Polarization operator of an electron-positron gas in a stationary external magnetic field

The polarization operator of an electron-positron gas in a stationary magnetic field is calculated in the one-loop approximation by taking into account exactly the external field in the temperature Green function technique. All components of the polarization tensor are evaluated separately and their behaviour with respect to Hermite and charge conjugations, coordinate and index inversion, rotation about the magnetic field etc. is considered. The requirement of four-dimensional transversality imposed by gauge invariance is explicitly fulfilled after carrying out the subtraction procedure. Expressions are presented for the dielectric permittivity when the waves propagate parallel or perpendicular to the external magnetic field. In conclusion the limits of the zero four-momentum and of an asymptotically strong magnetic field are discussed. The singularities of the polarization operator in connection with the absorption of the electromagnetic wave are also considered