= Solution
Use <natural units> $\hbar=c=1$ and the <Minkowski metric> $\eta=\operatorname{diag}(1,-1,-1,-1)$ throughout. An Abelian <gauge transformation> is $A_\mu\mapsto A_\mu+\partial_\mu\chi$. For a sufficiently regular $\chi$, commuting the <partial derivatives> gives $F_{\mu\nu}\mapsto F_{\mu\nu}$, so the <Maxwell Lagrangian> is invariant pointwise.
Vary the <action> with respect to the <electromagnetic four-potential>, using variations of <compact support> or vanishing on the <boundary>. The antisymmetry of the <electromagnetic field tensor> gives
$$
\delta\mathcal L=-\frac12F^{\mu\nu}\delta F_{\mu\nu}=-F^{\mu\nu}\partial_\mu\delta A_\nu.
$$
After <integration by parts>, $\delta S=\int d^4x\,(\partial_\mu F^{\mu\nu})\delta A_\nu$. The <principle of stationary action> therefore gives the source-free <Maxwell equations>,
$$
\boxed{\partial_\mu F^{\mu\nu}=0\iff\Box A^\nu-\partial^\nu(\partial_\mu A^\mu)=0.}
$$
In the <Lorenz gauge>, $\partial_\mu A^\mu=0$, this becomes $\Box A_\nu=0$ with $\Box=\partial_\mu\partial^\mu$. The PDF's name “Lorentz gauge” denotes the usual <Lorenz gauge>, named after Lorenz. Residual <gauge transformations> preserve it when $\Box\chi=0$; this condition is not itself a complete removal of the gauge freedom.
Back to article page
