Natural Symmetries of the Yang-Mills Equations

TL;DR

This paper classifies all natural symmetries of the Yang-Mills equations, showing they originate solely from gauge transformations, using a novel spinorial coordinate system on the jet bundle.

Contribution

It introduces a spinorial coordinate system on the jet bundle and provides a complete classification of natural symmetries of the Yang-Mills equations.

Findings

All natural symmetries derive from gauge transformations.

The spinorial coordinate system simplifies symmetry classification.

The classification is comprehensive and systematic.

Abstract

We define a natural generalized symmetry of the Yang-Mills equations as an infinitesimal transformation of the Yang-Mills field, built in a local, gauge invariant, and Poincar\'e invariant fashion from the Yang-Mills field strength and its derivatives to any order, which maps solutions of the field equations to other solutions. On the jet bundle of Yang-Mills connections we introduce a spinorial coordinate system that is adapted to the solution subspace defined by the Yang-Mills equations. In terms of this coordinate system the complete classification of natural symmetries is carried out in a straightforward manner. We find that all natural symmetries of the Yang-Mills equations stem from the gauge transformations admitted by the equations.