A Non-Abelian Fourier Transform for Gauge Theories

TL;DR

This paper introduces a non-Abelian Fourier transform for SU(2) gauge potentials in compactified spaces, transforming the gauge theory into an Abelian-like theory with additional fields, potentially simplifying analysis.

Contribution

It defines a gauge-invariant Fourier transform for non-Abelian gauge potentials, leading to a novel Abelian formulation of gauge theories with new field components.

Findings

Fourier coefficients are nearly gauge invariant.

The gauge theory becomes an Abelian theory with extra fields.

The formalism may simplify analysis of gauge theories.

Abstract

We consider SU(2) gauge potentials over a space with a compactified dimension. A non-Abelian Fourier transform of the gauge potential in the compactified dimension is defined in such a way that the Fourier coefficients are (almost) gauge invariant. The functional measure and the gauge field strengths are expressed in terms of these Fourier coefficients. The emerging formulation of the non-Abelian gauge theory turns out to be an Abelian gauge theory of a set of fields defined over the initial space with the compactified dimension excluded. The Abelian theory contains an Abelian gauge field, a scalar field, and an infinite tower of vector matter fields, some of which carry Abelian charges. Possible applications of this formalism are discussed briefly.