A question on the unification of electricity and magnetism As far as I understand it, Maxwell's equations unify the theories of electricity and magnetism, however, I don't see how they show that the electric field, $\mathbf{E}$ and the magnetic field, $\mathbf{B}$ are manifestations of a single entity, the electromagnetic field?! 
At least at face value, Maxwell's theory shows a mutual dependence between electric and magnetic fields, but doesn't explicitly show that they should be unified. (Adding to this point, I get that Maxwell's equations predict self propagating electromagnetic waves, but in this context they are still treated as transverse oscillations of the electric and magnetic fields).
Would it be correct to say that one cannot definitively conclude that the electric and magnetic fields should be considered as a single, unified entity, the electromagnetic field, until one takes into account special relativity?
To me this seems to be the case, since the Lorentz transformations "mix-up" the two fields, transforming one into the other, and so the notion of whether what is observed is an electric field or a magnetic field is observer dependent. The quantity that does maintain an objective reality, however, is the electromagnetic field strength, $F_{\mu\nu}$ which is Lorentz covariant, and indeed the quantity $F_{\mu\nu}F^{\mu\nu}\propto\mathbf{B}^{2}-\frac{1}{c^{2}}\mathbf{E}^{2}$ is Lorentz invariant. Since the components of $F_{\mu\nu}$ can be represented in terms of $\mathbf{E}$ and $\mathbf{B}$, one considers the electric field and the magnetic field to be different manifestations of a single unified field - the electromagnetic field.
In essence, my question is can one consider electricity and magnetism unified into a single concept of electromagnetism at the level of Maxwell's equations, or is it not until special relativity is taken into account that it becomes necessary to consider them as a single entity - the electromagnetic field?
 A: In early times only static electricity and magnetism were known. No connection between these phenomena was suspected. Indeed, static electric and magnetic fields are unconnected. Then the attention shifted to varying electric and magnetic fields. Electricity and magnetism were found to be connected and Faraday's and Ampère's laws were formulated. So experimentally, the electromagnetic unification was almost complete. Then Maxwell added the displacement current and the theoretical unification was complete, enabling the prediction of EM waves and the identification of light as an EM wave.
Why then are we still using E and B as physical quantities? That is because the Lorentz force is written in terms of E and B and is gauge invariant. This led to the idea that all EM quantities should be gauge invariant and thus be expressible in E and B. L.V. Lorenz introduced what later became the four vector potential (V,$\mathbf A$). However at the time it was understood that all EM phenomena should involve just E and B and Maxwell formulated his theory in terms of E and B.
Are E and B then not unified?  In my opinion the four potential unifies E and B. The potential is used in classical and quantum mechanics and you will have a hard time using E and B in its place. Also the Aharonov-Bohm effect cannot be explained by the B field. Nevertheless, due to the concept of gauge invariance it is believed that the potential is not physical. Therefor you could argue that the unification is incomplete. In my opinion the potential actually is entirely physical and unifies electromagnetism. I published a paper on this, which can be accessed on arxiv.org.
A: Maxwell equations unify both electric fields and magnetic fields.  They guarantee oscillations of both fields, simultaneously and orthogonally, at the place of generation of these fields.  But, they do not guarantee (mathematically) simultaneous propagation as waves.  Depending on the source of production of fields: (1) oscillation of electric fields becomes light waves (in general), when the voltage exceeds a certain level; (2) oscillation of magnetic fields becomes radio waves (in general), when the current exceeds a certain level; (3) oscillation these fields become independent light waves and radio waves when voltage level as well as current level exceed the respective limits, but they need not propagate jointly.
A: In electromagnetism there are three stories.
The first story is about the charged subatomic particles. So electron possesses an electric monopol and a magnetic dipole moment and by this intrinsic properties the electron is surrounded by both an electric field and a magnetic field. The electric field is symmetrical in all directions, the magnetic field has always an alignement in a direction.
The second story is about the macroscopic manifestations of this fields. Separating electrons from protons gives as electric fields which are easily observable (by the force with which the separated charges will attract each other or by a current in the case the charges will be connected by a wire). With the magnetic field it is not that easy. The magnetism of permanent magnets was not explainable as long as it was not clear that charged particles have intrinsic magnetic dipole moments and in permanent magnets this moment of (from?) the involved electrons is aligned and "frozen". An other way to to get a macroscopic magnetic field was the invention of a coil of a current carrying wire. Moving in circles (on a solenoidal path) the electrons are under permanent acceleration and by this their magnetic dipole moments get aligned all in the same direction and a macroscopic magnetic field arise.
The third story is about the (later so called) electromagnetic radiation of energy by the acceleration of charged particles. The flow of electrons inside a coil is accompanied with energy losses which are higher as the losses of a straight wire of the same length (due to Ohmic resistance). The radiated from the acccelerated electrons energy doesn't need a medium for movement away. Later instead of a coil electrons were accelerated back and forth in an (antenna) rod and this invention produces an oscillating radiation. The study of such radiation led to the conclusion that (strictly exact only in vacuum) this radiation consists an electric field component and a magnetic field component, both perpendicular to the direction of propagation and perpendicular to each other. So both field components of the electron over going to the EM radiation from this charged particles. 

As far as I understand it, Maxwell's equations unify the theories of electricity and magnetism, however, I don't see how they show that the electric field, E and the magnetic field, B are manifestations of a single entity, the electromagnetic field?!

Maybe this is not a direct answer to your questionnaire, but it gives explanations without any special relativity.

Let me add a post-sentence. An interesting area of scientific research would be the investigation About the inner structure of the electron which is unknown until now. But since the electron posseses permanently two fields to build a model of the constituents of this inner structure and by this of the fields should be good scientific practice. This model has to explain not only both field components of the electron but also both components of the EM radiation.
A: Electromagnetism is a $U(1)$ gauge theory. How much more unified can it be?
