Drastic violation of the U-matrix boundary condition by quantum-field theoretical on-shell renormalization and a new consistent renormalization concept involving bound-state problems

By generalizing the Gell-Mann–Low formula and applying the self-consistent projection-operator method, developed a decade ago by the present author, a connecting equation between the fermion propagator (involving the interaction with the quantum electromagnetic field in the presence of an external field) and the U-matrix elements, being valid to any order of the fermion–photon interaction, is derived. After proving rigorously, the drastic violation of the U-matrix boundary condition (U(t,t)=1) by the conventional on-shell renormalization of the fermion propagator (removal of free-fermion pole and residue corrections), the absolute necessity for the formulation of a new renormalization concept becomes self-evident. This latter is achieved in the present paper by elaborating a consistent renormalization procedure, based on the self-consistent projection-operator method (which, unlike the standard perturbation method, guarantees the consistency of the applied approximation to any order as to the strength of the interaction). The new complete renormalization, unlike the conventional one, removes completely the experimentally unobservable free-electron Dyson self-energy from the fermion propagator. By applying the new concept, for the first time, consistently renormalized contributions (up to the fourth order) to the bound-electron self-energy function with the corresponding Feynman diagrams are derived.