The Analytic Evolution of Dyson–Schwinger Equations via Homomorphism Densities

Abstract

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.

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Acknowledgements

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.

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Correspondence to Ali Shojaei-Fard.

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Shojaei-Fard, A. The Analytic Evolution of Dyson–Schwinger Equations via Homomorphism Densities. Math Phys Anal Geom 24, 18 (2021). https://doi.org/10.1007/s11040-021-09389-z

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Keywords

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

Mathematics Subject Classification (2010)

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