A gauge theory is non-abelian when its gauge group is non-commutative, \(g_1 g_2 \neq g_2 g_1\). QCD is the non-abelian theory with gauge group \(SU(3)_c\); Yang–Mills theory is the general name for such a construction. The contrast is the abelian \(U(1)\) of QED.

For \(SU(3)\) there are eight hermitian generators \(T^a\) (\(a=1,\dots,8\)) obeying the Lie algebra

\[ [T^a,T^b] = i f^{abc}\,T^c,\qquad \mathrm{Tr}(T^aT^b)=\tfrac12\,\delta^{ab}, \]

where \(f^{abc}\) are the (totally antisymmetric, non-zero) structure constants. It is precisely the non-vanishing of \(f^{abc}\) that makes the theory non-abelian and drives all the qualitatively new physics.

The non-abelian field strength picks up a quadratic term:

\[ F_{\mu\nu}^a = \partial_\mu A_\nu^a - \partial_\nu A_\mu^a + g\,f^{abc}A_\mu^b A_\nu^c . \]

Squaring it in \(-\tfrac14 F^a_{\mu\nu}F^{a\,\mu\nu}\) produces cubic and quartic self-couplings of the gluons. Physically, the gauge bosons themselves carry color charge and interact with one another — impossible in the abelian case.

• Asymptotic freedom: the one-loop running of the coupling is \[ \mu\frac{d g}{d\mu} = -\frac{g^3}{16\pi^2}\Big(\tfrac{11}{3}N_c - \tfrac{2}{3}N_f\Big), \]

which is negative for QCD (\(N_c=3\), \(N_f\le 16\)). The coupling weakens at short distance — the sign is a direct gift of the gluon self-interaction (the \(\tfrac{11}{3}N_c\) term).

• Confinement: at long distance the coupling grows strong, quarks are confined, and only color-singlet bound states exist.
• Non-trivial vacuum: instantons, the \(\theta\)-term and topological effects all follow from the rich structure of the non-abelian gauge group.

The transformation law of the gauge field,

\[ A_\mu \to g(x)\Big(A_\mu + \tfrac{i}{g}\partial_\mu\Big)g^\dagger(x), \]

now contains a genuine homogeneous rotation \(g A_\mu g^\dagger\) in addition to the inhomogeneous shift, reflecting that the \(A_\mu^a\) transform in the adjoint representation.