YangMillsteorier

Yang-Mills theories are a class of gauge theories based on non-abelian Lie groups that describe how elementary particles interact through gauge bosons. In these theories, the gauge fields A_μ^a act as connections on a principal G-bundle, and the dynamics are governed by the field strength F_μν^a = ∂_μ A_ν^a − ∂_ν A_μ^a + g f^{abc} A_μ^b A_ν^c. The covariant derivative D_μ = ∂_μ − i g A_μ^a T^a encodes local invariance under the Lie group G. The non-abelian structure implies that gauge bosons carry charge and interact with each other, producing self-interaction terms in the Lagrangian.

Historically, Yang-Mills theory was introduced in 1954 by Chen-Ning Yang and Robert Mills as a non-abelian generalization of electromagnetism. The framework underpins the Standard Model of particle physics: quantum chromodynamics uses the SU(3) gauge group with eight gluons, while the electroweak sector uses SU(2) × U(1) with W and Z bosons and the photon arising from spontaneous symmetry breaking.

Key properties of Yang-Mills theories include renormalizability and, for non-abelian groups, asymptotic freedom—the coupling becomes weaker at high energies. At low energies, phenomena such as confinement prevent isolated color charges from existing. Quantization requires gauge fixing and introduces ghost fields; BRST symmetry formalizes the resulting structure. Spontaneous symmetry breaking via the Higgs mechanism provides masses to gauge bosons in the electroweak sector.

Applications are wide, spanning calculations in the Standard Model and non-perturbative studies with lattice gauge theory. Experimental evidence supports Yang-Mills dynamics through jet phenomena, the discovery of W and Z bosons, and the observed running of coupling constants.