Non-perturbative techniques in QCD

The problems of a perturbative approach to Quantum Chromo Dynamics (QCD) are outlined and the Euclidean space - time lattice regularisation is introduced. We discuss the Monte Carlo procedure and use the mechanism to study the potential between charged particles in U(1) Lattice Gauge Theory in 3 dimensions, a theory having similarities with QCD. We compare two approaches to this problem and conclude that the standard source free method is inferior to the dual Polymer approach in attempting to extract a string tension for this theory. We make use of a technique due to Parisi to reduce the statistical fluctuations in the Monte Carlo data, applying it to a study of SU(2) Lattice Gauge Theory in 3 dimensions. The technique reduces our Monte Carlo errors considerably and we are able to dectect clear signs of scaling. We apply Mean Field Theory to a variant action representation of SU(2) Lattice Gauge Theory studied by Anthony. We find the method locates the position of the phase transition with remarkably good accuracy.