Non-abelian gauge theory
A gauge theory whose gauge group is non-commutative, so the gauge fields themselves carry charge under the group and the field strength tensor acquires a self-interaction term. Yang-Mills 1954 introduced the construction; QCD and the weak force are non-abelian; QED is the abelian exception.
Definition
A non-abelian gauge theory is a gauge theory whose gauge group does not commute — that is, two gauge transformations performed in different orders give different results. The condition T^a T^b ≠ T^b T^a translates into the structure-constant relation [T^a, T^b] = i f^{abc} T^c, with f^{abc} the non-zero structure constants of the gauge group's Lie algebra. Yang and Mills introduced the construction in 1954 by generalising Weyl's 1929 abelian gauge principle to the SU(2) symmetry of nuclear isospin.
The defining feature of non-abelian gauge theory is that the field strength tensor is no longer linear in the gauge potential: F^{a}_{μν} = ∂_μ A^a_ν − ∂_ν A^a_μ + g f^{abc} A^b_μ A^c_ν. The extra non-linear term g f^{abc} A^b_μ A^c_ν means that the gauge bosons themselves carry charge under the gauge group and self-interact at every order in the coupling. In QCD this self-interaction is the source of asymptotic freedom (the running coupling decreases at high energy because gluon-gluon loops dominate quark-antiquark loops in the beta function) and confinement (the same self-interaction prevents quarks from being separated to large distances). The mathematical machinery — covariant derivatives D_μ = ∂_μ + i g A^a_μ T^a, gauge-covariant field strengths, lattice regularisations — is universal across all non-abelian theories. The Yang-Mills 1954 paper is one of the most-cited results in twentieth-century theoretical physics; the entire Standard Model is built on its construction.