Yang-Mills and Beyond

A s recently as 2010, three groups received the J. J. Sakurai Prize for showing how spontaneous symmetry breaking explains the mechanism for generating vector boson masses, thus enabling the solution of one of the puzzles of the standard model of elementary particles. Since the Yang-Mills field strength incorporating a nonabelian feature is not only one of the cornerstones of the standard model, which is the basis of the ongoing search for the Higgs boson, but also since Yang-Mills theory influenced the development of algebraic and differential geometry (e.g., see Donaldson [1] and [2], and Dragomir et al. [3]), perhaps it is not out of order to explore the development of the formula for the field strength, F . In modern notation it is given in the form F = dB + B ∧ B, where B is the gauge field. Although Yang-Mills (YM) gauge theory can be done using fiber-bundle theory—see, for instance, the review articles of Daniel and Viallet [4] and Marateck [5]—it is historically useful to analyze the way it developed using classical gauge theory. In their seminal paper [6], Yang and Mills invented the non-abelian field strength to satisfy certain criteria but did not explain how it could be derived. In the penultimate section here we show how the Yang-Mills field strength derives from Yang’s gauge transformation. The preceding sections place Yang-Mills theory in historical perspective and cover material relating to the field