Yang-Mills equations
The non-abelian generalisation of Maxwell's equations: D_μ F^{aμν} = J^{aν}, where the covariant derivative D_μ = ∂_μ + i g [A_μ, ·] couples the gauge field to itself through the structure constants of the gauge group. The field equations of every non-abelian gauge theory in physics.
Definition
The Yang-Mills equations are the non-abelian generalisation of Maxwell's equations to gauge theories with arbitrary Lie-group structure. For a gauge group with generators T^a and structure constants f^{abc}, the field strength tensor is F^{a}_{μν} = ∂_μ A^a_ν − ∂_ν A^a_μ + g f^{abc} A^b_μ A^c_ν, and the equations of motion are D_μ F^{aμν} = J^{aν}, where the gauge-covariant derivative D_μ = ∂_μ + i g A^b_μ T^b acts on field-tensor components by the appropriate adjoint representation. The non-abelian part of D_μ contributes a term g f^{abc} A^b_μ F^{cμν} to the divergence — gluons sourcing gluons, W bosons sourcing W bosons.
For an abelian gauge group like U(1), the structure constants vanish, the covariant derivative reduces to the ordinary partial derivative, and the Yang-Mills equations reduce to Maxwell's: ∂_μ F^{μν} = J^ν. For non-abelian groups, the equations are highly non-linear and admit no general analytical solution; instanton solutions in Euclidean SU(2) Yang-Mills theory (Belavin-Polyakov-Schwarz-Tyupkin 1975) and lattice gauge-theory simulations are the workhorses. The Bianchi identity D_μ *F^{aμν} = 0 holds automatically as a consequence of F = dA + A∧A, and the gauge invariance under A_μ → U(x) A_μ U(x)⁻¹ + (i/g) U(x) ∂_μ U(x)⁻¹ generalises the abelian gauge transformation. The 2000 Clay Mathematics Institute "Yang-Mills mass gap" Millennium Prize Problem asks for a proof that pure SU(N) Yang-Mills theory in four dimensions has a mass gap; this remains one of the seven unsolved problems of mathematics, and a $1 million bounty awaits anyone who can prove that the lightest excitation has positive mass.