Correct No-Retardation Limit in Bethe-Salpeter Kernels

A careful discussion of the no-retardation (instantaneous) limit of the Bethe-Salpeter (BS) equation is carried out, as illustrated by one-photon exchange in the radiation gauge for positronium. Comparison is made with equations obtained by other authors, using the effective potential to order $m{\alpha }^4$ as a test. Detailed analysis, using a new approach to the BS equation, recently proposed by the authors, indicates that the equations do not give equivalent results to order $m{\alpha }^4$ and that the arguments leading to the experimentally well verified $m{\alpha }^4$ potentials are sometimes defective.