Existence of Confinement Phase in Quantum Electrodynamics

TL;DR

This paper demonstrates that four-dimensional U(1) gauge theory exhibits a confining phase at strong coupling, linked to vortex condensation and topological sectors, using a reformulation involving topological quantum field theory and dimensional reduction.

Contribution

It introduces a novel approach connecting continuum U(1) gauge theory confinement to the phase transition in the 2D O(2) nonlinear sigma model via topological quantum field theory reformulation.

Findings

Confinement occurs above a critical coupling g_c.

Confinement is associated with vortex condensation.

The phase transition aligns with the BKT transition in the O(2) model.

Abstract

We show that the four-dimensional U(1) gauge theory in the continuum formulation has a confining phase (exhibiting area law of the Wilson loop) in the strong coupling region above a critical coupling $g_c$. This result is obtained by taking into account topological non-trivial sectors in U(1) gauge theory. The derivation is based on the reformulation of gauge theory as a deformation of a topological quantum field theory and subsequent dimensional reduction of the D-dimensional topological quantum field theory to the (D-2)-dimensional nonlinear sigma model. The topological quantum field theory part of the four-dimensional U(1) gauge theory is exactly equivalent to the two-dimensional O(2) nonlinear sigma model. The confining (resp. Coulomb) phase of U(1) gauge theory corresponds to the high (resp. low) temperature phase of O(2) nonlinear sigma model and the critical point $g_c$ is determined by the Berezinskii-Kosterlitz-Thouless phase transition temperature. The quark (charge) confinement in the strong coupling phase is caused by the vortex condensation. Thus the continuum gauge theory has the direct correspondence to the compact formulation of lattice gauge theory.