Yang–Mills existence and mass gap

Builds on Yang–Mills theory Wightman axioms Simple Lie group

Problem

Let $G$ be a compact simple Lie group. Prove that a quantum Yang–Mills theory with gauge group $G$ exists on Minkowski space $\R^{1,3}$ and that it has a mass gap $\Delta > 0$ . Existence means a construction of the quantum theory that satisfies axioms at least as strong as the Wightman axioms or the Osterwalder–Schrader axioms. Its short-distance behaviour must agree with the predictions of perturbation theory. The mass gap means that the vacuum is the only state of zero energy. Every state orthogonal to the vacuum must have energy at least $\Delta$ . Equivalently, the spectrum of the energy operator $H$ is contained in $\set{0} \cup [\Delta, \infty)$ , and $0$ is a simple eigenvalue.

Remarks

The problem is one of the seven Millennium Prize Problems announced by the Clay Mathematics Institute in 2000. Arthur Jaffe and Edward Witten wrote its official description. It is open.

Physicists expect a mass gap from lattice computations and from the confinement of quarks. The particles of pure Yang–Mills theory, called glueballs, should all be massive, although the classical field equations have massless waves. Asymptotic freedom makes the theory well behaved at short distances, which is why a construction is believed possible in four dimensions. Constructive quantum field theory has built interacting theories satisfying the axioms in two and three spacetime dimensions, such as the $\phi^4$ model, but no interacting theory in four dimensions. Yang–Mills theory on a two-dimensional spacetime can be solved exactly.

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Simple Lie group concept
Wightman axioms concept
Yang–Mills theory concept
Yang–Mills existence and mass gap problem

Tag	`VXC1Y32FD6`
URI	`m2://VXC1Y32FD6`
Kind	problem
Writing	W0 stub
MSC	81T13, 81T08, 81T05
Status	This copy of the site has no build data. Run `m2 status VXC1Y32FD6`.
About	Yang–Mills theory
Uses	Yang–Mills theory, Wightman axioms, Simple Lie group
Same as	Wikidata Q1078496 Wikipedia Yang–Mills existence and mass gap
Changed	3 November 2026

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