Polarization of a QED plasma in a strong magnetic field

In this paper we study the polarization tensor of photons in a QED plasma in the presence of a magnetic field. We do it both at vanishing and at finite temperature. We use two different methods to compute the polarization tensor components ${\mathrm{\ensuremath{\Pi}}}^{\mathrm{\ensuremath{\mu}}\ensuremath{\nu}}$ within the one-loop approximation. The first starts from the effective Lagrangian of the system and relates ${\mathrm{\ensuremath{\Pi}}}^{\mathrm{\ensuremath{\mu}}\ensuremath{\nu}}$ to the thermodynamic quantities. The second makes use of the electron propagator in an external magnetic field. In this second approach we use an imaginary time formalism. These methods give consistent results in the first nonvanishing order in the photon four-momentum. Beyond this limit the first method is not applicable and the introduction of the second cannot be avoided. We give some physical interpretations of our results.