MathDB

The Yang–Mills mass gap conjecture

Let GG be a compact simple Lie group and let R1,3\mathbf{R}^{1,3} denote Minkowskian spacetime. A nontrivial Yang–Mills theory on this spacetime is a theory whose quantization satisfies the Wightman axiomatic properties of constructive quantum field theory.

Yang–Mills mass gap conjecture. For any compact simple Lie group GG there exists a nontrivial Yang–Mills theory on the Minkowskian R1,3\mathbf{R}^{1,3} whose quantization satisfies Wightman axiomatic properties of constructive quantum field theory and has a mass gap η>0\eta>0.

This is the seventh Clay Mathematics Institute Millennium Prize problem. The supplied source states that the conjecture was proved in 2024, subject to approval by the mathematical physics community.

Sources & referencesView supporting material

Primary source

Simone Farinelli, “Four Dimensional Quantum Yang-Mills Theory for Weak Coupling Strength: Mass Gap Implies Quark Confinement”, arXiv:2406.16881 (2024).

mass-gapquantum-field-theoryyang-mills-theory

Progress summary

Refreshed
Open

A 2024 preprint claims a proof, but no independent verification has established that the Yang–Mills mass gap problem is solved.

The Clay Millennium problem asks for a mathematically rigorous four-dimensional quantum Yang–Mills theory for every compact simple Lie group, with the required axioms and a positive mass gap η>0\eta>0. The Clay page says no proof is known, although experiments and simulations support the conjecture.

Known results

  • Exponential correlation decay and mass gaps are known in some lattice settings, including sufficiently strong coupling; this does not establish the continuum problem (2020).

April 2024 claimed proof

A preprint claims to construct the theory for weak coupling, verify the Osterwalder–Schrader and Wightman axioms, and prove ηg>0\eta_g>0 for g>0g>0. It explicitly treats the conjecture as unproved pending approval by the mathematical-physics community. Later repository items make additional unverified proof claims, without independent checking or referee reports.

Current status (as of August 2026): The conjecture remains unverified; claimed proofs exist, but no retrieved source establishes community-accepted resolution of the continuum existence and mass-gap problem.

Sources

Solutions 0

No solutions have been posted yet.