Gauge theories in three dimensions

Field theories in 2+ 1 space-time dimensions are of interest both intrinsically, due to their novel properties such as actions which are topologically non-trivial, and also due to their ability to explain of phenomena such as the fractional quantum Hall effect and certain behaviour of high Tc superconductors, and for their use in conformal field theory in 2D. This thesis begins by considering scalar and spinor QED in 2+ 1 dimensions, performing perturbation theory to study its behaviour (without allowing the presence or dynamical generation of a parity-violating photon mass). It is found, as first noted by Jackiw and Templeton, that an IR instability prohibits such a perturbative study. The gauge technique is adopted as a non-perturbative alternative, and the photon is allowed to be \dressed\" in a cloud of fermion loops yielding results which encompass the perturbation results in the UV region whilst remaining finite at IR momenta. Chern-Simons theory is then considered where the photon is allowed to acquire a parity-violating mass. In order to use dimensional regularization to handle the. apparently UV divergent integrals which appear a new formulation of the theory is proposed allowing the action to be written in arbitrary D dimensions so that the integrals can be safely evaluated. It is also found that the IR problems which plague the conventional theory are no longer present as the photon propagator behaviour has been \"softened\" by the photon mass allowing perturbation results to be obtained. Finally the idea of mass generation within these theories is considered in more detail where we see that the presence of a fermion mass will cause a photon mass to be dynamically generated and vice versa. These ideas are then generalized for arbitrary odd dimensional parity-violating theories."