Elimination of resonant divergences from QED in super-strong magnetic fields

Abstract

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.