The Yang–Mills mass gap conjecture
Let be a compact simple Lie group and let 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 there exists a nontrivial Yang–Mills theory on the Minkowskian whose quantization satisfies Wightman axiomatic properties of constructive quantum field theory and has a mass gap .
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.
References
Primary source
Simone Farinelli, “Four Dimensional Quantum Yang-Mills Theory for Weak Coupling Strength: Mass Gap Implies Quark Confinement”, arXiv:2406.16881 (2024).
Progress summary
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 . 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 for . 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.
Solutions 0
No solutions have been posted yet.