The relativistic electron gas: a low-frequency left-handed material

We show that a gas of relativistic electrons behaves as a left-handed material at low frequencies by computing its effective electric permittivity and effective magnetic permeability, and verifying that they are both negative below a threshold frequency. We confirm that electromagnetically induced propagation in the gas will either involve plasmons or an electromagnetic wave that propagates with the speed of light in vacuum. In the latter case, we find that no energy is dissipated, so that the medium is transparent. We obtain the value $n_{\rm eff}=-1$ for the effective index of refraction, discuss Snell's law, and provide expressions for the reflection and transmission coefficients of the gas.