Quantum Field Theory Between Rigor and Pragmatism

Abstract: Quantum Field Theory (QFT), the foundational framework of particle physics, has long existed in a state of tension between empirical success and mathematical rigor. Conventional QFT (CQFT), which underpins the Standard Model, offers unparalleled predictive accuracy but relies on inconsistent and ad hoc methods. In contrast, axiomatic QFT (AxQFT) aspires to a consistent, mathematically rigorous foundation, yet lacks empirical applicability. This paper introduces the heuristic vehicle of a realism quotient to model and navigate this tension, framing it as a dynamic balance between realism and instrumentalism in the mathematical structures of physical theories. By reconstructing the historical development of QFT and extrapolating its trajectory, the paper offers both a descriptive account of theoretical progress and a normative proposal: that CQFT and AxQFT must converge to address selected challenges of physics beyond the Standard Model. The model also contributes to broader debates in scientific realism, offering a structured framework for understanding the interplay between empirical adequacy and conceptual robustness and mathematical rigour in foundational physics.