Bound-state perturbation theory in four-dimensional momentum representation

Abstract
A four-dimensional perturbation theory for bound-state problems is developed here. It is shown that modified S -matrix techniques and graphical methods can be applied to bound-state problems. The correspondence of this theory to the ordinary perturbation theory is discussed, and by way of illustration, the four-dimensional theory is applied to the calculation of energy corrections from the exchange of two transverse photons by the electron and proton in hydrogen. This exchange is found to contribute only to the fine-structure and not to the spin-spin interaction in the first order it enters, α 3 (m/M) R .