  The absence of recognizable, low energy quantum gravitational effects
requires that some asymptotic series expansion be wonderfully accurate, but the
correct expansion might involve logarithms or fractional powers of Newton's
constant. That would explain why conventional perturbation theory shows
uncontrollable ultraviolet divergences. We explore this possibility in the
context of the mass of a charged, gravitating scalar. The classical limit of
this system was solved exactly in 1960 by Arnowitt, Deser and Misner, and their
solution does exhibit nonanalytic dependence on Newton's constant. We derive an
exact functional integral representation for the mass of the quantum field
theoretic system, and then develop an alternate expansion for it based on a
correct implementation of the method of stationary phase. The new expansion
entails adding an infinite class of new diagrams to each order and subtracting
them from higher orders. The zeroth order term of the new expansion has the
physical interpretation of a first quantized Klein-Gordon scalar which forms a
bound state in the gravitational and electromagnetic potentials sourced by its
own probability current. We show that such bound states exist and we obtain
numerical results for their masses.

General Relativity and Quantum Cosmology;High Energy Physics - Theory