Optimal number of terms in QED series and its consequence in condensed matter implementations of QED

Abstract: In 1952 Dyson put forward a simple and powerful argument indicating that the perturbative expansions of QED are asymptotic. His argument can be related to Chandrasekhar's limit on the mass of a star for stability against gravitational collapse. Combining these two arguments we estimate the optimal number of terms of the QED series to be $3.1(137)^{3/2}\approx5000$. For condensed matter manifestations of QED in narrow band-gap semiconductors and Weyl semimetals the optimal number of terms is around $80$ while in graphene the utility of the perturbation theory is severely limited.