Spatial Geometry and the Wu-Yang Ambiguity

TL;DR

This paper constructs continuous families of SU(2) gauge potentials in three dimensions that produce identical magnetic fields, revealing a geometric interpretation through a spatial metric and connection derived from the gauge fields.

Contribution

It introduces a geometric framework linking gauge potentials with spatial geometry, providing explicit families of potentials sharing the same magnetic field in SU(2) gauge theory.

Findings

Identifies continuous families of gauge potentials with identical magnetic fields.

Establishes a geometric mapping of gauge fields into a spatial metric and connection.

Provides explicit examples of the Wu-Yang ambiguity in SU(2) gauge theory.

Abstract

We display 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 $B$.