Yang-Mills theory

The Yang-Mills mass gap problem is one of the seven Millennium Prize Problems established by the Clay Mathematics Institute in 2000, carrying a $1 million prize for a valid proof. It asks mathematicians to rigorously prove that four-dimensional Yang-Mills gauge theory predicts a positive lower bound on particle masses, known as a mass gap.

Yang-Mills theory was introduced by physicists Chen-Ning Yang and Robert Mills in 1954 as a generalization of Maxwell's equations for electromagnetism. Yang later shared the 1957 Nobel Prize in Physics for related work on parity violation, though the Nobel committee did not specifically cite the Yang-Mills paper.

Yang-Mills theory provides the mathematical backbone for the Standard Model, specifically governing the strong nuclear force through quantum chromodynamics (QCD) and the electroweak interaction. The force-carrying particles in these interactions—gluons for the strong force and W and Z bosons for the weak force—are described by Yang-Mills gauge fields.

No, the Yang-Mills mass gap problem remains unsolved as of 2025, and the Clay Mathematics Institute's $1 million prize remains unclaimed. While physicists use Yang-Mills calculations successfully through techniques like lattice QCD, a rigorous mathematical proof of the mass gap in four dimensions has not yet been established.