Is there a way to write the Lorentz force in terms of one field, $L$, and one charge, $X$? I have heard that physicists like to write electromagnetism as one force (the Lorentz force) and define it as $\vec{F_L}\left(q, \vec{v}, \vec{E}, \vec{B}\right) = q\left(\vec{E} + \vec{v} \times \vec{B}\right)$. They also talk about electricity and magnetism as if they are one force. However, this doesn't look that much prettier to me, easier to use or unified.
Is there a way to write the Lorentz force in terms of one field, $L$, and one charge, $X$?
Obviously, one could just make $L$ and $X$ tuples and write $\vec{F_L}\left(X, L\right) = X_q \left(\vec{L_E} + \vec{X_v} \times \vec{L_B} \right)$ but that doesn't seem nice enough to me.
 A: Let us fix a reference frame $S$, where a particle of charge $q$ and velocity $v$ lies. It can be experimentally proven that, if another such particle $q'$ is present elsewhere in the universe, the initial one is subject to a force $\textbf{F}=q\textbf{E}$, where $\textbf{E}$ can be measured and addressed to the other body $q'$.
Likewise, if a current $i$ (or, equivalently, a magnet) exists somewhere in the universe, the initial particle is subject to a force $\textbf{F}=q\textbf{v}\times\textbf{B}$, where $\textbf{B}$ can be addressed to the current $i$. If both are present together, then the force is obviously the sum of the two pieces, thus $\textbf{F}=q(\textbf{E} + \textbf{v}\times\textbf{B})$.
Since, in principle, $\textbf{E}$ and $\textbf{B}$ seem to come from two different sources (the former being a charge $q'$ and the latter being a current $i$) and since they are measured in different ways, one is led to believe that they are indeed two different things, therefore one gives them two different names.
But then we realise that if, instead of choosing $S$ as reference frame, we choose $S'$ having the same velocity of the current $i$ with respect to $S$, then the two previous contributions $\textbf{E}$ and $\textbf{B}$ replace each other. Hence we understand that they are not really two different things, rather they are the same thing that only appears to be different just according to what reference frame we choose. This is indeed supported by the additional experimental results showing that whenever a variation in time of either of the two fields is present, the other gets automatically created. Again, this leads to think that they must somehow be the same underlying field having difference faces.
At this point we are quite sure that there is only one field, that we call $F$, whose representation is any reference frame of choice depends on the coordinate basis and can be described by a rank $2$ tensor (for some reasons). Doing a little re-ordering of the previous equations, together with the general Maxwell's equations for the field, one narrows things down to the following formulae for the fields:
$$
\partial_{\mu}F^{\mu\nu} = \frac{4\pi}{c}\,j^{\nu},\qquad \partial_{\lambda}F_{\mu\nu} + \textrm{permutations} = 0
$$
together with the equation of motion of a charged particle in such environment
$$
\frac{d}{ds}p_{\mu} = qF_{\mu\nu}u^{\nu}.
$$
