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