[#P36] Yang-Mills existence and mass gap
Problem. For every compact simple gauge group \(G\), construct a nontrivial quantum Yang-Mills theory on \(\mathbb{R}^4\) satisfying the required quantum field theory axioms and prove that its mass gap \(\Delta\) satisfies \(\Delta>0\).
1Context
The problem asks for a rigorous mathematical construction underlying a central quantum field theory and for a proof of its spectral gap.
2Problem setup
Definition 1 (A mass gap). A mass gap means there is a positive lower bound between the vacuum energy and the energies of all non-vacuum states in the reconstructed quantum theory.
Definition 2 (The required axioms, nontriviality condition, and relationship to classical Yang-Mills theory are those in the official problem formulation). The required axioms, nontriviality condition, and relationship to classical Yang-Mills theory are those in the official problem formulation.
Remark 1. The problem asks for a rigorous mathematical construction underlying a central quantum field theory and for a proof of its spectral gap.
3What counts as a solution
- Construct the quantum Yang-Mills theory for every compact simple G under the official axioms and prove a uniform positive mass gap in the required sense.
1Status
Current status (Dated status and exact unresolved remainder). Unresolved in this packet after the dated source check. Strongest checked result: No rigorous construction meeting the official four-dimensional axioms and positive-gap requirement is known. Experimental and lattice evidence does not satisfy the theorem. Exact unresolved remainder: Construct the theory for every compact simple G under the official axioms and prove the required positive mass gap.[1][2]
1Records
Notes and companion material
Original intake status. The cited authoritative source listed this problem as unsolved when checked on 2026-07-31. This is an admin-curated seed record, not an independent exhaustive literature review.
- Consult the cited official problem description for the exact axioms, admissible gauge groups, and meaning of the mass gap.
2See also
- Quantum PCP conjecturemathematical physics
- Exact heat-bath spectral gap on the six by six Ising torusmathematical physics
- Nearest hard-square partition-function zero for the sixteen gridmathematical physics
How to cite
TheoremDB contributors, “Yang-Mills existence and mass gap,” TheoremDB research memory, snapshot of July 31, 2026. https://theoremdb.org/statements/yang-mills-existence-and-mass-gapThis page as plain text: yang-mills-existence-and-mass-gap.md
This problem includes 2 records joined by 2 typed links, sourced from claymath.org[1], current as of July 31, 2026.
1References
- Packet source. Clay Mathematics Institute, Yang-Mills & the Mass Gap. claymath.org checked 2026-08-01. Official Problem Description by Arthur Jaffe and Edward Witten; listed under Unsolved Millennium Prize Problems. ↗website · primary source · checked 2026-07-31Source use: original summary.The cited authoritative source listed this problem as unsolved when checked on 2026-07-22. This is an admin-curated seed record, not an independent exhaustive literature review.Also cited at Unsolved status and overview.Also cited at Editorial research route recorded 2026-07-31.Source used to formulate or check the problem record.Source used to assess the problem's recorded status.Authoritative current status.Source named by the research packet.
- Arthur Jaffe and Edward Witten, Quantum Yang-Mills Theory, official Clay Mathematics Institute problem description. claymath.org checked 2026-08-01. Official problem statement. ↗website · primary source · checked 2026-08-01Source use: original summary.Defines the exact acceptance condition.
An original CC0 restatement prepared by TheoremDB maintainers.