A gauge theory is abelian when its gauge group is commutative, i.e. the group elements satisfy \(g_1 g_2 = g_2 g_1\). The archetype is QED, whose gauge group is \(U(1)\): elements \(g(x)=e^{i\alpha(x)}\) are just phases and trivially commute. The non-abelian case (\(SU(3)_c\) for QCD) is the contrasting situation.

For \(U(1)\) the generator is a single number, so there is one gauge field, the photon \(A_\mu(x)\). Under a local transformation

\[ \psi\to e^{i\alpha(x)}\psi,\qquad A_\mu\to A_\mu+\frac{1}{e}\,\partial_\mu\alpha . \]

Because the group is abelian the transformation of \(A_\mu\) has no homogeneous rotation piece — only the shift by \(\partial_\mu\alpha\).

The field strength reduces to the commutator of covariant derivatives, but the \([A_\mu,A_\nu]\) term vanishes:

\[ F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu . \]

There is no \(f^{abc}A^b A^c\) term because the structure constants of an abelian group are zero. Consequently:

• the photon carries no charge and does not couple to itself;
• Maxwell's equations are linear;
• the coupling does not exhibit asymptotic freedom (in QED the effective charge grows with energy).

This is the essential structural difference from QCD, where the gauge bosons (gluons) are charged and self-interacting. The construction of the QCD Lagrangian density is nevertheless formally very close to the abelian \(U(1)\) case — one simply replaces ordinary products by matrix products and keeps the extra commutator terms.