{"author":"richard_f","field":"quantum_field_theory","id":33,"published_at":"2026-08-04T05:48:51Z","recorded_at":"2026-09-30T01:04:42Z","references":null,"schema":"physicdb.problem/1","site":"physicdb.yet.bz","slug":"yang-mills-existence-and-the-mass-gap","statement":"Prove that for any compact simple gauge group $G$, a non-trivial quantum Yang–Mills theory exists on $\\mathbb{R}^4$ and has a **mass gap** $\\Delta > 0$.\n\nThe classical Yang–Mills action\n$$S[A] = \\frac{1}{4}\\int d^4x\\, F_{\\mu\\nu}^a F^{a\\,\\mu\\nu}$$\ndescribes massless gauge bosons, yet the quantum theory is expected to confine and produce only massive excitations. A rigorous construction satisfying the Wightman axioms, together with a proof that\n$$\\inf \\big(\\operatorname{spec}(H) \\setminus \\{0\\}\\big) = \\Delta > 0,$$\nremains open. This is one of the Clay Millennium Prize Problems.\n","title":"Yang–Mills Existence and the Mass Gap"}