The Yang-Mills field strength revisited

The Yang-Mills field strength incorporating a non-Abelian feature is one of the cornerstones of the standard model. Although Yang-Mills gauge theories have been around for over fifty years, surprisingly the derivation of the Yang-Mills field strength using classical gauge theory does not appear anywhere in the literature. In their 1954 paper, Yang and Mills had to invent a non-Abelian field strength to satisfy certain criteria. In Section 5 we use Yang’s gauge transformation in a heuristic derivation of the Yang-Mills field strength. The preceding sections cover material relating to the derivation. Section 3 shows where Pauli in the article cited by Yang and Mills gives an expression for the electro-magnetic field strength in terms of a commutator. For some reason, Yang and Mills did not use this approach.

💡 Research Summary

The paper “The Yang‑Mills field strength revisited” addresses a conspicuous gap in the literature: a purely classical gauge‑theoretic derivation of the non‑Abelian Yang‑Mills field strength tensor, (F_{\mu\nu}). Although Yang and Mills introduced their gauge theory in 1954 and the field strength has become a cornerstone of the Standard Model, textbooks and review articles typically present the result without showing how it follows directly from the gauge transformation law. The authors set out to fill this void by retracing the historical and mathematical steps that lead from the gauge transformation to the full non‑Abelian field strength.

The paper begins with a concise historical overview. It notes that while Yang and Mills invented a non‑Abelian field strength to satisfy certain physical criteria, they did not employ the commutator‑based expression that Wolfgang Pauli had already used for the electromagnetic field in 1932. Pauli’s formulation expressed the electromagnetic field tensor as a commutator of covariant derivatives, \