On pure Yang-Mills theory in 3+1 dimensions: Hamiltonian, vacuum and gauge invariant variables

TL;DR

This paper explores an analytic approach to solving pure Yang-Mills theory in 3+1 dimensions using gauge-invariant variables, extending methods successful in 2+1 dimensions to better understand QCD vacuum structure and strong coupling behavior.

Contribution

It introduces a generalization of Bars' gauge-invariant variables to 3+1 dimensions, develops a regularization preserving invariance, and analyzes the vacuum wave functional in this framework.

Findings

Regularization preserves generalized holomorphic invariance.

Vacuum wave functional behavior characterized at weak and strong coupling.

First local eigenstate of the kinetic Hamiltonian computed.

Abstract

In this work we discuss an analytic approach towards the solution of pure Yang-Mills theory in 3+1 dimensional spacetime which strongly suggests that the recent strategy already applied to pure Yang-Mills theory in 2+1 can be extended to 3+1 dimensions. We show that the local gauge invariant variables introduced by Bars gives a natural generalisation to any dimension of the formalism of Karabali and Nair which recently led to a new understanding of the physics of QCD in dimension 2+1. After discussing the kinematics of these variables, we compute the jacobian between the Yang-Mills and Bars variables and propose a regularization procedure which preserves a generalisation of holomorphic invariance. We discuss the construction of the QCD hamiltonian properly regularized and compute the behavior of the vacuum wave functional both at weak and strong coupling. We argue that this formalism allows the developpement of a strong coupling expansion in the continuum by computing the first local eigenstate of the kinetic part of Yang-Mills hamiltonian.