  Recently there has been great interest around quantum relativistic models for
plasmas. In particular striking advances have been obtained by means of the
Klein-Gordon-Maxwell system, which provides a first order approach to the
relativistic regimes of quantum plasmas. It is a reliable method as long as the
plasma spin dynamics is not a fundamental aspect, to be addressed using more
refined (and heavier) models involving the Pauli-Schr\"odinger or Dirac
equations. In this work a further simplification is considered, tracing back to
the early days of relativistic quantum theory. Namely, we revisit the
square-root Klein-Gordon-Poisson system, where the positive branch of the
relativistic energy-momentum relation is mapped to a quantum wave equation. The
associated linear wave propagation is analyzed and compared to the results in
the literature. We determine physical parameters where the simultaneous quantum
and relativistic effects can be noticeable in weakly coupled electrostatic
plasmas.
