This thesis investigates ultra-relativistic processes of thermal production of electrons and positrons, and the theories involved in their description. It first focuses on collisional ionisation and excitation rates in a partially-ionised plasma, presenting correction factors to them that account for special relativity in the free electron motion as a function of temperature and the threshold energy. These results are extended to de-excitation and three body recombination using detailed balance.

It then turns to the production of electron-positron pairs from Breit-Wheeler two-photon collision in the black-body field, and presents an analytic expression for the high-temperature limit of the total rate. It then examines the same physical process using the formalism of "external fields", in which the black-body is treated as as a thermal ensemble of classical field profiles. It presents a novel formal scheme for this calculation, and then presents the results of a numerical scheme that approximates this field ensemble by 1D spatial "Sauter pulses". It is found that pair-production calculated by this means exceeds that calculated by Breit-Wheeler by a large factor, and the energy spectrum of produced particles is presented.

Then, work comparing different theoretical formalisms of vacuum-destabilising background fields in QED is presented. It is first shown how solving the Dirac equation with peculiar boundary conditions can give probability amplitudes for fermion scattering and pair creation/annihilation. It is then demonstrated for a broad class of external fields that four fermion propagators used in the literature are equivalent: Schwinger’s "proper-time” propagator; the "causal propagator” used in the "Bogoliubov transformation” method; and two defined using analytic continuation. To do so, we re-derive Schwinger’s proper-time expression for the propagator as a statement relating solutions of the inhomogeneous Dirac equation to those of the inhomogeneous "proper-time Dirac equation”. We then show that all four propagators return solutions of the inhomogeneous Dirac equation that satisfy the same boundary condition.