On the Complexity of Quantum Field Theory

We initiate a study of the complexity of quantum field theories (QFTs) by proposing a measure of information contained in a QFT and its observables. We show that from minimal assertions, one is naturally led to measure complexity by two integers, called format and degree, which characterize the information content of the functions and domains required to specify a theory or an observable. The strength of this proposal is that it applies to any physical quantity, and can therefore be used for analyzing complexities within an individual QFT, as well as studying the entire space of QFTs. We discuss the physical interpretation of our approach in the context of perturbation theory, symmetries, and the renormalization group. Key applications include the detection of complexity reductions in observables, for example due to algebraic relations, and understanding the emergence of simplicity when considering limits. The mathematical foundations of our constructions lie in the framework of sharp o-minimality, which ensures that the proposed complexity measure exhibits general properties inferred from consistency and universality.