In this post, $\hbar=c=1$ units are used throughout.
It is well known that the action of classical electromagnetism is given by $$\mathcal S_{\text{Maxwell}} = \int d^4x\left\{-\frac{1}{4\mu_0}F_{\mu\nu}F^{\mu\nu}\right\}$$ which yields a Lagrangian density $$\mathscr{L}_{\text{Maxwell}} = -(4\mu_0)^{-1}F_{\mu\nu}F^{\mu\nu}.$$
In P&S, however, they instead use the density $$\mathscr L = -\frac 1 4F_{\mu\nu}F^{\mu\nu}.$$ This is a bit more natural than the real one, however to me it seems that this is insufficient to get $\mu_0$s and $\varepsilon_0$s into Maxwell's equations. I'm aware that in $c=1$ units $\varepsilon_0 = \mu_0^{-1}$ so we work with $\mu_0$s WLOG, but how does one get $\mu_0$ into the equation using P&S's Lagrangian?
