  We consider the propagation of photons in a model of a charged scalar
Bose-Einstein (BE) condensate. We determine the dispersion relations of the
collective modes, as well as the photon polarization tensor and the dielectric
constant in the model. Two modes correspond to the transverse photon
polarizations, with dispersion relations of the usual form for transverse
photons in a plasma. The other two modes, denoted as the $(\pm)$ modes, are
combinations of the longitudinal photon and the massive scalar field. Their
dispersion relations behave very differently as functions of momentum. The
$(+)$ mode dispersion relation increases steadily and remains greater than the
momentum as the momentum increases. The dispersion relation of the $(-)$ mode
decreases in a given momentum range, with the group velocity being negative in
that range, while in another range it increases steadily but remains smaller
than the momentum, akin to the situation in a medium with an index of
refraction greater than 1. We consider the non-relativistic limit of the
$(\pm)$ dispersion relations and discuss some aspects of the results. We also
determine the wavefunctions of the $(\pm)$ modes, which are useful to obtain
the corrections to the dispersion relations, e.g., imaginary parts due to the
damping effects and/or the effects of scattering, due to the interactions with
the excitations of the system. The results can be useful in various physical
contexts that have been considered in the literature involving the
electrodynamics of a charged scalar BE condensate.
