A relativistic quantum theory for processes in a plasma is developed in such a way that the theory reduces to that for a classical collisionless plasma in the classical limit and to conventional quantum electrodynamics in vacuo. The generalization involves two stages. The first stage is the use of quantum electrodynamics to calculate the linear and nonlinear responses of an electron gas. Explicit expressions are derived for the linear response of an electron gas in a static magnetic field, and for the lowest order nonlinear response of an electron gas in the absence of a static magnetic field. The second stage involves a generalization of the conventional diagrammatic technique to include the effects of the nonlinear responses of the medium. Additional diagrammatic structures called multiple-photon vertices are introduced. Rules are given for drawing the generalized diagrams and for writing down the associated amplitudes. The Cerenkov effect, photon splitting and emission in the presence of a static magnetic field are treated as illustrative examples.