Haag's Theorem and the Interpretation of Quantum Field Theories with Interactions

Quantum field theory (QFT) is the physical framework that integrates quantum mechanics and the special theory of relativity; it is the basis of many of our best physical theories. QFT's for interacting systems have yielded extraordinarily accurate predictions. Yet, in spite of unquestionable empirical success, the treatment of interactions in QFT raises serious issues for the foundations and interpretation of the theory. This dissertation takes Haag's theorem as a starting point for investigatingthese issues. It begins with a detailed exposition and analysis of different versions ofHaag's theorem. The theorem is cast as a reductio ad absurdum of canonical QFT prior to renormalization. It is possible to adopt different strategies in response to this reductio: (1) renormalizing the canonical framework; (2) introducing a volume i.e., long-distance) cutoff into the canonical framework; or (3) abandoning another assumption common to the canonical framework and Haag's theorem, which is the approach adopted by axiomatic and constructive field theorists. Haag's theorem doesnot entail that it is impossible to formulate a mathematically well-defined Hilbert space model for an interacting system on infinite, continuous space. Furthermore, Haag's theorem does not undermine the predictions of renormalized canonical QFT; canonical QFT with cutoffs and existing mathematically rigorous models for interactions are empirically equivalent to renormalized canonical QFT. The final two chapters explore the consequences of Haag's theorem for the interpretation of QFT with interactions. I argue that no mathematically rigorous model of QFT on infinite, continuous space admits an interpretation in terms of quanta (i.e., quantumparticles). Furthermore, I contend that extant mathematically rigorous models for physically unrealistic interactions serve as a better guide to the ontology of QFT than either of the other two formulations of QFT. Consequently, according to QFT, quanta do not belong in our ontology of fundamental entities