On the Part Played by Scalar and Longitudinal Photons in Ordinary Electromagnetic Fields

When electromagnetic interactions are calculated directly by second-order perturbation calculus without preceding separation of the Coulomb field from the transverse photon field by a canonical transformation, it is fundamentally not allowed to take a state without photons as zero-order approximation. A more correct zero-order approximation is determined, in which scalar and longitudinal photons are present in pairs. The probability distribution over the numbers of pairs present is given by the Schr\"odinger functional Eq. (20); these numbers are not limited. No transverse photons are present. The relativistic invariance of such a zero-order approximation is proved.This zero-order approximation is then used for the calculation of the self-energy of a free electron at rest, of the M{}ller matrix element for electron-electron scattering and of the Breit interaction energy between electrons in an atom. Apart from mathematical ambiguities in the derivation of the self-energy, this leads to the same results as can also be obtained by the usual separation of the Coulomb field from the transverse photon field.