The transformations of the electromagnetic potentials under translations

Abstract

I consider infinitesimal translations $x'^α=x^α+δx^α$ 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 $A'_γ (x') = A_γ(x)+\partial_γ [δx_β A^β(x)]$, which differs from the usually assumed invariance $A'_γ (x') = A_γ(x)$, by the gauge transformation $\partial_γ [δx_β A^β(x)]$.