We compute the real and imaginary parts of the electric permittivities and magnetic permeabilities for relativistic electrons from quantum electrodynamics at finite temperature and density. A semiclassical approximation establishes the conditions for neglecting nonlinear terms in the external electromagnetic fields as well as electron-electron interactions. We obtain both the electric and magnetic responses in a unified manner and relate them to longitudinal and transverse collective plasma oscillations. We demonstrate that such collective modes are thresholds for a metamaterial regime of the electron plasma which exhibits simultaneously negative longitudinal permittities and permeabilities. For nonzero temperatures, we obtain electromagnetic responses given by one-dimensional integrals to be numerically calculated, whereas for zero temperature we find analytic expressions for both their real/dispersive and imaginary/absorptive parts.
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