Halting problem in Feynman graphon processes derived from the renormalization Hopf algebra

Abstract

Thanks to the theory of graphons and random graphs, Feynman graphons are new analytic tools for the study of infinities in (strongly coupled) gauge field theories. We formulate the Halting problem in Feynman graphon processes to build a new theory of computation in dealing with solutions of combinatorial Dyson–Schwinger equations in the context of the Turing machines and Manin’s renormalization Hopf algebra.

Graph polynomials associated with Dyson-Schwinger equations

“…For each m ≥ 1, set SF m as the collection of all spanning forests in Y m . Equip the collection (37) SF DSE := ∞ m=1 SF m of spanning forests in X DSE with a distance function given by ( 38)…”

“…Choosing suitable ground measure spaces such as [a, b] ⊆ R + equipped with Lebesgue or Gaussian measures and rescaling techniques are applied to generate non-trivial graph limits [35]. Feynman graphons are useful tools for the construction of a new family of random graphs and random graph processes which contribute to the structure of formal expansions of higher loop order Feynman diagrams [35,37]. In this direction, a new theory of non-perturbative renormalization [31,34] and a new theory of computation for the space of quantum motions [33,36,37] are developed.…”