Continuum Regularized Yang-Mills Theory.

Using the machinery of stochastic quantization, Z. Bern, M. B. Halpern, C. Taubes and I recently proposed a continuum regularization technique for quantum field theory. This regularization may be implemented by applying a regulator to either the (d + 1)-dimensional Parisi-Wu Langevin equation or, equivalently, to the d -dimensional second order Schwinger-Dyson (SD) equations. This technique is non-perturbative, respects all gauge and Lorentz symmetries, and is consistent with a ghost-free gauge fixing (Zwanziger's). This thesis is a detailed study of this regulator, and of regularized Yang-Mills theory, using both perturbative and non-perturbative techniques. The perturbative analysis comes first. The mechanism of stochastic quantization is reviewed, and a perturbative expansion based on second-order SD equations is developed. A diagrammatic method ("SD diagrams") for evaluating terms of this expansion is developed. We apply the continuum regulator to a scalar field theory. Using SD diagrams, we show that all Green functions can be rendered finite to all orders in perturbation theory. Even non-renormalizable theories can be regularized. The continuum regulator is then applied to Yang -Mills theory, in conjunction with Zwanziger's gauge fixing. A perturbative expansion of the regulator is incorporated into the diagrammatic method. SD diagrams are used to calculate the 2-point function to one loop order, and the gluon mass is seen to vanish. A non-perturbative analysis of the Langevin equation for regularized Yang-Mills follows. The existence of a solution in 3 dimensions is argued. Specifically, the noise and drift terms are separately shown to be incapable of producing UV divergences in finite time. The analysis is then repeated for an arbitrary compact smooth 3-manifold (instead of R ^3), where no IR divergences can occur, with similar results. It is hoped that the techniques discussed in this thesis will contribute to the construction of a renormalized Yang-Mills theory in 3 and 4 dimensions.