The transformations of the electromagnetic potentials under translations

Abstract

I consider infinitesimal translations $x^{\prime \alpha }=x^{\alpha }+\delta x^{\alpha }$ 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 $A_{\gamma }^{\prime }(x^{\prime })=A_{\gamma }(x)+{\partial }_{\gamma }[\delta x_{\beta }{A}^{\beta }(x)]$, which differs from the usually assumed invariance $A_{\gamma }^{\prime }(x^{\prime })=A_{\gamma }(x)$, by the gauge transformation ${\partial }_{\gamma }[\delta x_{\beta }{A}^{\beta }(x)]$.

References to work by Jackiw and others are added