{"paper":{"title":"Renormalization in quantum field theory and the Riemann-Hilbert problem","license":"","headline":"","cross_cats":["hep-ph","math-ph","math.MP","math.QA"],"primary_cat":"hep-th","authors_text":"Alain Connes, Dirk Kreimer","submitted_at":"1999-09-17T20:05:43Z","abstract_excerpt":"We show that renormalization in quantum field theory is a special instance of a general mathematical procedure of multiplicative extraction of finite values based on the Riemann-Hilbert problem. Given a loop $\\gamma(z), | z |=1$ of elements of a complex Lie group G the general procedure is given by evaluation of $ \\gamma_{+}(z)$ at z=0 after performing the Birkhoff decomposition $ \\gamma(z)=\\gamma_{-}(z)^{-1} \\gamma_{+}(z)$ where $ \\gamma_{\\pm}(z) \\in G$ are loops holomorphic in the inner and outer domains of the Riemann sphere (with $\\gamma_{-}(\\infty)=1$). We show that, using dimensional reg"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"hep-th/9909126","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/hep-th/9909126/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}