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The Lorenz Gauge can be written (in QED) as $\partial^{\mu}A_{\mu} = 0$ or equivalently as $p^{\mu}A_{\mu} = 0$. The Fock-Schwinger gauge is similar: $x^{\mu}A_{\mu} = 0$.

Can it be written equivalently as $\frac{\partial}{\partial p^{\mu}}A^{\mu} = 0$, or is this naive?

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