  Owing to effects arising from quantum electrodynamics (QED),
magnetohydrodynamical fast modes of sufficient strength will break down to form
electron-positron pairs while traversing the magnetospheres of strongly
magnetised neutron stars. The bulk of the energy of the fast mode fuels the
development of an electron-positron fireball. However, a small, but potentially
observable, fraction of the energy ($\sim 10^{33}$ ergs) can generate a
non-thermal distribution of electrons and positrons far from the star. This
paper examines the cooling and radiative output of these particles. Small-scale
waves may produce only the non-thermal emission. The properties of this
non-thermal emission in the absence of a fireball match those of the quiescent,
non-thermal radiation recently observed non-thermal emission from several
anomalous X-ray pulsars and soft-gamma repeaters. Initial estimates of the
emission as a function of angle indicate that the non-thermal emission should
be beamed and therefore one would expect this emission to be pulsed as well.
According to this model the pulsation of the non-thermal emission should be
between 90 and 180 degrees out of phase from the thermal emission from the
stellar surface.
