What good is Haag's no-go theorem? What axiomatic methods can teach us about particle physics

Haag's theorem is a no-go theorem for the interaction picture in relativistic quantum field theory. However, the interaction picture is still widely used in conventional perturbative calculations. But how exactly is the no-go theorem thereby avoided, and what do these formal results tell us about the physical systems we study, if anything? I argue that the value of axiomatic quantum field theory for modelling particle physics systems lies in understanding the structural relationships between certain features of a quantum field description. For Haag's theorem, we learn that unitary inequivalence is related to the infinite spacetime volume idealization. We can evade Haag's theorem using more realistic idealizations, or we can ignore it for all practical purposes when we know the root cause. Various solution strategies from axiomatic quantum field theory make this conclusion clear, where conventional Lagrangian approaches are too conceptually coarse to pinpoint the infinite spacetime idealization. However, the two approaches are consistent, indicating the value of axiomatic methods for quantum field theory