Quantum chromodynamics

Remarks

Gauge invariance: for a function $U$ with values in $SU(3)$, the transformation $\psi_f \mapsto U\psi_f$, $A_\mu \mapsto U A_\mu U^{-1} - \frac{i}{g} (\partial_\mu U) U^{-1}$ gives $D_\mu \psi_f \mapsto U D_\mu \psi_f$ and $F_{\mu\nu} \mapsto U F_{\mu\nu} U^{-1}$. So each term of $\mathcal{L}$ is unchanged, because the trace is invariant under conjugation. In a basis $T^a$ of the Hermitian traceless matrices with $\mathrm{tr}(T^a T^b) = \frac12 \delta^{ab}$, the gluon term is $-\frac14 F^a_{\mu\nu} F^{a\,\mu\nu}$, with $8$ gluon fields $A^a_\mu$.

Non-example: replacing $SU(3)$ by $U(1)$ gives a theory of the form of quantum electrodynamics, whose gauge field has no self-interaction, because the commutator term in $F_{\mu\nu}$ vanishes.

In nature $N_f = 6$, with the up, down, strange, charm, bottom and top quarks. The theory is asymptotically free when $N_f \le 16$, as found by Gross, Wilczek and Politzer in 1973. A rigorous construction of QCD is open, like the Yang–Mills existence problem.