Abstract
It is shown that an approximate summation of the crossed-ladder Feynman diagrams for the scattering of two charged particles leads to the formula $s={{m}_{1}}^{2}+{{m}_{2}}^{2}+2{m}_{1}{m}_{2}{[1+\frac{{Z}^{2}{\ensuremath{\alpha}}^{2}}{{(n\ensuremath{-}{\ensuremath{\epsilon}}_{j})}^{2}}]}^{\ensuremath{-}\frac{1}{2}}$ for the squared mass of bound states. This formula neglects radiative corrections. It includes recoil effects properly, and reduces in the limit of one infinite mass to the corresponding spectrum of a relativistic particle in a static Coulomb potential. In the particular case of positronium, its expansion in powers of $\ensuremath{\alpha}$ coincides up to order ${\ensuremath{\alpha}}^{4}$ with the singlet energy levels. In an appendix we investigate some properties (gauge invariance, static limit) of this series of graphs.
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