Boletín de la Sociedad Matemática Mexicana

, Volume 24, Issue 2, pp 507–533 | Cite as

A measure theoretic perspective on the space of Feynman diagrams

  • Ali Shojaei-FardEmail author
Original Article


The article applies Connes–Kreimer Hopf algebra of Feynman diagrams and theory of graphons to build an operational calculus machinery on the basis of measure theory for Green’s functions of quantum field theory.


Quantum field theory Graphons Feynman diagrams Measure theory Operational calculus 

Mathematics Subject Classification

81T18 28A25 05C63 34A25 81T16 


  1. 1.
    Bollobás, B.: Extremal Graph Theory. Academic Press, London (1978)zbMATHGoogle Scholar
  2. 2.
    Bollobás, B.: Modern Graph Theory. Springer, New York (1998)CrossRefGoogle Scholar
  3. 3.
    Borgs, C., Chayes, J.T., Lovász, L.: Moments of two-variable functions and the uniqueness of graph limits. Geom. Funct. Anal. 19(6), 1597–1619 (2010)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Borgs, C., Chayes, J.T., Lovász, L., Sos, V.T., Vesztergombi, K.: Convergent sequences of dense graphs I. Subgraph frequencies, metric properties and testing. Adv. Math. 219(6), 1801–1851 (2008)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Borgs, C., Chayes, J.T., Lovász, L., Sos, V.T., Vesztergombi, K.: Convergent sequences of dense graphs II. multiway cuts and statistical physics. Ann. Math. (2) 176(1), 151–219 (2012)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bergbauer, C., Kreimer, D.: Hopf algebras in renormalization theory: locality and Dyson–Schwinger equations from Hochschild cohomology. IRMA Lect. Math. Theor. Phys. 10, 133–164 (2006)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Broadhurst, D.J., Kreimer, D.: Renormalization automated by Hopf algebra. J. Symb. Comput. 27(6), 581–600 (1999)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Cohn, D.L.: Measure Theory. Birkhauser, Boston (1980)CrossRefGoogle Scholar
  9. 9.
    Collins, J.C.: Renormalization, Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge (1984)Google Scholar
  10. 10.
    Cheney, E.W.: Analysis for Applied Mathematics, Series Vol. 208. Springer, New York (2001)CrossRefGoogle Scholar
  11. 11.
    Connes, A.: Gravity coupled with matter and the foundation of noncommutative geometry. Commun. Math. Phys. 182, 155 (1996)CrossRefGoogle Scholar
  12. 12.
    Connes, A.: Noncommutative geometry and the standard model with neutrino mixing. J. High Energy Phys. 2006(11), 081 (2006)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Chamseddine, A., Connes, A.: Universal formula for noncommutative geometry actions: unification of gravity and the standard model. Phys. Rev. Lett. 77, 486804871 (1996)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Chamseddine, A., Connes, A.: The spectral action principle. Commun. Math. Phys. 186, 731–750 (1997)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Chamseddine, A., Connes, A.: Why the standard model. J. Geom. Phys. 58, 38–47 (2008)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Chamseddine, A., Connes, A., Marcolli, M.: Gravity and the standard model neutrino mixing. Adv. Theor. Math. Phys. 11, 991–1089 (2007)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Connes, A., Kreimer, D.: Hopf algebras, renormalization and noncommutative geometry. Commun. Math. Phys. 199(1), 203–242 (1998)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Connes, A., Marcolli, M.: Noncommutative Geometry, Quantum Fields and Motives, vol. 55. American Mathematical Society, Providence (2007)zbMATHGoogle Scholar
  19. 19.
    Diestel, R.: Directions in Infinite Graph Theory and Combinatorics, Topics in Discrete Mathematics 3, Elsevier, New York (1992) (ISBN 0444894144)Google Scholar
  20. 20.
    Diao, P., Guillot, D., Khare, A., Rajaratnam, B.: Differential calculus on graphon space. J. Comb. Theory Ser. A 133, 183–227 (2015) arXiv:1403.3736v2 MathSciNetCrossRefGoogle Scholar
  21. 21.
    Diaconis, P., Holmes, S., Janson, S.: Interval graph limits. Ann. Comb. 17(1), 27–52 (2013)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Diaconis, P., Janson, S.: Graph limits and exchangeable random graphs. Rend. Mat. Appl. (7) 28(1), 33–61 (2008)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Ebrahimi-Fard, K., Guo, L., Kreimer, D.: Integrable renormalization II: the general case. Ann. Henri Poincare 6, 369 (2005)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Erdos, P., Lovász, L., Spencer, J.: Strong independence of graphcopy functions. In: Graph theory and related topics (Proc. Conf., Univ. Waterloo, Waterloo, Ont., 1977), pp. 165–172. Academic Press, New York (1979)Google Scholar
  25. 25.
    Feynman, R.P.: An operator calculus having applications in quantum electrodynamics. Phys. Rev. 84, 108–128 (1951)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Feynman, R.P., Hibbs, A.R.: Quantum Mechanics and Path Integrals. McGraw-Hill, New York (1965)zbMATHGoogle Scholar
  27. 27.
    Gelfand, I.M., Fomin, S.V.: Calculus of Variations. Dover, USA (2000)zbMATHGoogle Scholar
  28. 28.
    Glimm, J., Jaffe, A.: Quantum Physics. Springer, Berlin (1981)CrossRefGoogle Scholar
  29. 29.
    Itzykson, C., Zuber, J.B.: Quantum Field Theory. McGraw Hill, New York (1980)zbMATHGoogle Scholar
  30. 30.
    Janson, S.: Graphons, Cut Norm and Distance, Couplings and Rearrangements, NYJM Monographs, Vol. 4 (2013)Google Scholar
  31. 31.
    Johnson, G.W., Lapidus, M.L.: Generalized Dyson series, generalized Feynman diagrams, the Feynman integral and Feynman’s operational calculus, vol. 351. American Mathematical Society (1986)Google Scholar
  32. 32.
    Jacod, J., Protter, P.: Probability Essentials, Universitext. Springer, Berlin (2004)CrossRefGoogle Scholar
  33. 33.
    Kane, G.: Modern Elementary Particle Physics. Addison-Wesley, Boston (1987)Google Scholar
  34. 34.
    Kreimer, D.: Anatomy of a gauge theory. Ann. Phys. 321, 2757–2781 (2006)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Kreimer, D.: Structures in Feynman graphs: Hopf algebras and symmetries. Proc. Symp. Pure Math. 73, 43–78 (2005)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Kreimer, D.: On overlapping divergences. Commun. Math. Phys. 204, 669 (1999)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Khare, A., Rajaratnam, B.: Integration and Measures on the Space of Countable Labelled Graphs. arXiv:1506.01439 [math.CA] (2015)
  38. 38.
    Khare, A., Rajaratnam, B.: Differential calculus on the space of countable labelled graphs, Technical Report, Departments of Mathematics and Statistics, Stanford University (2014). arXiv:1410.6214
  39. 39.
    Krajewski, T., Wulkenhaar, R.: On Kreimer’s Hopf algebra structure on Feynman graphs. Eur. Phys. J. C 7(4), 697–708 (1999)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Lapidus, M.L.: The Feynman–Kac formula with a Lebesgue–Stieltjes measure and Feynman’s operational calculus, I. Berkeley, II, Mathematical Sciences Research Institute (1986) (Preprint)Google Scholar
  41. 41.
    Lovász, L.: Large Networks and Graph Limits, American Mathematical Society Colloquium Publications, vol. 60. American Mathematical Society, Providence (2012)Google Scholar
  42. 42.
    Lindenstrauss, J., Preiss, D.: On Fréchet differentiability of Lipschitz maps between Banach spaces. Ann. Math. 157(1), 257–288 (2003)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Marino, M.: Nonperturbative effects and nonperturbative definitions in matrix models and topological strings. J. High Energy Phys. 2008(12), 114 (2008)MathSciNetCrossRefGoogle Scholar
  44. 44.
    Milnor, J., Moore, J.: On the structure of Hopf algebras. Ann. Math. 81, 211–264 (1965)MathSciNetCrossRefGoogle Scholar
  45. 45.
    Peskin, M.E., Schroeder, D.V.: An Introduction to Quantum Field Theory. Addison-Wesley Publishing Company, Boston (1995)Google Scholar
  46. 46.
    Rao, M.M.: Measure Theory and Integration (pure and applied mathematics). Wiley, New York (1987)Google Scholar
  47. 47.
    Royden, H.L.: Real Analysis. MacMillan, London (1968)zbMATHGoogle Scholar
  48. 48.
    Shojaei-Fard, A.: A geometric perspective on counterterms related to Dyson–Schwinger equations. Int. J. Mod. Phys. A 28(32), 1350170 (2013)MathSciNetCrossRefGoogle Scholar
  49. 49.
    Shojaei-Fard, A.: The global \(\beta -\)functions from solutions of Dyson–Schwinger equations. Mod. Phys. Lett. A 28(34), 1350152 (2013)MathSciNetCrossRefGoogle Scholar
  50. 50.
    Strocchi, F.: An introduction to non-perturbative foundations of quantum field theory. Int. Ser. Monogr. Phys. 158, 1–272 (2013)MathSciNetzbMATHGoogle Scholar
  51. 51.
    Yeats, K.: Rearranging Dyson–Schwinger equations, vol. 211. American Mathematical Society, Providence (2011)MathSciNetCrossRefGoogle Scholar
  52. 52.
    Weinzierl, S.: Introduction to Feynman Integrals, Geometric and Topological Methods for Quantum Field Theory, pp. 144–187. Cambridge University Press, Cambridge (2013)Google Scholar
  53. 53.
    Zimmermann, W.: Convergence of Bogoliubov’s method of renormalization in momentum space. Commun. Math. Phys. 15, 208 (1969)MathSciNetCrossRefGoogle Scholar

Copyright information

© Sociedad Matemática Mexicana 2017

Authors and Affiliations

  1. 1.TehranIran

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