  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.

Quantum Physics