  We show that a gas of relativistic electrons is a left-handed material at low
frequencies by computing the effective electric permittivity and effective
magnetic permeability that appear in Maxwell's equations in terms of the
responses appearing in the constitutive relations, and showing that the former
are both negative below the {\it same} frequency, which coincides with the
zero-momentum frequency of longitudinal plasmons. We also show, by explicit
computation, that the photonic mode of the electromagnetic radiation does not
dissipate energy, confirming that it propagates in the gas with the speed of
light in vacuum, and that the medium is transparent to it. We then combine
those results to show that the gas has a negative effective index of refraction
$n_{\rm eff}=-1$. We illustrate the consequences of this fact for Snell's law,
and for the reflection and transmission coefficients of the gas.
