  We perform the covariant canonical quantization of the CPT- and
Lorentz-symmetry-violating photon sector of the minimal Standard-Model
Extension, which contains a general (timelike, lightlike, or spacelike) fixed
background tensor $k_{AF}^\mu$. Well-known stability issues, arising from
complex-valued energy states, are solved by introducing a small photon mass,
orders of magnitude below current experimental bounds. We explicitly construct
a covariant basis of polarization vectors, in which the photon field can be
expanded. We proceed to derive the Feynman propagator and show that the theory
is microcausal. Despite the occurrence of negative energies and
vacuum-Cherenkov radiation, we do not find any runaway stability issues,
because the energy remains bounded from below. An important observation is that
the ordering of the roots of the dispersion relations is the same in any
observer frame, which allows for a frame-independent condition that selects the
correct branch of the dispersion relation. This turns out to be critical for
the consistency of the quantization. To our knowledge, this is the first system
for which quantization has consistently been performed, in spite of the fact
that the theory contains negative energies in some observer frames.
