{"record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"0b960c0295ecb241f2da51a8b169269a2e29882be16c76320124fb2eb67f886d","created":"2026-10-03T07:18:06Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["5475cf5dc5b1a72283d26dc95dc6e00d7ea4e00096a7c6492e409a1816eeb011"],"salt":"2d86079dda9fde404ad92a7247d8c686cbcb3784e3f16b3be25cb9f7c22571ca","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"proof","assisted_by":[],"external_id":"mathph.sigma-models.continuum-sigma-model","kind":"well-posed","literature_status":"open","n":"1","parents":[],"plain":"Zooming out at the scale of the correlation length while the temperature goes to zero should give a continuum quantum field theory with massive interacting particles. Building it rigorously would be a two-dimensional model of the Yang-Mills construction problem.","posed_since":"","precise":"Show that lattice $O(N)$ correlations ($N \\ge 3$) at separations $x = y\\,\\xi(\\beta),\\ y\\ \\text{fixed}$, converge as $\\beta \\to \\infty$ to the Schwinger functions of a non-Gaussian Euclidean field theory on $R^2$ satisfying the Osterwalder-Schrader axioms with a mass gap, and identify its scattering with the factorized S-matrix of A. B. and Al. B. Zamolodchikov (1979). Answer: a construction with proof.","problem_ref":null,"references":"","settled_by":"A proof of convergence of the rescaled lattice correlations to an Osterwalder-Schrader theory with non-Gaussian four-point function.","status_note":"","title":"Construct the continuum $2\\mathrm{D}$ $O(N)$ sigma model as an interacting theory","topic_ref":"b4bb48d722ff2368d355af570ae350e795f3cc5cea7a0142509f239da3c98279"},"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"4d658cfc5ae23ad68dff309eb8ff4b242c3fcfb6327d97b8b930ef08375ea5db","schema":"pubphys.attested/1"},"envelope":{"attested_hash":"616ec2493915e50b3f9a8c599e0ef2b875e9f641b061354ff7e756ae81100d99","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"DHDKOncZEV9G_FiE_r0VEioNoROfmBKQMokdQhz_MEUdWTMqmJgem3sowFK6Asfu6NW4PrlQMNtqmUYUp5MzAg"},"schema":"pubphys.envelope/1"},"record_hash":"4d658cfc5ae23ad68dff309eb8ff4b242c3fcfb6327d97b8b930ef08375ea5db","leaf_index":1657}