  For inner magnetospheric models of hard X-ray and gamma-ray emission in
high-field pulsars and magnetars, resonant Compton upscattering is anticipated
to be the most efficient process for generating continuum radiation. This is
due in part to the proximity of a hot soft photon bath from the stellar surface
to putative radiation dissipation regions in the inner magnetosphere. Moreover,
because the scattering process becomes resonant at the cyclotron frequency, the
effective cross section exceeds the classical Thomson value by over two orders
of magnitude, thereby enhancing the efficiency of continuum production and the
cooling of relativistic electrons. This paper presents computations of the
electron cooling rates for this process, which are needed for resonant Compton
models of non-thermal radiation from such highly-magnetized pulsars. The
computed rates extend previous calculations of magnetic Thomson cooling to the
domain of relativistic quantum effects, sampled near and above the quantum
critical magnetic field of 44.13 TeraGauss. This is the first exposition of
fully relativistic, quantum magnetic Compton cooling rates for electrons, and
it employs both the traditional Johnson and Lippman cross section, and a newer
Sokolov and Ternov (ST) formulation of Compton scattering in strong magnetic
fields. Such ST formalism is formally correct for treating spin-dependent
effects that are important in the cyclotron resonance, and has not been
addressed before in the context of cooling by Compton scattering. The QED
effects are observed to profoundly lower the rates below extrapolations of the
familiar magnetic Thomson results, as expected, when recoil and Klein-Nishina
reductions become important.
