The transformations of the electromagnetic potentials under translations

Abstract

I consider infinitesimal translations $x'^{\alpha}=x^{\alpha}+\delta x^{\alpha}$ and demand that Noether's approach gives a symmetric electromagnetic energy-momentum tensor as it is required for gravitational sources. This argument determines the transformations of the electromagnetic potentials under infinitesimal translations to be $A'_{\gamma} (x') = A_{\gamma}(x)+\partial_{\gamma} [\delta x_{\beta} A^{\beta}(x)]$, which differs from the usually assumed invariance $A'_{\gamma} (x') = A_{\gamma}(x)$, by the gauge transformation $\partial_{\gamma} [\delta x_{\beta} A^{\beta}(x)]$.