My understanding is that QFT is not axiomatized - it works but there's a bit of hand-waving at the level of actual proofs. There are people working on this, but currently physics is physics and maths is maths.
QFT also becomes increasingly unwieldy for large systems. E.g., try calculating bond energies of complex molecules. You can do it for hydrogen and "hydrogen like" molecules, but once you get beyond a few interactions, the differential equation does not have closed form solutions. This is not a total loss as you can resort to numerical solutions in at least some cases.
Science is mostly about producing models that "work" whether this involves reductionism to more fundamental principles or not. At worst however, the model which describes the larger system should not conflict with the description of the smaller one.
Science is mostly about producing models that "work" whether this involves reductionism to more fundamental principles or not. At worst however, the model which describes the larger system should not conflict with the description of the smaller one.