Lorenz-gauge reduction of the electromagnetic action

The electromagnetic field tensor action differs from the component-gradient potential action by $(2{\mu }_0{)}^{-1}\int {\partial }^b{A}^a{\partial }_a{A}_b{d}^{4}x$. Integration by parts expresses this term as a boundary integral plus $(2{\mu }_0{)}^{-1}\int ({\partial }_a{A}^a{)}^2{d}^{4}x$. 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.