Note on Positronium

Abstract

By means of an inequality providing an upper bound for the norm of integral operators, it is shown that the Bethe-Salpeter equation for bound states of the electron-positron system (in the ladder approximation) admits solutions associated with a discrete spectrum of binding energies. It is found that in the weak coupling limit the spectrum B(α) approaches asymptotically the Coulomb spectrum in the sense lim lim α→0(B(α)/mα2)={(1/4n2) | n=1, 2, 3, … ∞},where α is the fine-structure constant and m the electron rest mass.