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Yang-Mills Theory

QCD is a manifestation of Yang-Mills theory, describing fundamental forces mediated by gauge fields (gluons) and involving symmetry groups.
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The statement of the theorem

Let GG be a compact, semi-simple Lie group (e.g., SU(N)SU(N)). The Yang-Mills Lagrangian density LYM\mathcal{L}_{YM} is defined for the gauge field Aμ=AμaTaA_{\mu} = A^a_{\mu} T^a as: LYM=−14Tr(FμνFμν)\mathcal{L}_{YM} = -\frac{1}{4} \text{Tr}(F_{\mu\nu} F^{\mu\nu}) where the field strength tensor FμνF_{\mu\nu} is given by the commutator: Fμν=∂μAν−∂νAμ+ig2g[Aμ,Aν]F_{\mu\nu} = \partial_{\mu} A_{\nu} - \partial_{\nu} A_{\mu} + i g \frac{\sqrt{2}}{g} [A_{\mu}, A_{\nu}].