Yang–Mills existence and mass gap
For every compact simple gauge group , construct a nontrivial quantum Yang–Mills theory on satisfying suitable axioms and prove its Hamiltonian has a mass gap .
Problem statement
For every compact simple gauge group , construct a nontrivial quantum Yang–Mills theory on satisfying suitable axioms and prove its Hamiltonian has a mass gap .
Lean 4
No Lean statement or proof is attached yet. Have a formalization? Share its source to add it to this problem.
Background
This is a Clay Millennium Problem. One must construct four-dimensional quantum Yang–Mills theory as a well-defined quantum field theory (satisfying, e.g., the Wightman or Osterwalder–Schrader axioms) and prove a positive excitation threshold above the vacuum despite the massless classical gauge fields. This spectral gap must not be conflated with screening of external colour charges or the confinement criterion in QF2.
What is known: Two-dimensional Yang–Mills has rigorous constructions. Strong-coupling lattice results and lower-dimensional constructions provide important benchmarks, but a lattice mass gap at fixed spacing is not a positive gap in the four-dimensional continuum limit. Asymptotic freedom (Gross–Wilczek–Politzer) controls short-distance perturbation theory; it does not construct the theory or prove its nonperturbative gap.
The Clay problem remains open: control both continuum and infinite-volume limits, preserve suitable axioms, and obtain a strictly positive physical mass scale. The cited Chandra–Chevyrev–Hairer–Shen work concerns the 2D Yang–Mills measure, not a 3D or 4D continuum construction. Confinement (QF2) is a separate property and does not follow from a mass gap alone. Related: QF2, QF6.
Progress & source assessment
Source catalogue review: 4 September 2026. Improved indicates partial progress, including qualified preprint claims.
Recorded progress
Add yours →Nothing recorded yet. A pointer to a paper the entry is missing counts.
Machine attempts
0 recordedAttempts run against the statement below and are recorded in full, including the failures. The point is to stop the next model — and the next person — from re-walking a road that is already known to end. Each attempt costs 25 credits.
No machine attempts recorded.
Sign in to run an attempt. New accounts start with enough credits for ten.
Discussion
0 notesNotes are for clarifying the statement, flagging a missing reference, or recording why an approach fails. Claimed solutions belong in a submission, where they can be reviewed.
No notes yet.
Sign in to add a note.