Comment on 'There is No Quantum World' by Jeffrey Bub

Comment on “There is No Quantum World” by Jeffrey Bub. Philippe Grangier Laboratoire Charles Fabry, IOGS, CNRS, Universit´e Paris Saclay, F91127 Palaiseau, France. In a recent preprint [1] Jeffrey Bub presents a discussion of neo-Bohrian interpretations of quantum mechanics, and also of von Neumann’s work on infinite tensor products [2]. He rightfully writes that this work provides a theoretical framework that deflates the measurement problem and justifies Bohr’s insistence on the primacy of classical concepts. But then he rejects these ideas, on the basis that the infinity limit is “never reached for any real system composed of a finite number of elementary systems”. In this note we present opposite views on two major points: first, admitting mathematical infinities in a physical theory is not a problem, if properly done; second, the critics of [3–5] comes with a major misunderstanding of these papers: they don’t ask about “the significance of the transition from classical to quantum mechanics”, but they start from a physical ontology where classical and quantum physics need each other from the beginning. This is because they postulate that a microscopic physical object (or degree of freedom) always appears as a quantum system, within a classical context. Here we argue why this (neo-Bohrian) position makes sense. I. INTRODUCTION: DISAGREEMENTS AND PROPOSED CLARIFICATIONS. Bub’s paper [1] raises important issues about the interpretation of quantum mechanics and about the status of classical concepts. We agree with much of his historical and philosophical discussion, and share his interest in von Neumann’s infinite-product framework. However we disagree on two major issues. The first disagreement is about the methodological statement that the infinite limit would be irrelevant to physics because no real system attains it exactly. Despite this, it is standard practice in physics to use mathematical limits as idealizations that reveal robust, qualitative behavior of large but finite systems. Well-known examples include the thermodynamic limit in statistical mechanics, which explains phase transitions and emergent order, and also continuum limits in mechanics and field theory, which allow differential equations to model media composed of discrete atoms. Therefore, arguing that an infinite construction is irrelevant because it is never literally attained conflates ontological literalism with methodological usefulness. The infinite tensor product can be used in the same spirit: it provides a mathematically controlled idealization that describes the structural features (super-selection sectors, stable records) that macroscopic systems exhibit. More details will be given below. The second disagreement stems from quite differing ontological emphases. In our CSM (Contexts, Systems and Modalities) approach [3–5] the basic ontology is not “a purely quantum world” or “a purely classical world” but quantum systems that are described within classical contexts. This is neither a retreat to naive classical realism, nor a mere revival of Bohr’s complementarity: it is a realist stance that recognizes the irreducible role of contexts (macroscopic apparatus, classical records) in the operational meaning of quantum states and probabilities. From this starting point, the mathematical framework that includes infinite tensor products and the associated sector structure becomes a natural tool to model how macroscopic contexts stabilize definite outcomes. Then quan- tum and classical physics can be described in a unified algebraic formalism - without one “emerging” from the other. In order to give some flesh to the statements above, we quote in italics some paragraphs extracted from Bub’s paper [1], and provide specific comments and references for each one. II. COMMENTED QUOTES FROM BUB’S PREPRINT A. A quote from von Neumann [6] Another way to put it is that if you take the case of an orthogonal space, those mappings of this space on itself, which leave orthogonality intact, leave all angles intact, in other words, in those systems which can be used as models of the logical background for quantum theory, it is true that as soon as all the ordinary concepts of logic are fixed under some isomorphic transformation, all of probability theory is already fixed. Comment: This enigmatic sentence by John von Neumann was visionary, given that he told this in 1954, before knowing Uhlhorn’s theorem (about leaving orthogonality intact) and Gleason’s theorems (about an isomorphic transformation fixing all of (quantum) probability theory). These issues are discussed in details in [7, 8]. arXiv:2512.22965v1 [quant-ph] 28 Dec 2025 2 B. The sort of realist explanation we are familiar with - or not On this view, quantum mechanics does not provide a representational explanation of events. Noncommutativity or non-Booleanity makes quantum mechanics quite unlike any theory we have dealt with before, and there is no reason, apart from tradition, to assume

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