This two-volume Element reconstructs and analyzes the historical debates on whether renormalized quantum field theory is a mathematically consistent theory. This volume covers the years immediately following the development of renormalized quantum electrodynamics. It begins with the realization that perturbation theory cannot serve as the foundation for a proof of consistency, due to the nonconvergence of the perturbation series. Various attempts at a nonperturbative formulation of quantum field theory are discussed, including the Schwinger–Dyson equations, Gunnar Källén's non-perturbative renormalization, the renormalization group of Murray Gell-Mann and Francis Low, and, in the last section, early axiomatic quantum field theory. The volume concludes with the establishment of Haag's theorem, which proved that even the Hilbert space of perturbation theory is an inadequate foundation for a consistent theory. This title is also available as Open Access on Cambridge Core.