_{2}\wedge
d\bar{z}_{1}-f_{\bar{2}}dz_{1}\wedge d\bar{z}_{1}\wedge d\bar{z}_{2}\right).$

It follows that

$\bar{\partial}_{1}f_{1}+\bar{\partial}_{2}f_{2}+\partial_{1}f_{\bar{1}}+\partial_{2}f_{\bar{2}}=-\frac{1}{2}d^{*}\omega.$

The sums of partial derivatives appearing in (3.8) is the real variable
version of that in Theorem 4.15. In the Minkowski metric over
$\mathbb{C}^{2}$, we have

$d^{*}\omega=\star d\star\omega.$ (5.1)

Under the Minkowski metric, using Example 2.9 we have

$\displaystyle\star\omega=\frac{1}{2}$ $\displaystyle(f_{1}dz_{2}\wedge
d\bar{z}_{1}\wedge d\bar{z}_{2}+f_{2}dz_{1}\wedge dz_{2}\wedge
d\bar{z}_{1}+f_{\bar{1}}dz_{1}\wedge dz_{2}\wedge d\bar{z}_{2}$ (5.2)
$\displaystyle+f_{\bar{2}}dz_{1}\wedge d\bar{z}_{1}\wedge d\bar{z}_{2}).$

Now applying the exterior derivative to (5.2), after simplifying and
rearranging terms, we have

$\displaystyle d\star\omega$
$\displaystyle=\frac{1}{2}\left(\partial_{1}f_{1}-\bar{\partial}_{2}f_{2}+\bar{\partial}_{1}f_{\bar{1}}-\partial_{2}f_{\bar{2}}\right)\left(dz_{1}\wedge
d\bar{z}_{1}\wedge dz_{2}\wedge d\bar{z}_{2}\right)$ (5.3)
$\displaystyle=-2\left(\partial_{1}f_{1}-\bar{\partial}_{2}f_{2}+\bar{\partial}_{1}f_{\bar{1}}-\partial_{2}f_{\bar{2}}\right)\left(dx_{0}\wedge
dx_{1}\wedge dx_{2}\wedge dx_{3}\right).$

Since $\star\left(dx_{0}\wedge dx_{1}\wedge dx_{2}\wedge dx_{3}\right)=-1$ in
the Minkowski metric, it follows that

$d^{*}\omega=2\left(\partial_{1}f_{1}-\bar{\partial}_{2}f_{2}+\bar{\partial}_{1}f_{\bar{1}}-\partial_{2}f_{\bar{2}}\right).$
(5.4)

∎

In the case we write $\omega$ in the real form $\phi
dx_{0}-A_{1}dx_{1}-A_{2}dx_{2}-A_{3}dx_{3},$ then (5.4) implies

$d^{*}\omega=-\left(\frac{\partial\phi}{\partial x_{0}}+\frac{\partial
A_{1}}{\partial x_{1}}+\frac{\partial A_{2}}{\partial x_{2}}+\frac{\partial
A_{3}}{\partial x_{3}}\right).$

If $d^{*}\omega$ is a constant, say $k$, then one can easily modify
$f_{i},f_{\bar{i}},i=1,2$ such that $d^{*}\omega$ becomes $0$. For example, in
the Euclidean metric we can replace $f_{1}$ by $f_{1}+\frac{k}{2}\bar{z}_{1}$
and keep other functions unchanged. Similar modification can be done in the
Minkowski metric. In this view, the Lorenz gauge condition is just a trivial
strengthening of the condition $d^{*}\omega$ being constant. Therefore, we
shall say that a smooth $1$-form $\omega$ satisfies the Lorenz gauge condition
if $d^{*}\omega$ is constant.

With the foregoing observation, in the Euclidean metric case one can write
(4.15) as

$\displaystyle d\star F_{\omega}$
$\displaystyle=\left(2\nabla^{2}f_{2}+\frac{1}{2}\partial_{2}d^{*}\omega\right)dz_{1}\wedge
dz_{2}\wedge
d\bar{z}_{1}-\left(2\nabla^{2}f_{1}+\frac{1}{2}\partial_{1}d^{*}\omega\right)dz_{1}\wedge
dz_{2}\wedge d\bar{z}_{2}$
$\displaystyle+\left(2\nabla^{2}f_{\bar{2}}+\frac{1}{2}\bar{\partial_{2}}d^{*}\omega\right)dz_{1}\wedge
d\bar{z}_{1}\wedge
d\bar{z}_{2}-\left(2\nabla^{2}f_{\bar{1}}+\frac{1}{2}\bar{\partial_{1}}d^{*}\omega\right)dz_{2}\wedge
d\bar{z}_{1}\wedge d\bar{z}_{2}.$

And in the Minkowski metric case, one can write (4.26) as

$\displaystyle 2d\star F_{\omega}$
$\displaystyle=\left(-\Box^{2}f_{2}+\partial_{2}d^{*}\omega\right)dz_{1}\wedge
dz_{2}\wedge
d\bar{z}_{1}+\left(-\Box^{2}f_{1}+\bar{\partial_{1}}d^{*}\omega\right)dz_{1}\wedge
dz_{2}\wedge d\bar{z}_{2}$
$\displaystyle+\left(-\Box^{2}f_{\bar{2}}+\bar{\partial_{2}}d^{*}\omega\right)dz_{1}\wedge
d\bar{z}_{1}\wedge
d\bar{z}_{2}+\left(-\Box^{2}f_{\bar{1}}+{\partial_{1}}d^{*}\omega\right)dz_{2}\wedge
d\bar{z}_{1}\wedge d\bar{z}_{2}.$

Moreover, if $\omega$ is a solution to the Maxwell’s equations, we have
$d^{*}d\omega=d^{*}F_{\omega}=0$. Hence $\Delta\omega=(dd^{*}+d^{*}d)\omega=0$
if and only if $dd^{*}\omega=0$, i.e., $d^{*}\omega$ is a constant, or in
other words $\omega$ satisfies the Lorenz gauge condition. We summarize
Theorem 4.5, Theorem 4.15, and the foregoing observations in the following
corollary.

######  Corollary 5.2.

Let $\omega$ be a solution to the Maxwell’s equations in vacuum under the
Euclidean or Minkowski metric. Then the following are equivalent.

  1. (a)

$\omega$ satisfies the Lorenz gauge condition.

  2. (b)

$\omega$ is harmonic or respectively wavelike.

  3. (c)

$\omega$ is HL-harmonic.

If $\omega$ as defined in (4.1) is a solution to the Maxwell’s equations in
vacuum under the Euclidean or Minkowski metric, then for every smooth function
$u$ the form $\omega^{\prime}=\omega+du$ is also a solution because

$F_{\omega^{\prime}}=d\omega^{\prime}=d\omega+d^{2}u=d\omega=F_{\omega}.$

This is the gauge invariance of the Maxwell’s equations in differential forms.
We write the above gauge transformations as

$f^{\prime}_{j}=f_{j}+\partial_{j}u,\ \ \
f^{\prime}_{\bar{j}}=f_{\bar{j}}+\bar{\partial}_{j}u,\ j=1,2.$

Then with respect to Theorem 4.5 direct computations give

$\bar{\partial}_{1}f^{\prime}_{1}+\bar{\partial}_{2}f^{\prime}_{2}+\partial_{1}f^{\prime}_{\bar{1}}+\partial_{2}f^{\prime}_{\bar{2}}=\bar{\partial}_{1}f_{1}+\bar{\partial}_{2}f_{2}+\partial_{1}f_{\bar{1}}+\partial_{2}f_{\bar{2}}+\frac{1}{2}\nabla^{2}u,$
(5.5)

and likewise with respect to Theorem 4.15 we have

${\partial}_{1}f^{\prime}_{1}-\bar{\partial}_{2}f^{\prime}_{2}+\bar{\partial}_{1}f^{\prime}_{\bar{1}}-\partial_{2}f^{\prime}_{\bar{2}}={\partial}_{1}f_{1}-\bar{\partial}_{2}f_{2}+\bar{\partial}_{1}f_{\bar{1}}-\partial_{2}f_{\bar{2}}+\frac{1}{2}\Box
u.$ (5.6)

It is known that ([4, 8]) for every smooth function $h$ on ${\mathbb{R}}^{4}$,
the equations

$\nabla^{2}u=h,\ \ \text{and}\ \ \Box u=h$

both have solutions (non-unique). Hence there exists a smooth function $u$
such that $\omega^{\prime}=\omega+du$ satisfies the Lorenz gauge condition
with respect to the Euclidean metric (or the Minkowski metric). Hence by
Corollary 5.2 the curvature field
$F_{\omega}=F_{\omega^{\prime}}=d\omega^{\prime}$ is harmonic (or respectively
wavelike). We summarize this observation as follows.

######  Corollary 5.3.

Let $F_{\omega}$ be a solution to the Maxwell’s equations in vacuum under the
Euclidean or Minkowski metric. Then $F_{\omega}$ is harmonic, or respectively
wavelike.

In particular, this indicates that there is no non-wavelike solution to the
Maxwell’s equations in vacuum.

##  6\. Concluding Remarks

Complex analysis is a core component in mathematics, and it has also played an
increasingly important role in modern physics. It is thus meaningful to
reinterpret some fundamental theories in physics from a complex perspective,
for instance special relativity, Maxwell’s equations, and Yang-Mills
equations, whose original fomulations were in real variables. This
reinterpretation will not only provide a complex formulation of the theories,
but also give rise to new and natural observations from this point of view.
The exploration in this direction has been made in literature, see for example
[10, 12], but it is far from being complete. This paper shall serve as a
starting point for the authors to explore greater applications of complex
analysis to physics theories.

Acknowledgments. The authors would like to thank Marius Beceanu and Oleg Lunin
for valuable comments on the initial draft of this paper. This paper is in
part based on the first author’s doctoral dissertation ([14]) submitted to
SUNY at Albany, and he is grateful to the Department of Mathematics and
Statistics for providing him an opportunity to pursue his research interests.

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