  In the current paper the properties of a birefringent Lorentz-violating
extension of quantum electrodynamics is considered. The theory results from
coupling modified Maxwell theory, which is a CPT-even Lorentz-violating
extension of the photon sector, to a Dirac theory of standard spin-1/2
particles. It is then restricted to a special birefringent case with one
nonzero Lorentz-violating coefficient. The modified dispersion laws of
electromagnetic waves are obtained plus their phase and group velocities are
considered. After deriving the photon propagator and the polarization vectors
for a special momentum configuration we prove both unitarity at tree-level and
microcausality for the quantum field theory based on this Lorentz-violating
modification. These analytical proofs are done for a spatial momentum with two
vanishing components and the proof of unitarity is supported by numerical
investigations in case all components are nonvanishing. The upshot is that the
theory is well-behaved within the framework of our assumptions where there is a
possible issue for negative Lorentz-violating coefficients. The paper shall
provide a basis for the future analysis of alternative birefringent quantum
field theories.
