Lecture X: Non-Abelian Gauge Theory and Strong InteractionIntroductionThe generalization of an Abelian gauge theory to an non-Abelian gauge theory is straightforward.However, its application in particle physics, which was first made by Yang and Mills, has a revolutionaryimpact. It means that gauge symmetry is not something accidentally appearing in EM field theory, but aguiding principle for describing fundamental interactions or constructing their theories.Non-Abelian Gauge TheoryFor a single Dirac spinor filedψ, the general Lagrangian with non-Abelian gauge symmetry isgiven byL=¯ψ(i γμDμ-m)ψ-14XaFaμνFaμν=ψ(i γμ∂μ-m)ψ-gXaAaμJaμ-14XaFaμνFaμν(1)Herearuns over all gauge fields. In this Lagrangian, “g” is gauge coupling which is an analogue of“e” in the EM theory. “ta” is a generator which is an analogue of electric charge number “q” in the EMtheory.Jaμ=¯ψγμtaψ(2)is the gauge current of the gauge fieldAaμ. The gauge transformation is defined asψ→(1-iXaθa(x)ta+O(θ2))ψAaμ→Aaμ-1g∂μθa(x)-Xb,cfabcAbμθc(x)(3)Again, the gauge fields are massless in an non-Abelian gauge theory, because their mass terms are notinvariant under gauge transformation. What is the difference between an Abelian and a non-Abeliangauge theory? Let’s make a comparison.