Continuum Regularized YangMills Theory.
Abstract
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 ParisiWu Langevin equation or, equivalently, to the d dimensional second order SchwingerDyson (SD) equations. This technique is nonperturbative, respects all gauge and Lorentz symmetries, and is consistent with a ghostfree gauge fixing (Zwanziger's). This thesis is a detailed study of this regulator, and of regularized YangMills theory, using both perturbative and nonperturbative techniques. The perturbative analysis comes first. The mechanism of stochastic quantization is reviewed, and a perturbative expansion based on secondorder 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 nonrenormalizable 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 2point function to one loop order, and the gluon mass is seen to vanish. A nonperturbative analysis of the Langevin equation for regularized YangMills 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 3manifold (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 YangMills theory in 3 and 4 dimensions.
 Publication:

Ph.D. Thesis
 Pub Date:
 1987
 Bibcode:
 1987PhDT.......173S
 Keywords:

 Physics: Elementary Particles and High Energy