Yang-Mills fields as optical media

TL;DR

This paper explores a geometric interpretation of Yang-Mills fields as optical systems, linking gauge theories to 3-space geometries and examining implications of curvature and torsion in these representations.

Contribution

It introduces a novel geometrization of SU(2) gauge theory, connecting it to 3-space geometries and analyzing the role of torsion and curvature in representing gauge fields.

Findings

Yang-Mills fields can be represented by 3-space geometries.

The Wu-Yang ambiguity relates to multiple torsions of metric-preserving connections.

Symmetric homogeneous spaces yield Christoffel connections as solutions to Yang-Mills equations.

Abstract

A geometrization of the Yang-Mills field, by which an SU(2) gauge theory becomes equivalent to a 3-space geometry - or optical system - is examined. In a first step, ambient space remains Euclidean and current problems on flat space can be looked at from a new point of view. The Wu-Yang ambiguity, for example, appears related to the multiple possible torsions of distinct metric-preserving connections. In a second step, also the ambient space becomes curved. In general, the strictly Riemannian, metric sector plays the role of an arbitrary host space, with the gauge field represented by a contorsion. For some field configurations, however, it is possible to obtain a purely metric representation. In those cases, if the space is symmetric homogeneous the Christoffel connections are automatically solutions of the Yang-Mills equations.