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Axiomatic Quantum Field Theory in the Philosophy of Quantum Field Theory

March 14, 2014

Bihui Li

Abstract: The philosophy of quantum field theory (QFT) in North America is dominated by philosophers working in the algebraic QFT tradition. They justify their choice of this tradition by a methodological requirement that when philosophers of physics study a theory T, the referent of T must be some mathematically well-defined structure. At the same time, philosophers think that their preferred axiomatic versions of QFT are best suited for “foundational”, “fundamental”, or “ontological” inquiries. I argue that the mathematical structures satisfying the methodological requirement are not always the best starting points for foundational, fundamental, or ontological inquiries. Many foundational, fundamental, and ontological questions are best addressed by constructive QFT, which starts with the ill-defined formalisms of Lagrangian QFT and tries to make them well-defined.  Because different interactions in constructive QFT require different mathematical constructions, the mathematical structures required cannot be determined prior to constructing a solution for a particular interaction. In contrast, the advantage of axiomatic QFT is its independence from the specific interactions being modeled, but this independence also means that it cannot provide many kinds of foundational, fundamental, or ontological information.

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