  We use a recently developed kinetic model derived from the Dirac equation, in
order to study electromagnetic wave propagation in superstrong magnetic fields,
such as in magnetars, where relativistic Landau quantization is prominent. The
leading contribution to the conductivity tensor in such a plasma is calculated.
It is found that the electron Hall current has an anomalous contribution, in
the quantum relativistic regime, where the effective particle energy has a
significant contribution from the diamagnetic and Zeeman energy. As a result, a
new quantum resonance frequency appears, and the dispersion relation for the
left- and right-hand polarized modes are strongly modified for long and
moderate wavelengths. The implications for magnetar physics are discussed.
