The Analytic Evolution of Dyson–Schwinger Equations via Homomorphism Densities


Feynman graphon representations of Feynman diagrams lead us to build a new separable Banach space \(\mathcal {S}^{\Phi ,g}_{\approx }\) originated from the collection of all Dyson–Schwinger equations in a given (strongly coupled) gauge field theory Φ with the bare coupling constant g. We study the Gâteaux differential calculus on the space of functionals on \(\mathcal {S}^{\Phi ,g}_{\approx }\) in terms of a new class of homomorphism densities. We then show that Taylor series representations of smooth functionals on \(\mathcal {S}^{\Phi ,g}_{\approx }\) provide a new analytic description for solutions of combinatorial Dyson–Schwinger equations.

This is a preview of subscription content, access via your institution.


  1. 1.

    Borgs, C., Chayes, J.T., Cohn, H., Holden, N.: Sparse exchangeable graphs and their limits via graphon processes. J. Mach. Learn. Res. 18 (Paper No. 210), 71 (2017)

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Bartocci, C., Bruzzo, U., Cianci, R. (eds.): Differential Geometric Methods in Theoretical Physics. Proceed. 19th International Conference in Rapallo. Springer, Italy (1990)

  3. 3.

    Bollobas, B., Riordan, O.: Metrics for sparse graphs, in surveys in combinatorics. LMS Lecture Notes Series 365(CUP 2009), 211–287 (2009)

    MATH  Google Scholar 

  4. 4.

    Chen, W.F.: Differential geometry from Quantum Field Theory. Intern. J. Geom. Meth. Mod. Phys. 10(4), 1350003 (2013)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Diao, P., Guillot, D., Khare, A., Rajaratnam, B.: Differential calculus on graphon space. J. Comb. Theory Ser. A 133, 183–227 (2015)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Gutierrez, E., Ahmad, A., Ayala, A., Bashir, A., Raya, A.: The QCD phase diagram from Schwinger–Dyson equations. J. Phys. G:, Nucl. Part. Phys. 41, 075002 (2014)

    ADS  Article  Google Scholar 

  7. 7.

    Hell, P., Nesetril, J.: Graphs and Homomorphisms. Oxford University Press (2004)

  8. 8.

    Janson, S.: Graphons, Cut Norm and Distance, Couplings and Rearrangements, New York Journal of Mathematics NYJM Monographs, vol. 4. State University of New York, University at Albany, Albany, NY (2013)

  9. 9.

    Kohayakawa, Y.: Szemeredi’s regularity lemma for sparse graphs. In: Cucker, F., Shub, M. (eds.) Foundations of Computational Mathematics, pp 216–230 (1997)

  10. 10.

    Kreimer, D.: Structures in Feynman graphs: Hopf algebras and symmetries. Proc. Symp. Pure Math. 73, 43–78 (2005)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Kreimer, D.: Combinatorics of perturbative quantum field theory. Phys. Rept. 363, 387–424 (2002)

    ADS  MathSciNet  Article  Google Scholar 

  12. 12.

    Kreimer, D.: On overlapping divergences. Commun. Math. Phys. 204, 669–689 (1999)

    ADS  MathSciNet  Article  Google Scholar 

  13. 13.

    Kreimer, D.: Anatomy of a gauge theory. Annals Phys. 321, 2757–2781 (2006)

    ADS  MathSciNet  Article  Google Scholar 

  14. 14.

    Krajewski, T., Wulkenhaar, R.: On Kreimer’s Hopf algebra structure of Feynman graphs. Eur. Phys. J. C 7, 697–708 (1999)

    ADS  MathSciNet  Article  Google Scholar 

  15. 15.

    Lovasz, L.: Very large graphs. Curr. Dev. Math. 2008, 67–128 (2009)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Lovasz, L.: Large Networks and Graph Limits American Mathematical Society Colloquium Publications, vol. 60. Amer. Math. Soc., Providence, RI (2012)

    Google Scholar 

  17. 17.

    Martinetti, P.: Beyond the standard model with noncommutative geometry, strolling towards quantum gravity. J. Phys.: Conference Series 634, 012001 (2015)

    Google Scholar 

  18. 18.

    Masson, T.: Gauge theories in noncommutative geometry. AIP Conference Proceedings 1446, 73 (2012)

    ADS  Article  Google Scholar 

  19. 19.

    Rund, H.: Differential-geometric and variational background of classical gauge field theories. Aeq. Math. 24, 121–174 (1982)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Roberts, C.D., Schmidt, S.M.: Dyson–schwinger equations: density, temperature and continuum strong QCD. Prog. Part. Nucl.Phys. 45, S1–S103 (2000)

    ADS  Article  Google Scholar 

  21. 21.

    Roberts, C.D., Williams, A.G.: Dyson–Schwinger equations and the application to hadronic physics. Prog. Part. Nucl Phys. 33, 477–575 (1994)

    ADS  Article  Google Scholar 

  22. 22.

    Scott, A.: Szemeredi’s regularity lemma for matrices and sparse graphs. Combin. Probab. Comput. 20(3), 455–466 (2011)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Shojaei-Fard, A.: A measure theoretic perspective on the space of Feynman diagrams. Bol. Soc. Mat. Mex. (3) 24(2), 507–533 (2018)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Shojaei-Fard, A.: Graphons and renormalization of large Feynman diagrams. Opuscula Mathematica 38(3), 427–455 (2018)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Shojaei-Fard, A.: Non-perturbative β-functions via Feynman graphons. Modern Phys. Lett. A 34(14), 1950109(10) (2019)

    ADS  MathSciNet  Article  Google Scholar 

  26. 26.

    Shojaei-Fard, A.: Formal aspects of non-perturbative Quantum Field Theory via an operator theoretic setting. Intern. J. Geom. Methods Mod. Phys. 16 (12), 1950192(23) (2019)

    ADS  MathSciNet  Article  Google Scholar 

  27. 27.

    Zimmermann, W.: Convergence of Bogoliubov’s method of renormalization in momentum space. Commun. Math. Phys. 15, 208 (1969)

    ADS  MathSciNet  Article  Google Scholar 

  28. 28.

    Zorn, M.A.: Characterization of analytic functions in Banach spaces. Annals of Mathematics, Second Series 46(4), 585–593 (1945)

    MathSciNet  Article  Google Scholar 

Download references


The author is grateful to Institut des Hautes Etudes Scientifiques for the support and hospitality. In addition, the author would like to thank the Reviewer because of fundamental comments which were helpful to clarify the results of this work.

Author information



Corresponding author

Correspondence to Ali Shojaei-Fard.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Shojaei-Fard, A. The Analytic Evolution of Dyson–Schwinger Equations via Homomorphism Densities. Math Phys Anal Geom 24, 18 (2021).

Download citation


  • Combinatorial Dyson–Schwinger equations
  • Feynman graphons
  • Homomorphism densities of graphons
  • Non-perturbative QFT
  • Taylor series

Mathematics Subject Classification (2010)

  • 81T16
  • 46G05
  • 46B09
  • 05C63
  • 81T27