{"record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"18d61780ada771f4c82ce7c29e817d6afd879a4ccdba0d299d49bc7939ac4c04","created":"2026-10-15T23:43:22Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["59f5a39798c9d34582c88fc7916969379ec1acf473ca0f0f75f8a3d2b3c5ff26","ce59fb6cd812fc05906d0c60a71c75ea22bc686bcf38e7c3ef7010834d98ee7b"],"salt":"4443f026f04d6dbf0c7f07c7b2fcc17c425a6b171095f9c40b3af9a6464efc63","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"proof","assisted_by":[],"external_id":"mathph.yang-mills.continuum-limit-4d","kind":"well-posed","literature_status":"open","n":"2","parents":[],"plain":"Yang-Mills theory can be defined on a grid of points, and the task is to show that it has a sensible limit as the grid spacing goes to zero. The limit should be interacting, not a free theory.","posed_since":"","precise":"Take $\\mathrm{SU}(N)$ lattice gauge theory with the Wilson action on $(\\epsilon \\mathrm{Z})^4$, with bare coupling chosen by asymptotic freedom, $g(\\epsilon)^2 \\sim 1/(2 b_0 \\log(1/(\\epsilon \\Lambda)))$ with $b_0 = 11 N/(48 \\pi^2)$, i.e. $1/g^2 \\sim (11 N/(24 \\pi^2)) \\log(1/(\\epsilon \\Lambda))$. Prove that correlations of gauge-invariant observables (smoothed Wilson loops or smeared $F^2$) converge as $\\epsilon \\to 0$ in infinite volume to a non-Gaussian limit satisfying the Osterwalder-Schrader axioms. Answer: a proof.","problem_ref":"59f5a39798c9d34582c88fc7916969379ec1acf473ca0f0f75f8a3d2b3c5ff26","references":"","settled_by":"A rigorous renormalization-group or probabilistic proof of the infinite-volume continuum limit with non-Gaussian correlations.","status_note":"Balaban's renormalization-group program of the 1980s gave ultraviolet stability bounds in finite volume but did not complete the construction.","title":"Continuum limit of four-dimensional lattice Yang-Mills theory","topic_ref":null},"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"b17ed0f1a13b79c98fecf553158718f7f0af5229d16b200a95fb13255681ad69","schema":"pubphys.attested/1"},"envelope":{"attested_hash":"22eb646ae1a03c01e4b5bfdb135b958078ed9b07e11539929bb21bd2ac4843d3","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"0pKclZjCfqW99V83EJTg_LGKObW20qO7MkksDEYBMUYoZi7G-epPQrtrkS7LW4Gkn70-euobytlt5ZrfppB1Bw"},"schema":"pubphys.envelope/1"},"record_hash":"b17ed0f1a13b79c98fecf553158718f7f0af5229d16b200a95fb13255681ad69","leaf_index":3456}