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

TL;DR

This paper advances the analytical understanding of the Yang-Mills vacuum wave functional in 2+1 dimensions within the Schrödinger representation, focusing on perturbative calculations, regularization, and consistency between methods.

Contribution

It computes the vacuum wave functional for SU(N_c) gauge theories up to order e^2, clarifies regularization procedures, and develops a new regularization approach applicable to quantum field theories.

Findings

Computed vacuum wave functional up to O(e^2) for SU(N_c).

Resolved inconsistencies in regularization methods.

Proposed a general regularization approach for Schrödinger picture.

Abstract

We investigate Yang-Mills theory in 2+1 dimensions in the Schroedinger representation. The Schroedinger 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 Schroedinger 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 Schroedinger representation on more solid ground by computing the vacuum wave functional for a general gauge group SU$(N_c)$ up to O$(e^2)$, utilizing modifications of these two methods. This is important since it provides us with a tool for testing non-perturbative approaches, which should reproduce the perturbative result in an appropriate limit. Furthermore, regularization and renormalization are also not well understood in the Schroedinger 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, providing 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 Schroedinger picture.