= Lorenz-gauge reduction of the electromagnetic action
{c}
The <electromagnetic field tensor> action differs from the component-gradient potential action by $(2\mu_0)^{-1}\int \partial^bA^a\partial_aA_b\,d^4x$. Integration by parts expresses this term as a boundary integral plus $(2\mu_0)^{-1}\int(\partial_aA^a)^2\,d^4x$. With the <Lorenz gauge> and fixed or vanishing boundary data, the two actions therefore differ only by a boundary term. Varying the component-gradient action without restriction gives the potential wave equations; the Lorenz condition must be supplied to recover all <Maxwell equations>.
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