Covariant calculations at finite temperature: The relativistic plasma

It is shown that finite-temperature calculations in field theory are manifestly Lorentz covariant at all stages if the Minkowski-space form of the temperature-dependent propagators is used and if the four-velocity u_μ of the heat bath is taken into account. New tensor structures involving u_μ generally arise but are severely constrained by covariant current conservation. A complete high-temperature (Tm) expansion of the vacuum polarization tensor for non-Abelian gauge theories is computed to order g² and displays the separate dependence on frequency ω and wave number k that occurs at finite temperature. A covariant phenomenology of "electric" and "magnetic" properties is applied to the collective plasma effects, characterized by a plasma frequency ωₚ²=(Nf+2N)g²T²6 for SU(N) with Nf fermions. The longitudinal normal modes of the "electric" field exist only for ω>ωₚ; for ω<ωₚ all "electric" fields are screened. The transverse normal modes are plane waves along ̃ \~E× ̃ \~B for ω>ωₚ; for ω<ωₚ both transverse "electric" and "magnetic" fields are shielded except for the static (ω=0) case.