[Paper Review] Reconstructing the formalism of quantum mechanics in the ``contextual objectivity" point of view

Philippe Grangier|arXiv (Cornell University)|Nov 29, 2001

Quantum Mechanics and Applications3 citations

TL;DR

This paper proposes a geometric, axiomatic reconstruction of quantum mechanics based on 'contextual objectivity,' where quantum states are defined by predictable, repeatable measurements on a system. By treating the number of exclusive modalities (N) and non-exclusive modalities (K) as fundamental, and using group-theoretic geometry of measurement apparatus transformations, it derives the Hilbert space structure and shows that K = N²—characteristic of quantum theory—emerges naturally from non-commutative geometry rather than probabilistic axioms.

ABSTRACT

In a previous preprint (quant-ph/0012122) we introduced a ``contextual objectivity" formulation of quantum mechanics (QM). A central feature of this approach is to define the quantum state in physical rather than in mathematical terms, in such a way that it may be given an "objective reality". Here we use some ideas about the system dimensionality, taken from quant-ph/0101012, to propose a possible axiomatic approach to QM. In this approach the structure of QM appears as a direct consequence of the non-commutative character of the (classical geometrical) group of "knobs transformations", that relate between themselves the different positions of the measurement apparatus.

Motivation & Objective

• To reconstruct the formalism of quantum mechanics from a physical, operational definition of the quantum state based on repeatable, certain measurements.
• To address the foundational issue of why quantum theory requires K = N² non-exclusive modalities (where N is the number of exclusive modalities), rather than K = N as in classical probability.
• To provide a geometric justification for the axioms of quantum mechanics—particularly the non-trivial K = K_A K_B and K = N²—by replacing probabilistic assumptions with geometric constraints on measurement apparatus transformations.
• To show that the quantum formalism emerges from the non-commutative structure of classical geometrical transformations of measurement settings, rather than from probabilistic continuity axioms.

Proposed method

• Defines a quantum state operationally as a set of physical quantities that can be predicted with certainty and measured repeatedly without disturbance, forming a complete set of commuting observables (CSCO).
• Introduces the distinction between exclusive modalities (N) — mutually exclusive outcomes of a measurement context — and non-exclusive modalities (K) — pure states of non-commuting observables.
• Uses the group of 'knob transformations' (classical geometric transformations of measurement apparatus settings) to model how different measurement contexts are related, with non-commutativity implying quantum structure.
• Applies geometric reasoning to show that the number of non-exclusive modalities (K) must scale as K = N², derived from the requirement that the transformation group acts irreducibly on the state space.
• Replaces Hardy’s probabilistic axioms (especially H2 and H4b on K) with geometric constraints, arguing that the K = K_A K_B and K = N² relations follow from the structure of the transformation group.
• Derives the Liouville space formulation of QM (including trace and time evolution) from the geometric structure of K and N, without assuming Hilbert space a priori.

Experimental results

Research questions

• RQ1 Can the formalism of quantum mechanics be reconstructed from a physical, operational definition of the quantum state, without assuming Hilbert space or probabilities a priori?
• RQ2 Why does the number of non-exclusive modalities (K) scale as K = N² rather than K = N, and what physical or geometric principle justifies this scaling?
• RQ3 Can the axioms of quantum mechanics—particularly those governing K and composite systems—be derived from geometric properties of measurement apparatus transformations rather than probabilistic continuity?
• RQ4 How does the non-commutativity of measurement apparatus transformations lead to the quantum structure of state space and the existence of superpositions?
• RQ5 What is the role of 'contextual objectivity' in grounding quantum states as objective features of reality, independent of observers or consciousness?

Key findings

• The quantum state is defined physically as a set of repeatable, certain measurements, which provides a foundation for 'contextual objectivity'—an objective reality for quantum states defined through macroscopic measurement contexts.
• The distinction between exclusive modalities (N) and non-exclusive modalities (K) captures a core quantum feature: that not all physical quantities can be simultaneously predicted with certainty, and that pure states cannot be completed.
• The paper shows that K = N² emerges naturally from the non-commutative geometry of measurement apparatus transformations, providing a geometric justification for the quantum scaling of degrees of freedom.
• The requirement that transformations between measurement contexts form a non-abelian group leads to the structure of quantum mechanics, with K = K_A K_B for composite systems, implying a tensor product structure.
• By replacing Hardy’s probabilistic continuity axiom (H5) with geometric continuity of transformations, the paper derives the same K = N² result, suggesting that quantum theory arises from geometric constraints on measurement settings.
• The Liouville space formulation of quantum mechanics—including the trace and time evolution—can be reconstructed from the geometric structure of K and N, without assuming Hilbert space a priori.