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Current-Conserving Relativistic Linear Response for Collisional Plasmas

We investigate the response of a relativistic plasma to electromagnetic fields in the framework of the Boltzmann equation incorporating a collision term in the relaxation rate approximation selected in a form assuring current conservation. We obtain an explicit solution for the linearized perturbation of the Fermi-Dirac equilibrium distribution in terms of the average relaxation rate $κ$. We study the resulting covariant, gauge invariant, and current conserving form of the polarization tensor in the ultrarelativistic and non-relativistic limits. We evaluate the susceptibility in the ultrarelativistic limit and explore their dependence on $κ$. Finally, we study the dispersion relations for the longitudinal and transverse poles of the propagator. We show that for $κ> 2ω_p$, where $ω_p$ is the plasma frequency, the plasma wave modes are overdamped. In the opposite case, $κ\ll ω_p$, the propagating plasma modes are weakly damped.