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 $\{0\}\cup[\Delta, \infty)$ 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: it names recent work and was partly written without a live literature search; it awaits a check by an editor.
No accepted proof as of 2026; several claimed proofs posted on arXiv have not been accepted.
Parent problems
- Special case of Construct any interacting quantum field theory in four spacetime dimensions · open · from revision 1
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