The Berry Phase and Monopoles in Non-Abelian Gauge Theories

TL;DR

This paper explores the geometric (Berry) phase in SU(2) gauge theory, introduces monopole-like defects via Wilson loops, and discusses their properties and physical implications in lattice simulations.

Contribution

It presents a novel gauge-dependent construction of monopoles in non-Abelian gauge theories using Berry phases and Wilson loops, with a focus on the Lorenz gauge.

Findings

Eigenvalues of Wilson loops decompose into geometric and dynamical phases.

Unique U(1) gauge rotations leave the Berry phase invariant.

Constructed monopoles exhibit correct continuum limit in Lorenz gauge.

Abstract

We consider the quantum mechanical notion of the geometrical (Berry) phase in SU(2) gauge theory, both in the continuum and on the lattice. It is shown that in the coherent state basis eigenvalues of the Wilson loop operator naturally decompose into the geometrical and dynamical phase factors. Moreover, for each Wilson loop there is a unique choice of U(1) gauge rotations which do not change the value of the Berry phase. Determining this U(1) locally in terms of infinitesimal Wilson loops we define monopole-like defects and study their properties in numerical simulations on the lattice. The construction is gauge dependent, as is common for all known definitions of monopoles. We argue that for physical applications the use of the Lorenz gauge is most appropriate. And, indeed, the constructed monopoles have the correct continuum limit in this gauge. Physical consequences are briefly discussed.