The Tameness of Quantum Field Theory, Part I -- Amplitudes

We propose a generalized finiteness principle for physical theories, in terms of the concept of tameness in mathematical logic. A tame function or space can only have a finite amount of structure, in a precise sense which we explain. Tameness generalizes the notion of an analytic function to include certain non-analytic limits, and we show that this includes many limits which are known to arise in physics. For renormalizable quantum field theories, we give a general proof that amplitudes at each order in the loop expansion are tame functions of the external momenta and the couplings. We then consider a variety of exact non-perturbative results and show that they are tame but only given constraints on the UV definition of the theory. This provides further evidence for the recent conjecture of the second author that all effective theories that can be coupled to quantum gravity are tame. We also discuss whether renormalization group flow is tame, and comment on the applicability of our results to effective theories.