The covariant derivative \(D_\mu\) is the modification of the ordinary derivative \(\partial_\mu\) that transforms homogeneously under a local gauge transformation, i.e. in the same way as the field it acts on. It is what makes a gauge theory possible: it repairs the invariance that \(\partial_\mu\) alone would spoil.

For QCD, with gauge field \(A_\mu=A_\mu^aT^a\) and coupling \(g\),

\[ D_\mu = \partial_\mu - i g A_\mu . \]

Under \(\psi\to g(x)\psi\) with \(g(x)=e^{i\alpha^a(x)T^a}\), one demands

\[ D_\mu\psi\ \to\ g(x)\,D_\mu\psi , \]

which fixes the gauge-field transformation law

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

Because \(D_\mu\psi\) rotates like \(\psi\), the bilinear \(\overline\psi\,i\gamma^\mu D_\mu\psi\) is gauge invariant.

\(D_\mu\) compares the field at neighbouring points after parallel-transporting the color frame between them. This transport is implemented by the comparator (Wilson line) \(U(y,x)\) used in Continuum QCD:

\[ \eta^\mu D_\mu\psi(x)=\lim_{\epsilon\to0}\frac{\psi(x+\eta\epsilon)-U(x+\eta\epsilon,x)\,\psi(x)}{\epsilon},\qquad U(x+\eta\epsilon,x)=\mathbb{1}+ig\epsilon\,\eta^\mu A_\mu+\mathcal{O}(\epsilon^2). \]

\(A_\mu\) is thus a connection, and the field strength is its curvature,

\[ F_{\mu\nu}=\frac{-i}{g}[D_\mu,D_\nu], \]

measuring the non-commutativity of transport around a closed loop — the seed of the gluon self-interaction in the non-abelian theory.