The Analytic Evolution of Dyson–Schwinger Equations via Homomorphism Densities
Résumé
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 $\Phi$ 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.