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 Gateaux 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.