Relativistic electron gas: A candidate for nature’s left-handed materials

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The electric permittivities and magnetic permeabilities for a relativistic electron gas are calculated from quantum electrodynamics at finite temperature and density as functions of temperature, chemical potential, frequency, and wave vector. The polarization and the magnetization depend linearly on both electric and magnetic fields, and are the sum of a zero-temperature and zero-density vacuum part with a temperature- and chemical-potential-dependent medium part. Analytic calculations lead to generalized expressions that depend on three scalar functions. In the nonrelativistic limit, results reproduce the Lindhard formula. In the relativistic case, and in the long wavelength limit, we obtain the following: (i) for ω=0, generalized susceptibilities that reduce to known nonrelativistic limits; (ii) for ω≠0, Drude-type responses at zero temperature. The latter implies that both the electric permittivity ε and the magnetic permeability μ may be simultaneously negative, a behavior characteristic of metamaterials. This unambiguously indicates that the relativistic electron gas is one of nature's candidates for the realization of a negative index of refraction system. Moreover, Maxwell's equations in the medium yield the dispersion relation and the index of refraction of the electron gas. Present results should be relevant for plasma physics, astrophysical observations, synchrotrons, and other environments with fast-moving electrons.