Mathematical Physics, Analysis and Geometry | 2021
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 S ≈ Φ , g $\\mathcal {S}^{\\Phi ,g}_{\\approx }$ 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 S ≈ Φ , g $\\mathcal {S}^{\\Phi ,g}_{\\approx }$ in terms of a new class of homomorphism densities. We then show that Taylor series representations of smooth functionals on S ≈ Φ , g $\\mathcal {S}^{\\Phi ,g}_{\\approx }$ provide a new analytic description for solutions of combinatorial Dyson–Schwinger equations.