Gauge invariant method for maximum simplification of the field strength in non-Abelian Yang-Mills theories

TL;DR

This work presents a local, gauge-invariant framework to maximally simplify the $SU(2)$ Yang–Mills field strength by extending a tetrad-based diagonalization approach from electromagnetism to non-Abelian theories. It introduces a non-Abelian extremal field and a duality-rotation construction that yields gauge-invariant tetrad skeletons, along with three new local gauge-invariant objects that guide a four-condition block-diagonalization procedure acting on isospin projections. The approach results in a locally block-diagonal field strength and identifies six local observables derived from three isospin directions, potentially extending to $SU(3)$ and offering a geometrical perspective on YM degrees of freedom. Overall, the method harmonizes gauge and gravitational structures in four-dimensional spacetime, enabling clearer analytical handling of YM fields and providing a foundation for further exploration of their Dirac–Bergmann properties.

Abstract

A new local gauge invariant method is introduced in order to maximally simplify the expression for a SU(2) non-Abelian field strength. The new tetrads introduced in previous works are going to play a fundamental role in the algorithm presented in this manuscript. Three new local gauge invariant objects are going to guide us through the process of making a field strength block diagonal. The process is also covariant. Any non-trivial isospace field strength projection will become block diagonal through this gauge invariant algorithm. As an application we will find new local observables in Yang-Mills theories.