  The purpose of this paper is to explain the discrepancies existing in the
literature relative to $e^+e^-$ pair production in peripheral heavy ion
collisions at ultra-relativistic energies. A controversial issue is the
possible cancellation of Coulomb corrections to the Born term in the pair
production cross-section. Such a cancellation has been observed in a recent
approach based on finding retarded solutions of the Dirac equation, but does
not seem to hold in a perturbative approach. We show in this paper that the two
approaches are in fact calculating different observables: the perturbative
approach gives the exclusive cross-section of single pair production, while the
other method gives the inclusive cross-section.
  We have also performed a thorough study of the electron propagator in the
non-static background field of the two nuclei, the conclusion of which is that
the retarded propagator is in the ultra-relativistic limit a much simpler
object than the Feynman propagator, and can be calculated exactly.
