Relativistic quantum plasma dispersion functions (RQPDFs) are defined and the longitudinal and transverse response functions for an electron (plus positron) gas are written in terms of them. The dispersion is separated into Landau-damping, pair-creation and dissipationless regimes. Explicit forms are given for the RQPDFs in the cases of a completely degenerate distribution and a nondegenerate thermal (Jüttner) distribution. Particular emphasis is placed on the relation between dissipation and dispersion, with the dissipation treated in terms of the imaginary parts of RQPDFs. Comparing the dissipation calculated in this way with the existing treatments leads to the identification of errors in the literature, which we correct. We also comment on a controversy as to whether the dispersion curves in a superdense plasma pass through the region where pair creation is allowed.

Relativistic effects on dispersion in a degenerate electron gas are discussed by comparing known response functions derived relativistically and non-relativistically. The main distinguishing feature is one-photon pair creation, which leads to logarithmic singularities in the response functions. Dispersion curves for longitudinal waves have a similar tongue-like appearance in the relativistic and non-relativistic case, with the main relativistic effects being on the Fermi speed and the cutoff frequency. For transverse waves the non-relativistic treatment has a non-physical feature near the cutoff frequency for large Fermi momenta, and this is attributed to an incorrect treatment of the electron spin. We find (with two important provisos) that one-photon pair creation is allowed in superdense plasmas, implying relatively strong coupling between transverse waves and pair creation.

Pair modes were identified by Hines and collaborators as longitudinal waves in a relativistic quantum plasma of spin 0 particles in the completely degenerate limit. Pair modes with both longitudinal and transverse polarization also exist in spin 1 plasmas in the completely degenerate limit. We consider the possible existence of pair modes in completely degenerate, spin ½ plasmas, with a negative conclusion. It is argued that pair modes are associated with a pole (in the response function) that is present only in the presence of a Bose condensate. The properties of these wave modes, including their energetics, are discussed.