Covariant calculations at finite temperature: The relativistic plasma

Abstract

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 $u_{\mu }$ of the heat bath is taken into account. New tensor structures involving $u_{\mu }$ generally arise but are severely constrained by covariant current conservation. A complete high-temperature ($T\gg m$) expansion of the vacuum polarization tensor for non-Abelian gauge theories is computed to order $g^2$ and displays the separate dependence on frequency $\omega$ 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 ${{\omega }_p}^2=\frac{(N_f+2N)g^2{T}^2}{6}$ for $\mathrm{SU}\mathrm{}\left(N\right)\mathrm{}$ with $N_f$ fermions. The longitudinal normal modes of the "electric" field exist only for $\omega >{\omega }_p$; for $\omega <{\omega }_p$ all "electric" fields are screened. The transverse normal modes are plane waves along $\tilde{E}\times \tilde{B}$ for $\omega >{\omega }_p$; for $\omega <{\omega }_p$ both transverse "electric" and "magnetic" fields are shielded except for the static ($\omega =0\mathrm{}$) case.