New Look at QED$_4$: the Photon as a Goldstone Boson and the Topological Interpretation of Electric Charge

TL;DR

This paper presents a dual formulation of 3+1D QED, showing the photon as a Goldstone boson resulting from magnetic symmetry breaking, with electric charge interpreted topologically via magnetic vortices.

Contribution

It introduces a dual picture of QED where the photon emerges as a Goldstone boson and interprets electric charge topologically through magnetic vortex winding numbers.

Findings

Photon is massless in the Coulomb phase due to magnetic symmetry breaking.

Electric charge is topologically quantized as a winding number.

Higgs-Coulomb transition corresponds to magnetic vortex condensation.

Abstract

We develop the dual picture for Quantum Electrodynamics in 3+1 dimensions. It is shown that the photon is massless in the Coulomb phase due to spontaneous breaking of the magnetic symmetry group. The generators of this group are the magnetic fluxes through any infinite surface $\Phi_S$. The order parameter for this symmetry breaking is the operator $V(C)$ which creates an infinitely long magnetic vortex. We show that although the order parameter is a stringlike rather than a local operator, the Goldstone theorem is applicable if $<V(C)>\ne 0$. If the system is properly regularized in the infrared, we find $<V(C)>\ne 0$ in the Coulomb phase and $V(C)=0$ in the Higgs phase. The Higgs - Coulomb phase transition is therefore understood as condensation of magnetic vortices. The electric charge in terms of $V(C)$ is topological and is equal to the winding number of the mapping from a circle at spatial infinity into the manifold of possible vacuum expectation values of a magnetic vortex in a given direction. Since the vortex operator takes values in $S^1$ and $\Pi_1(S^1)={\cal Z}$, the electric charge is quantized topologically.