Gauge independence of interelectronic potentials

It has been shown recently by Lindgren that gauge independent energy shifts in the effective potential formalism can only be obtained when all the diagrams to the same order of perturbation are taken into account. In this work, we extend Lindgren’s proof in two generalized gauge conditions. It is shown explicitly that in the generalized Landau gauge and axial gauge, by neglecting radiative corrections, the energy shifts calculated in these gauges equal the Feynman gauge results to the leading relativistic order.