Quantum Theory and Mathematical Rigor

Cécile de Witt-Morette

Yuan Horng Lin

This paper proposes a research program aimed at exploring the possibility of establishing a rigorous mathematical foundation for quantum information ontology. We systematically study the path of incorporating the entanglement structure of κ-qubit networks into an extended set-theoretic framework. Based on the principles of Jiuzhang Constructive Math ematics (JCM), we explore the conceptual foundations of constructing a Quantum Informa tion Set Theory (QIST). This paper provides an in-depth analysis of the mathematical and physical challenges faced by this attempt, establishes close connections with fields such as quantum logic, Boolean-valued models, categorical quantum mechanics, and the holographic principle, and proposes a roadmap for its gradual realization. We pay special attention to the dual role of the κ parameter (as a mathematical scale parameter and a physical constant) and its key role in connecting mathematical structures with physical reality. Although a complete theory has not yet been established, this paper provides a systematic research framework and a clear direction for the mathematical rigorization of quantum information ontology.

Johannes Branahl

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.

Berthold‐Georg Englert

Bihui Li

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.

Ted Bastin