YangMillsteori

Yang-Mills theory is a class of gauge theories that describe interactions through non-Abelian gauge fields and generalize Maxwell's electromagnetism. Developed by Chen-Ning Yang and Robert Mills in 1954, it provides a mathematical framework for describing fundamental forces in terms of gauge symmetry with non-commuting group generators.

The theory is built from a gauge group G with gauge fields A_μ = A_μ^a T^a, where T^a are generators of the Lie algebra. The field strength is F_{μν} = ∂_μ A_ν - ∂_ν A_μ - i g [A_μ, A_ν], and the standard Lagrangian is L = -1/2 Tr(F_{μν} F^{μν}) (often with additional matter fields). Gauge invariance under local transformations U(x) in G implies self-interactions of the gauge fields, a feature distinguishing non-Abelian theories from Abelian ones. The classical equations of motion are D_μ F^{μν} = J^ν, where D_μ is the covariant derivative. In vacuum, J^ν=0.

Quantum mechanically, non-Abelian Yang-Mills theories are renormalizable in four dimensions and exhibit asymptotic freedom, meaning interactions become weak at high energies. This underpins the Standard Model, with quantum chromodynamics described by SU(3) Yang-Mills theory and the electroweak sector by SU(2)×U(1) gauge theory, later broken by the Higgs mechanism to give mass to W and Z bosons while the photon remains massless.

Topological configurations, such as instantons, and terms like the theta term enrich the theory's vacuum structure. Yang-Mills theories have been extensively tested in high-energy experiments and studied nonperturbatively by lattice gauge theory, providing insights into confinement and hadron structure.