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arXiv:2505.15577

Quantum Field Theory Between Rigor and Pragmatism

Johannes Branahl

Published May 21, 2025• 0 citations
arXivHTMLPDF

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.

DOI
10.48550/arxiv.2505.15577

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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.
The debate between Fraser and Wallace (2011) over the foundations of quantum field theory (QFT) has spawned increased focus on both the axiomatic and conventional formalisms. The debate has set the tone for future foundational analysis, and has forced philosophers to “pick a side”. The two are seen as competing research programs, and the major divide between the two manifests in how each handles renormalization. In this paper I argue that the terms set by the Fraser-Wallace debate are misleading. AQFT and CQFT should be viewed as complementary formalisms that start from the same physical basis. Further, the focus on cutoffs as demarcating the two approaches is also highly misleading. Though their methods differ, both axiomatic and conventional QFT seek to use the same physical principles to explain the same domain of phenomena.
Quantum field theory (QFT) presents a genuine example of the underdetermination of theory by empirical evidence. There are variants of QFT—for example, the standard textbook formulation and the rigorous axiomatic formulation—that are empirically indistinguishable yet support different interpretations. This case is of particular interest to philosophers of physics because, before the philosophical work of interpreting QFT can proceed, the question of which variant should be subject to interpretation must be settled. New arguments are offered for basing the interpretation of QFT on a rigorous axiomatic variant of the theory. The pivotal considerations are the roles that consistency and idealization play in this case.
A standard approach towards interpreting physical theories proceeds by first identifying the theory with a set of mathematical objects, where such objects are defined according to mathematicians’ standards of rigor. In making this identification, philosophers rule out the relevance of many inferential methods that physicists use, as these often do not meet mathematicians’ standards of rigor. Philosophers thus sanitize physical theories of all math- ematically messy or ambiguous parts before interpreting them. My dissertation argues against this sanitized approach towards interpreting theories using the example of quantum field theory (QFT). When we look at the details of QFT, we find that the mathematical objects it requires differ according to the specific systems the theory is being applied to in ways that advocates of the sanitized approach do not anticipate. Furthermore, the mathematical objects required for successful application are still being developed in some applicational contexts, so it would be unwise to determine in advance which objects constitute the theory. During this ongoing developmental process, physicists interpret the mathematics using strategies that violate the standards of pure mathematics. In contrast to the sanitized approach, these strategies are more sensitive to the ways in which the mathematics required for the relevant contexts is still under development. I argue that these strategies are not merely instrumental. They suggest alternative approaches to interpretation that philosophers should take into account.
Quantum field theory accurately describes the world on the finest scales to which we have empirical access. There has been significant disagreement, however,about which mathematical structures ought to be taken as constitutive of the theory, and thus over which structures should serve as the basis for its interpretation. Perturbative methods allow for successful empirical prediction but require mathematical manipulations that are at odds with the canonical approach to interpreting physical theories that has been passed down from the logical positivists. Axiomatic characterizations of the theory, on the other hand, have not been shown to admit empirically interesting models. This dissertation shows how to understand the empirical success of quantum field theory by reconsidering widely held commitments about how physical meaning accrues to mathematical structure.
Quantum field theory (QFT) can be analyzed in terms of its mathematical structure, its conceptual system for physical descriptions, or its basic ontology. The analysis can be done logically or historically. In this chapter, only the genesis of the conceptual foundations of QFT relevant to its basic ontology will be treated; no discussion of its mathematical structures or its epistemological underpinnings will be given. Some conceptual problems, such as those related to probability and measurement, will be discussed, but only because of their relevance to the basic ontology of QFT, rather than their intrinsic philosophical interest. Here, by basic ontology I mean the irreducible entities that QFT is invented to describe. The often mentioned candidates for the basic ontology of QFT, in fact of the physical world, are the discrete particle and the continuous field. Another possible candidate (the spacetime point) has also been suggested recently (Redhead, 1983). Since the aim of this chapter is to analyze the historical process in which the conceptual foundations of QFT were laid down, rather than the logical structure of QFT which philosophers of the present day treat, no discussion of the last possibility will be given. The content of this chapter involves the formation and interpretation, in a roughly chronological order, of the concepts of the wave function, quantization, quantization of the field, the vacuum, interactions between fields, and renormalization. The first two topics will be discussed in relation to the quantization of the field, with their role being taken as the starting point of the conceptual development of QFT.
Quantum Field Theory (QFT) is among the most successful theoretical frameworks in modern science, providing extremely accurate predictions for particle interactions and underpinning much of contemporary technology. Despite this success, QFT remains conceptually incomplete, relying on mathematical procedures such as renormalization and probabilistic interpretation without a unified explanatory foundation. This paper presents a systems-level interpretation of QFT using the Universal Formula, grounded in three natural laws: system integrity, balance, and feedback mechanisms. Rather than replacing QFT, this framework explains why QFT works and why its conceptual difficulties persist. The analysis reframes quantum phenomena as lawful processes of balance regulation within natural systems.