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On axiomatization and rigour in physics

Miklós Rédei, Zalán Gyenis

Published Jan 1, 2026• 0 citations
OpenAlex

Abstract

We propose that axiomatization in physics is a specific sort of mathematically conscientious and empirically mindful systematic organization of a physical theory describing a specific domain of physical systems and phenomena. It is shown how such a notion of axiomatization can be specified using category theory. Suitable concepts of completeness and categoricity of the proposed axiomatization are defined and discussed. The axiomatization is illustrated on the example of non-relativistic quantum mechanics of finite degrees of freedom. It is argued that if the suggested notion of axiomatization in physics is accepted as manifestation of rigour in physics, then rigour is not a single property but a structured notion.

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We propose that axiomatization in physics is a specific sort of mathematically conscientious and empirically mindful systematic organization of a physical theory describing a specific domain of physical systems and phenomena. It is shown how such a notion of axiomatization can be specified using category theory. Suitable concepts of completeness and categoricity of the proposed axiomatization are defined and discussed. The axiomatization is illustrated on the example of non-relativistic quantum mechanics of finite degrees of freedom. It is argued that if the suggested notion of axiomatization in physics is accepted as manifestation of rigour in physics, then rigour is not a single property but a structured notion.
An axiomatic mathematical structure is presented in terms of which arbitrary mathematical models of a physical system can be rigorously formulated and studied. Physical systems, observables, and states all arise in a natural way from the primitive notion of a physical operation. In particular, the structure is independent of the special mathematical completion defined by the quantum mechanical model, since no lattice theoretical or Hilbert space assumptions are used. The precise relationship between the present structure and the usual quantum mechanical model is investigated in a succeeding paper.
The emergence of an objective reality in line with the laws of the microscopic world has been the focus of longstanding debates. Recent approaches seem to have reached a consensus at least with respect to one aspect, namely, that the encoding of information about a given observable in a physical degree freedom is a necessary condition for such an observable to become an element of the physical reality. Taking this as a fundamental premise and inspired by quantum information theory, here we build an axiomatization for quantum realism---a notion of realism compatible with quantum theory. Our strategy consists of listing some physically motivated principles able to characterize quantum realism in a ``metric'' independent manner. We introduce some criteria defining monotones and measures of realism and then search for potential candidates within some celebrated information theories---those induced by the von Neumann, R\'enyi, and Tsallis entropies. We explicitly construct some classes of entropic quantifiers, among which some are shown to satisfy all the proposed axioms and hence can be taken as faithful estimates for the degree of reality (or definiteness) of a given physical observable. Hopefully, our framework may offer a formal ground for further discussions on foundational aspects of quantum mechanics.
Authoritative appraisals qualified this book as an axiomatic theory. However, being its essential content no more than an analogy, its theoretical organization cannot be an axiomatic one. In fact, in the first edition Dirac declares to avoid an axiomatic presentation. Moreover, I show that the text is aimed at solving a basic problem: How quantum mechanics is similar to classical mechanics. A previous paper analyzed all past theories of physics, chemistry and mathematics, presented by the respective authors in a non-axiomatic way. Four characteristic features of a new model of organizing a theory have been recognized. An accurate inspection of Dirac text proves that it actually applied this kind of organization of a theory. This fact gives a formal reason of what Kronz and Lupher suggested through intuitive categories, pragmatism and rigor, i.e. Dirac formulation of Quantum mechanics represents a distinct theoretical attitude from von Neumann axiomatic attitude. However, since the second edition Dirac changed his attitude; although relying again on an analogy, his theory was referred to the axiomatic method. Some considerations on the odd paths which led to the present formulation of QM are added. They suggest to look for a new foundation of this theory.
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.