Quark confinement and topology of gauge theories

Abstract

The phenomenon of quark confinement is known to be connected with the restoration of apparently broken gauge symmetry. In this paper we focus on a special mechanism which is responsible for such restoration. The major suggestion is that in the treatment of infrared problems certain classical field trajectories are of paramount importance, which trajectories connect seemingly degenerate vacua thereby eliminating the degeneracy. Initially, we demonstrate this mechanism in certain simple non-gauge models. In order of increasing difficulty, we next examine compact quantum electro-dynamics in 2 + 1 space-time dimensions- essentially a variant of unified models of the Georgi-Glashow type. In this case we prove that charge is confined and show that the force between two charges is independent of the distance between them. For small values of the fine structure constant this force is explicitly evaluated. Finally, we turn to the more realistic case of a 3 + 1 dimensional non-Abelian gauge theory, and analyse the contribution of a single pseudoparticle to the correlation functions. It is proposed that the quantum fluctuations of the pseudoparticle are inessential, and that the one-loop approximation is effectively correct even for the large scale pseudoparticles. The emergent conclusion then is that the renormalized Yang-Mills theory is reduced to the problem of evaluating the effects of those configurations which involve many pseudoparticles, due account being taken of the interactions between them. Some aspects of this last problem are also discussed.