The Yang-Mills Vacuum Wave Functional in 2+1 Dimensions

Abstract

We investigate Yang-Mills theory in 2+1 dimensions in the Schrödinger representation. Three dimensional Yang-Mills theory is relevant on the one hand, because it is the lowest dimensional Yang-Mills theory with propagating degrees of freedom,_x000D_ on the other hand, because it constitutes the high temperature limit of four dimensional QCD. The Schrödinger picture is interesting because it is well suited to explore properties of the vacuum state in the non-perturbative regime. Yet, not much analytical work has been done on this subject, and even the topic of perturbation theory in the Schrödinger representation is not well developed, especially in the case of gauge theories. In a paper by Hatfield [Phys. Lett. B 147, 435 (1984)] the vacuum wave functional for SU(2) theory was computed to O(e). In the non-perturbative regime, the most sophisticated analytical approach has been developed by Karabali et al. in a series of papers (see [Nucl. Phys. B 824, 387 (2010)] and references therein). This thesis aims to put perturbation theory in the Schrödinger representation on more solid ground by computing the vacuum wave functional for a general gauge group SU(N) up to O(e^2), utilizing modifications of these two methods. This is important since it provides us with a tool for the test of non-perturbative approaches, which should reproduce the perturbative result in an appropriate limit. _x000D_ In addition, regularization and renormalization are also not well understood in the Schrödinger picture. The regularization method proposed by Karabali et al. leads to conflicting results when applied to the computation of the vacuum wave functional with the two different methods mentioned above. We aim to clarify how regularization should be implemented and develop a new regularization approach, which brings these two expressions into agreement, giving a strong check of the regularization employed. We argue that this regularization procedure is not specific to the cases studied here. It should be applied in the same way to any quantum field theory in any dimension in the Schrödinger picture. This is the main result of the thesis._x000D_ We then go on to illustrate how physical observables can be computed in the non-perturbative regime, using the trial wave functional proposed in [Nucl. Phys. B 824, 387 (2010)]._x000D_ Among other observables, we compute the static potential at long distances, for which we find corrections not compatible with a linear potential. _x000D_ Finally, we also discuss the possibility of extending this approach to 3+1 dimensions.

Date of Award	25 Feb 2014
Original language	Undefined/Unknown
Supervisor	Antonio Miguel Pineda Ruiz (Director)