The Wu-Yang Ambiguity Revisited

TL;DR

The paper explores the Wu-Yang ambiguity by providing examples of SU(2) vector potentials that produce identical magnetic fields, utilizing a geometric approach linking gauge fields to spatial geometry with torsion.

Contribution

It revisits the Wu-Yang ambiguity through explicit examples and a geometric framework connecting gauge potentials to a spatial geometry with torsion.

Findings

Multiple vector potentials yield the same magnetic field.

A geometric mapping relates gauge fields to a spatial geometry with torsion.

The approach offers new insights into gauge field ambiguities.

Abstract

Several examples are given of continuous families of SU(2) vector potentials $A_i^a(x)$ in 3 space dimensions which generate the same magnetic field $B^{ai}(x)$ (with det $B\neq 0$). These Wu-Yang families are obtained from the Einstein equation $R_{ij}=-2G_{ij}$ derived recently via a local map of the gauge field system into a spatial geometry with $2$-tensor $G_{ij}=B^a{}_i B^a{}_j\det B$ and connection $\Gamma_{jk}^i$ with torsion defined from gauge covariant derivatives of $B$.