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

TL;DR

This paper critically examines a proposed non-linear superposition rule in quantum mechanics, demonstrating it cannot alter physical predictions or replace the standard linear superposition due to fundamental mathematical constraints.

Contribution

It provides a detailed analysis showing that the proposed M"obius-type composition law cannot serve as a valid modification of quantum superposition.

Findings

The new rule differs from the standard sum only by a scalar factor for two components.

For three or more components, the rule becomes order-dependent and can alter the quantum state.

The inclusion-exclusion principle and optics analogy do not justify changing Hilbert-space linearity.

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.