electromagnetism - Deriving Lagrangian density for electromagnetic field

In considering the (special) relativistic EM field, I understand that assuming a Lagrangian density of the form

$$\mathcal{L} =-\frac{1}{4}F_{\mu\nu}F^{\mu\nu} + \frac{1}{c}j_\mu A^\mu$$

and following the Euler-Lagrange equations recovers Maxwell's equations.

Does there exist a first-principles derivation of this Lagrangian? A reference or explanation would be greatly appreciated!

Answer

Ultimately the reasoning must be that (as you stated) it must be constructed so the Euler-Lagrange equations are Maxwell's equations. So in a sense you have to guess the Lagrangian that produces this as is done here for example.

However you can get some guidance from the fact that we need to construct a Lagrangian for a massless non self interacting field. So we need a gauge and lorentz invariant combination of the 4-vector potential which only has a kinetic term (quadratic in derivatives of the fields). You are then not left with many options apart from $F^{\mu\nu}F_{\mu\nu}$. The source term is then trivial to add in if needed.