In the first part of my talk, I will review a bunch of no-go theorems that directly refute the central ideas of the canonical (that is, operator-theoretic) formalism of quantum field theory and discuss possible expedients. In particular, I will argue that the most vicious one, namely Haag’s theorem – which should come as no big surprise to physics – is, in fact, in a sense circumvented by renormalisation.

The second part deals with Dyson-Schwinger and renormalisation group equations in QED and Yukawa theory which imply ODEs and fixed point equations for anomalous dimensions. I present some nonperturbative results regarding an ODE for the photon’s anomalous dimension and furthermore show that the fixed point equations encode under certain conditions what is known as the ‘principle of resurgence’, meaning that the perturbative sector of a resurgent transseries determines the nonperturbative one.