  A relativistic quantum field theory with nontrivial background fields is
developed and applied to study waves in plasmas. The effective action of the
electromagnetic 4-potential is calculated ab initio from the standard action of
scalar QED using path integrals. The resultant effective action is gauge
invariant and contains nonlocal interactions, from which gauge bosons acquire
masses without breaking the local gauge symmetry. To demonstrate how the
general theory can be applied, we study a cold unmagnetized plasma and a cold
uniformly magnetized plasma. Using these two examples, we show that all linear
waves well-known in classical plasma physics can be recovered from relativistic
quantum results when taking the classical limit. In the opposite limit,
classical wave dispersion relations are modified substantially. In unmagnetized
plasmas, longitudinal waves propagate with nonzero group velocities even when
plasmas are cold. In magnetized plasmas, anharmonically spaced Bernstein waves
persist even when plasmas are cold. These waves account for cyclotron
absorption features observed in spectra of X-ray pulsars. Moreover, cutoff
frequencies of the two non-degenerate electromagnetic waves are redshifted by
different amounts. These corrections need to be taken into account in order to
correctly interpret diagnostic results in laser plasma experiments.
