{"record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"9538372866077b05cd15eab22b1c11657ca4cec796b1dcde58d757407292b3d2","created":"2026-10-03T07:18:06Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["5475cf5dc5b1a72283d26dc95dc6e00d7ea4e00096a7c6492e409a1816eeb011"],"salt":"1a7f451f7687f9c64fae40e26961798d740ae0c994ba76c23619098f1e085483","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"proof","assisted_by":[],"external_id":"mathph.yang-mills.area-law","kind":"well-posed","literature_status":"open","n":"1","parents":[],"plain":"Quark confinement shows up mathematically as a loop observable that decays with the area enclosed by the loop. This is proven only when the coupling is strong, far from the regime that describes nature.","posed_since":"","precise":"For 4D lattice $\\mathrm{SU}(2)$ or $\\mathrm{SU}(3)$ gauge theory with Wilson action at large $\\beta = 2N/g^{2}\\ (N = 2\\ \\mathrm{or}\\ 3)$, prove $\\langle W(C)\\rangle \\le \\exp(-\\sigma(\\beta)\\operatorname{Area}(C))$ with string tension $\\sigma(\\beta) > 0$ for all rectangular loops $C$, uniformly in the infinite-volume limit. Answer: a proof, ideally with $\\sigma(\\beta) \\sim \\exp(-c \\beta)$ as predicted by asymptotic freedom.","problem_ref":null,"references":"","settled_by":"A proof of a positive string tension for all $\\beta$ in four dimensions.","status_note":"The area law is proven only at strong coupling (small $\\beta$), going back to Osterwalder and Seiler (1978).","title":"Area law for Wilson loops in 4D lattice gauge theory at weak coupling","topic_ref":"c1a7c5336cf6e262c1ec8a96010ceec0183209142167af7077f763d0eba2c1d0"},"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"eff6e3b8158ba736395166d654cc20a87c79bc802384cbaa914163116f5cb49d","schema":"pubphys.attested/1"},"envelope":{"attested_hash":"f7f06abb180f0cf62e9d1ef8c2044dcfc7770e937ac8a66be87f2525f393175b","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"foawfpOI2J3iguRoJO6gnj0DW_Fq8yZ-CL4C0wXbPYNmrytvgZQ9HR9WCZfBAbjlhY-mlMexwlFSw4HAEO0BAQ"},"schema":"pubphys.envelope/1"},"record_hash":"eff6e3b8158ba736395166d654cc20a87c79bc802384cbaa914163116f5cb49d","leaf_index":1660}