  We study the resonant divergences that occur in quantum scattering
cross-sections in the presence of a strong external magnetic field. We
demonstrate that all such divergences may be eliminated by introducing
radiative corrections to the leading-order scattering amplitudes. These
corrections impose a choice of basis states that must be used in scattering
calculations: electron states must diagonalize the mass operator, while photon
states must diagonalize the polarization operator. The radiative corrections
introduce natural line-widths into the energy denominators of all propagators,
as well as into the time-development exponentials of all scattering states
corresponding to external lines. Since initial and final scattering states may
now decay, it is logically necessary to compute scattering amplitudes for a
finite time-lapse between the preparation of the initial state and the
measurement of the final state. Strict energy conservation, which appeared in
previous formulations of the theory, must thus be abandoned. We exhibit the
generic formulae for the scattering cross-sections in two useful limits,
corresponding to the cases where either the initial states or the final states
are stable, and discuss the application of the general formula when neither of
these limits applies.
