MATHPH In the literature: open

Existence and mass gap of four-dimensional quantum Yang-Mills theory

In plain words

Prove that the quantum theory of gluons exists as a well-defined mathematical object and that its lightest particle has a strictly positive mass. This is the Clay Millennium problem on Yang-Mills theory.

Precise statement

For every compact simple gauge group $G$ ($SU(2)$ is the minimal case), construct a quantum Yang-Mills theory on $R^4$ satisfying the Osterwalder-Schrader or Wightman axioms, and prove a mass gap: the Hamiltonian $H$ has spectrum contained in $\{0\}\operatorname{union}[\Delta, \infty)$ for some $\Delta > 0$. Answer: a proof of both statements (Jaffe-Witten formulation for the Clay Mathematics Institute).

What would settle it

An accepted proof of existence and $\Delta > 0$ for every compact simple $G$, as the Clay statement requires; a proof for $\mathrm{SU}(2)$ alone would be a major partial result.

Status in the literature

Unverified note

No accepted proof as of 2026; several claimed proofs posted on arXiv have not been accepted.