Remarks on the effect of bound states and threshold in $g$-2

The appearance of positronium poles in a photon propagator in quantum electrodynamics (QED) formally requires a summation of an infinite series of terms in perturbation theory. For physical quantities that are determined by this propagator in high orders of perturbation theory—for example for the electron anomalous magnetic moment—this observation may cause doubts on the validity of perturbative results obtained in traditional ways that ignore potential nonperturbative contributions due to the QED bound states. In this paper we show how these nonperturbative contributions disappear, using the case of the electron anomalous magnetic moment as an example. We also give arguments that emphasize the general nature of this result.