Opuscula Math. 38, no. 3 (2018), 427-455

Opuscula Mathematica

Graphons and renormalization of large Feynman diagrams

Ali Shojaei-Fard

Abstract. The article builds a new enrichment of the Connes-Kreimer renormalization Hopf algebra of Feynman diagrams in the language of graph functions.

Keywords: graph functions, Dyson-Schwinger equations, Connes-Kreimer renormalization Hopf algebra.

Mathematics Subject Classification: 05C05, 05C63, 81T16, 81T18.

Full text (pdf)

  1. C. Bergbauer, D. Kreimer, Hopf algebras in renormalization theory: Locality and Dyson-Schwinger equations from Hochschild cohomology, IRMA Lect. Math. Theor. Phys. 10 (2006), 133-164.
  2. B. Bollobas, O. Riordan, Metrics for sparse graphs, Surveys in Combinatorics 2009, 211-287, LMS Lecture Notes Series, vol. 365, Cambridge Univ. Press, Cambridge, 2009.
  3. C. Borgs, J.T. Chayes, L. Lovász, Moments of two-variable functions and the uniqueness of graph limits, Geom. Funct. Anal. 19 (2010) 6, 1597-1619.
  4. C. Borgs, J.T. Chayes, L. Lovász, V.T. Sos, K. Vesztergombi, Convergent sequences of dense graphs I. Subgraph frequencies, metric properties and testing, Adv. Math. 219 (2008) 6, 1801-1851.
  5. C. Borgs, J.T. Chayes, L. Lovász, V.T. Sos, K. Vesztergombi, Convergent sequences of dense graphs II. Multiway cuts and statistical physics, Ann. Math. (2) 176 (2012) 1, 151-219.
  6. P. Breitenlohner, D. Maison (eds.), Quantum Field Theory, Proceedings of the Ringberg Workshop Held at Tegernsee, Germany, 21-24 June 1998, on the Occasion of Wolfhart Zimmermann's 70th Birthday, Springer, 2000.
  7. E. Brezin, C. Itzykson, G. Parisi, J.B. Zuber, Planar diagrams, Comm. Math. Phys. 59 (1978) 1, 35-51.
  8. D.J. Broadhurst, D. Kreimer, Renormalization automated by Hopf algebra, J. Symb. Comput. 27 (1999) 6, 581-600.
  9. C. Brouder, On the trees of quantum fields, Eur. Phys. J. C 12 (2000), 535-549.
  10. C. Brouder, A. Frabetti, Renormalization of QED with planar binary trees, Eur. Phys. J. C 19 (2001), 715-741.
  11. C. Brouder, A. Frabetti, QED Hopf algebras on planar binary trees, J. Alg. 267 (2003), 298-322.
  12. C. Brouder, A. Frabetti, F. Menous, Combinatorial Hopf algebras from renormalization, J. Algebraic Combin. 32 (2010) 4, 557-578.
  13. D. Calaque, T. Strobl (eds.), Mathematical Aspects of Quantum Field Theories, Mathematical Physics Studies, Springer, 2015.
  14. A. Connes, D. Kreimer, Renormalization in quantum field theory and the Riemann-Hilbert problem. I: The Hopf algebra structure of graphs and the main theorem, Comm. Math. Phys. 210 (2000) 1, 249-273.
  15. A. Connes, D. Kreimer, Renormalization in quantum field theory and the Riemann-Hilbert problem. 2. The \(\beta\)-function, diffeomorphisms and the renormalization group, Comm. Math. Phys. 216 (2001) 1, 215-241.
  16. P. Diaconis, S. Holmes, S. Janson, Interval graph limits, Ann. Comb. 17 (2013) 1, 27-52.
  17. P. Diaconis, S. Janson, Graph limits and exchangeable random graphs, Rend. Mat. Appl. (7) 28 (2008) 1, 33-61.
  18. L. Foissy, Faà di Bruno subalgebras of the Hopf algebra of planar trees from combinatorial Dyson-Schwinger equations, Adv. Math. 218 (2008) 1, 136-162.
  19. L. Foissy, General Dyson-Schwinger equations and systems, Comm. Math. Phys. 327 (2014) 1, 151-179.
  20. M.E. Hoffman, Combinatorics of rooted trees and Hopf algebras, Trans. Amer. Math. Soc. 355 (2003) 9, 3795-3811.
  21. R. Holtkamp, Comparison of Hopf algebras on trees, Arch. Math. 80 (2003) 4, 368-383.
  22. S. Janson, Graphons, cut norm and distance, couplings and rearrangements, NYJM Monographs, vol. 4, 2013.
  23. T. Krajewski, R. Wulkenhaar, On Kreimer's Hopf algebra structure on Feynman graphs, Eur. Phys. J. C 7 (1999) 4, 697-708.
  24. D. Kreimer, Structures in Feynman graphs: Hopf algebras and symmetries, Proc. Symp. Pure Math. 73 (2005), 43-78.
  25. D. Kreimer, Anatomy of a gauge theory, Ann. Phys. 321 (2006), 27-57.
  26. D. Kreimer, Dyson-Schwinger equations: from Hopf algebras to number theory, [in:] Universality and renormalization, Fields Inst. Commun. 50, Amer. Math. Soc., Providence, RI, 2007, pp. 225-248.
  27. L. Lovász, Large networks and graph limits, American Mathematical Society Colloquium Publications, vol. 60, Amer. Math. Soc., Providence, RI, 2012.
  28. L. Lovász, B. Szegedy, Limits of dense graph sequences, J. Combin. Theory Ser. B 96 (2006) 6, 933-957.
  29. I. Moerdijk, On the Connes-Kreimer construction of Hopf algebras, Cont. Math. 271 (2001), 311-321.
  30. F. Panaite, Relating the Connes-Kreimer and Grossman-Larson Hopf algebras built on rooted trees, Lett. Math. Phys. 51 (2000) 3, 211-219.
  31. V. Parameswaran Nair, Quantum Field Theory: A Modern Perspective, Graduate Texts in Contemporary Physics, Springer, 2005.
  32. F. Paugam, Towards the Mathematics of Quantum Field Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, Springer, vol. 59, 2014.
  33. A. Shojaei-Fard, The global \(\beta\)-functions from solutions of Dyson-Schwinger equations, Modern Phys. Lett. A 28 (2013) 34, 1350152, 12 pp.
  34. A. Shojaei-Fard, Counterterms in the context of the universal Hopf algebra of renormalization, Internat. J. Modern Phys. A 29 (2014) 8, 1450045, 15 pp.
  35. A. Shojaei-Fard, A new perspective on intermediate algorithms via the Riemann-Hilbert correspondence, Quantum Stud. Math. Found. 4 (2017) 2, 127-148.
  36. A. Shojaei-Fard, A measure theoretic perspective on the space of Feynman diagrams, Bol. Soc. Mat. Mex., DOI: 10.1007/s40590-017-0166-6. https://doi.org/10.1007/s40590-017-0166-6
  37. A. Tanasa, Overview of the parametric representation of renormalizable noncommutative field theory, J. Phys.: Conf. Ser. 103 (2008), 012012.
  38. W.D. van Suijlekom, Renormalization of gauge fields: A Hopf algebra approach, Comm. Math. Phys. 276 (2007) 3, 773-798.
  39. S. Weinzierl, Introduction to Feynman integrals, [in:] Geometric and topological methods for quantum field theory, Proceedings of the 2009 Villa de Leyva Summer School, Cambridge Univ. Press, 2013, pp. 144-187.
  40. S. Weinzierl, Hopf algebras and Dyson-Schwinger equations, Front. Phys. 11 (2016), 111206.
  41. K. Yeats, A Combinatorial Perspective on Quantum Field Theory, Springer Briefs in Mathematical Physics, vol. 15, Springer, 2017.
  • Ali Shojaei-Fard
  • 1461863596 Marzdaran Blvd., Tehran, Iran
  • Communicated by Palle E.T. Jorgensen.
  • Received: 2017-12-07.
  • Revised: 2018-01-02.
  • Accepted: 2018-01-07.
  • Published online: 2018-03-19.
Opuscula Mathematica - cover

Cite this article as:
Ali Shojaei-Fard, Graphons and renormalization of large Feynman diagrams, Opuscula Math. 38, no. 3 (2018), 427-455, https://doi.org/10.7494/OpMath.2018.38.3.427

Download this article's citation as:
a .bib file (BibTeX),
a .ris file (RefMan),
a .enw file (EndNote)
or export to RefWorks.

In accordance with EU legislation we advise you this website uses cookies to allow us to see how the site is used. All data is anonymized.
All recent versions of popular browsers give users a level of control over cookies. Users can set their browsers to accept or reject all, or certain, cookies.