Degenerate Systems and Mass Singularities

For a system with degenerate energies, the power series expansions of the $S$-matrix elements may become singular. An elementary theorem in quantum mechanics is proved which shows that under certain general conditions such singularities do not appear in the power series expansions of the transition probabilities, provided these are averaged over an appropriate ensemble of degenerate states. Application of this theorem leads to the cancellations of mass singularities and infrared divergences in quantum electrodynamics. The question of whether a charged particle can have zero mass is studied.