Infra-red divergences in 2+1 dimensional gauge theories
Quantum field theories generally exhibit divergences. Ultra-violet divergences are treated through the renormalisation programme. Infra-red divergences, which accompany massless particles, are a characteristic of unbroken gauge theories and make it difficult to extract physical predictions. In this thesis we analyse various approaches to the infra-red problem and apply them to 2+1 dimensional gauge theories. These are useful as toy models, are related to the high temperature limit and are important in condensed matter physics. After briefly reviewing various responses to the infra-red problem in 3H-1 dimensions, we begin our study of gauge theories in 2+1 dimensions by performing a one loop renormalisation of various on-shell Green's functions. Both the fermionic and scalar theories are employed to study the spin dependence of the infra-red structures. Ward identities are explicitly verified and gauge dependence is analysed by calculating in different gauges. Following arguments due to Kulish and Faddeev we see that the asymptotic interaction in QED cannot be neglected before or after scattering. This means that, even at asymptotic times, QED has a non-trivial gauge symmetry and so the Lagrangian fermion cannot be identified with a physical field. We then introduce a systematic method to construct locally gauge invariant dressed fields which describe particles moving with a well-defined velocity. We then find that the mass shift and the wave function renormalisation constants are infra-red finite when these dressed solutions are used. The infra-red structure of scattering is also analysed. Finally, the Bloch-Nordsieck method is used to study the IR problem at the level of the inclusive cross-section. It is seen that this method breaks down in 2+1 dimensions. Some suggestions for future work conclude this thesis.