Is it possible to unify Maxwell's four equations into one single equation? The ultimate goal of physics is to unify all the four fundamental forces together. These forces basically are laws of nature.
This makes me ask if it is possible or not to unify Maxwell's four laws?

 A: Maxwell's equations reflect several unrelated facts:

*

*electromagnetism is a massless field emerging from charges and currents

*the matter from which it emerges obeys charge-current conservation

*fields in matter can be described using linear response theory

In vacuum point 3 is not relevant and a single equation can be written
$$\partial_\mu F^{\mu\nu} =  j^\nu / \epsilon_0\,.$$
The field tensor $F^{\mu\nu}$ can be written as $\partial^\mu A^\nu-\partial^\nu A^\mu$ so this becomes
$$\partial_\mu \partial^\mu A^\nu - \partial_\nu \partial^\mu A^\mu =  j^\nu / \epsilon_0\,.$$
If you take $\partial_\mu A^\mu = 0$, the Lorenz condition, then
$$\partial_\mu \partial^\mu A^\nu =  j^\nu / \epsilon_0\,.$$
The potential $A^\mu$ unifies $\bf E$ and $\bf B$.
Standard physics considers the Lorenz condition to be a choice of gauge.
A: I will say it is difficult without hiding some equations elsewhere. If you want to keep $\mathbf B$ and $\mathbf E$ it will be very complicated. With relativistic electromagnetism you can write two covariant equations
$$F^{\alpha\beta}_{~~~~~,\alpha}=\mu_0 J^\beta$$
where $J^\mu$ is the 4-current and
$$\partial_{\alpha}(\tfrac{1}{2}\epsilon^{\alpha\beta\gamma\delta}F_{\gamma\delta}) = 0$$
but then you have two define electromagnetic tensor $F^{\alpha\beta}$ in terms of $\mathbf E$ and $\mathbf B$ which accounts for additional equations.
You could use electromagnetic potentials instead and write just
$$\partial_\beta\partial^\beta A^\alpha = \mu_0 J^\alpha$$
where $A^\alpha$ is the 4-potential. However this equation works only in the Lorenz gauge which adds an extra equation, and you need to have two other equations to explain how to derive $\mathbf B$ and $\mathbf E$ from $A^\alpha$.
If you consider the Euler-Lagrange equations a different equation you could write the Lagrangian density as
$$\mathcal{L} ={\epsilon_0 \over 2} {E}^2 - {1 \over {2 \mu_0}} {B}^2   + \frac{1}{c}J_\alpha A^\alpha$$
but it is the same issue as the previous case because you still have $A^\alpha$.
More advanced mathematical notations may reduce it down to 2 (? not sure) if you are accustomed to their notation and properties of the operators. Note that you can always invent some new operator $T$, such that
$$T(\mathbf B ,\mathbf E,J^\alpha)=0$$
implies Maxwell's equations. However you will have to prove that $T$ is useful for other stuff and not a convoluted way to write 4 equations. Preferably, the operator should also add some new understanding to the physics.
A: It basically boils down to how much maths you want to put into the notation.  For example in Space-Time Algebra notation, Maxwell's equations reduce to the single equation [1]
$$
\nabla F = \mu_0 c J
$$
The geometric algebra approach is comparable to introducing one complex-valued equation instead of two real-valued ones, which requires that the problem to be "encoded" has the right structure.
Whether or not such condensed notation adds to the understanding is definitely a matter of taste and of debate, and it depends on how fluent a reader is in the respective notation.
