Yang-Mills theory

Yang-Mills theory is the generalization of electromagnetism to non-abelian gauge groups, formulated by Chen-Ning Yang and Robert Mills in 1954. Where quantum electrodynamics describes a single massless gauge boson (the photon) interacting with electrically charged matter, Yang-Mills theory describes multiple gauge bosons that interact with each other as well as with matter — a consequence of the non-commutativity of the gauge group. The result is a theory of self-interacting force carriers that is both mathematically richer and physically more powerful than its abelian predecessor.

The predictive content of Yang-Mills theory is remarkable. Quantum chromodynamics, the SU(3) Yang-Mills theory of the strong nuclear force, successfully accounts for the spectrum of hadrons, the behavior of quarks at high energy, and the phenomenon of asymptotic freedom — the counterintuitive property that the strong force weakens at short distances. The electroweak theory unifies the electromagnetic and weak forces into an SU(2) × U(1) Yang-Mills theory, spontaneously broken by the Higgs mechanism.

Yet Yang-Mills theory contains unsolved problems of immense depth. The Yang-Mills existence and mass gap problem — one of the seven Millennium Prize Problems — asks whether a rigorous mathematical construction of the theory exists and whether the ground state has a mass gap. The physical evidence says yes; the mathematical proof says nothing. The gap between what physicists calculate and what mathematicians can prove is a chasm that has persisted for seventy years, and it is not clear which discipline owns the failure.