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Comment on arXiv:2601.04248v1: Superposition of states in quantum theory (J.-M. Vigoureux)

Mikołaj Sienicki, Krzysztof Sienicki

TL;DR

Vigoureux's Möbius-type composition ⊕ is evaluated as a replacement for quantum superposition. For two components, ⊕ reduces to a scalar multiple of the ordinary sum, so after normalization it leaves predictions unchanged; for three or more components, a natural recursive extension becomes non-associative and bracketing-dependent, yielding different rays and implying a non-unique state. The paper also argues that the inclusion–exclusion analogy and the optics Fabry–Perot resummation do not justify a foundational shift away from linear Hilbert-space structure. Overall, the critique shows that ⊕ fails as a viable modification to quantum theory, because it does not preserve unambiguous state preparation and leads to ill-defined or path-dependent outcomes, while the optical analogy reflects linear resummation rather than a replacement of superposition.

Abstract

Vigoureux suggests replacing the usual linear superposition rule of quantum mechanics with a M"obius-type "composition law" $\oplus$, motivated by (i) bounded-domain composition laws in special relativity, (ii) familiar transfer-matrix formulas in multilayer optics, and (iii) an analogy with the inclusion-exclusion rule for classical probabilities. In this note we explain why the proposal does not work as a modification of quantum theory. For two components, the new rule differs from the ordinary sum only by an overall scalar factor, so after normalization it represents the same ray and cannot change any physical prediction. For three or more components, if one extends the two-term prescription in the natural recursive way, the result becomes bracket/order dependent and can even change the ray, so a "state" is no longer uniquely determined by a given preparation. We also clarify why the inclusion-exclusion argument and the optics analogy do not support a foundational change to Hilbert-space linearity.

Comment on arXiv:2601.04248v1: Superposition of states in quantum theory (J.-M. Vigoureux)

TL;DR

Vigoureux's Möbius-type composition ⊕ is evaluated as a replacement for quantum superposition. For two components, ⊕ reduces to a scalar multiple of the ordinary sum, so after normalization it leaves predictions unchanged; for three or more components, a natural recursive extension becomes non-associative and bracketing-dependent, yielding different rays and implying a non-unique state. The paper also argues that the inclusion–exclusion analogy and the optics Fabry–Perot resummation do not justify a foundational shift away from linear Hilbert-space structure. Overall, the critique shows that ⊕ fails as a viable modification to quantum theory, because it does not preserve unambiguous state preparation and leads to ill-defined or path-dependent outcomes, while the optical analogy reflects linear resummation rather than a replacement of superposition.

Abstract

Vigoureux suggests replacing the usual linear superposition rule of quantum mechanics with a M"obius-type "composition law" , motivated by (i) bounded-domain composition laws in special relativity, (ii) familiar transfer-matrix formulas in multilayer optics, and (iii) an analogy with the inclusion-exclusion rule for classical probabilities. In this note we explain why the proposal does not work as a modification of quantum theory. For two components, the new rule differs from the ordinary sum only by an overall scalar factor, so after normalization it represents the same ray and cannot change any physical prediction. For three or more components, if one extends the two-term prescription in the natural recursive way, the result becomes bracket/order dependent and can even change the ray, so a "state" is no longer uniquely determined by a given preparation. We also clarify why the inclusion-exclusion argument and the optics analogy do not support a foundational change to Hilbert-space linearity.
Paper Structure (12 sections, 1 theorem, 16 equations)

This paper contains 12 sections, 1 theorem, 16 equations.

Table of Contents

  1. Introduction: "deformations" as representations vs. new physics
  2. What Vigoureux proposes
  3. Fatal point #1: for two components the proposal is ray-trivial (or contradicts its own claims)
  4. Ray equivalence: $\mathbin{\oplus}$ differs from $+$ only by an overall scalar
  5. The denominator is not a normalization factor
  6. Ill-definedness: poles and cancellation limits
  7. Fatal point #2: for three or more components the rule is bracket/order dependent
  8. The "bounded amplitudes" motivation is a category error
  9. The inclusion--exclusion probability analogy does not carry to quantum events
  10. Why the optics analogy cuts the other way
  11. The Schrödinger-equation remark is true only in a way that makes $\mathbin{\oplus}$ vacuous
  12. Conclusion

Key Result

Proposition 1

There exist $u,v,w$ in a finite-dimensional Hilbert space such that $(v\mathbin{\oplus} w)\mathbin{\oplus} u$ and $v\mathbin{\oplus}(w\mathbin{\oplus} u)$ are not proportional, hence represent different rays.

Theorems & Definitions (2)

  • Proposition 1: Bracket dependence can change the ray
  • proof