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The parliament of observers - Against the observable axiom of quantum mechanics

This paper challenges the foundational quantum mechanical axiom that all observables correspond to self-adjoint operators, arguing that the noncommutative algebra of bounded operators fails to adequately formalize the empirical reality of measurement procedures when considered collectively.

Original authors: Markus Frembs, Andreas Döring

Published 2026-10-07

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Markus Frembs, Andreas Döring

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Modern physics rests on a foundation of mathematics that describes the behavior of the smallest things in the universe. For nearly a century, the most successful theory we have for this microscopic world has been quantum mechanics. It predicts the behavior of atoms and particles with a precision that is almost impossible to imagine, allowing us to build lasers, computers, and medical imaging devices. At the heart of this theory is a specific rule that connects the physical world we can measure to the abstract math we use to describe it. This rule states that every physical quantity we can measure, such as the spin of an electron or the energy of a photon, corresponds directly to a specific type of mathematical object called a self-adjoint operator. In the standard view, these mathematical objects are the perfect, one-to-one representatives of the physical things we observe in a laboratory.

However, a new analysis by physicists Markus Frembs and Andreas Döring suggests that this fundamental rule might be broken. They argue that while the mathematics of quantum mechanics works incredibly well for making predictions, the idea that these mathematical objects can serve as a consistent, unified list of everything we can measure is actually impossible. Their work does not claim that quantum mechanics is wrong in its predictions; rather, it claims that the way we currently understand the connection between the physical act of measuring and the mathematical symbols we use is flawed. If their argument holds, it means we have been trying to force a single, rigid mathematical structure onto a set of physical procedures that simply cannot fit into it without losing their meaning.

To understand the problem, one must first look at what a measurement actually is in the real world. When a scientist performs an experiment, they set up a specific arrangement of equipment, choose a specific way to read the results, and record the data. This entire setup is what the authors call an "experimental context." Crucially, a scientist can communicate this setup to others. They can write down exactly how they labeled the outcomes, how they calibrated their instruments, and how they moved from one measurement to another. This ability to record and communicate the details of an experiment in a clear, consistent language is what the authors call "consistent recordability." It is a basic requirement for any physical theory: if we cannot agree on what we are measuring and how we are measuring it, we cannot build a coherent science.

The standard rule of quantum mechanics tries to simplify this complex reality by saying that all these different experimental setups can be mapped directly onto a single collection of mathematical operators. In this view, if two different experiments are designed to measure the same physical quantity, they are treated as identical in the math, regardless of the specific equipment used. The authors argue that this simplification is too aggressive. They show that when you try to map the entire collection of possible measurements onto this mathematical structure, you run into a logical contradiction. The mathematical objects, when taken all together, do not allow for the kind of consistent labeling and communication that real scientists must perform.

The researchers demonstrate that this contradiction arises specifically when a system has three or more dimensions of complexity, which covers almost all interesting quantum systems. They prove that if you assume the standard rule is true, you reach a point where you can no longer be certain which mathematical object represents which physical measurement. It is as if you had a library where the books were arranged in a way that made it impossible to know which story was which, even though the stories themselves were perfectly clear. The mathematical structure forces different experimental contexts to merge in a way that erases the unique identity of each measurement. This means that the "algebra of observables," the mathematical list of everything we can measure, cannot actually exist as a single, consistent entity that faithfully represents the physical world.

This finding forces a re-evaluation of how we interpret the success of quantum mechanics. The theory has been incredibly accurate for a hundred years, but the authors suggest that this accuracy might be hiding a deeper inconsistency in its foundations. The problem is not with the calculations or the predictions, which remain valid. The problem is with the interpretation of what those calculations represent. The standard view assumes that the mathematical operators are the things themselves, but the authors argue that the operators are merely tools that work well in limited contexts but fail when we try to view them as a complete, unified description of reality. The mathematical structure is too rigid to accommodate the fluid, context-dependent nature of actual physical measurements.

The implications of this work reach back to the earliest days of quantum theory and the debates between its founders. The authors suggest that their result supports the idea that we must always keep a clear distinction between the mathematical formalism and the physical reality it describes. They argue that we cannot simply assume that the math tells the whole story of what is happening in the lab. Instead, we must acknowledge that the physical act of measurement, with all its specific details and the ability to communicate those details, is primary. The mathematical objects are secondary, useful for calculation but not a perfect mirror of the physical world. This perspective aligns with the views of early physicists who insisted that experimental results must always be described in clear, classical terms, even if the underlying theory is strange.

Ultimately, this paper does not overturn quantum mechanics, but it challenges the way we think about its basic building blocks. It suggests that the "observable axiom," the rule that links physical measurements to mathematical operators, is not a perfect bridge. The bridge exists and allows us to cross from theory to experiment, but it is not a solid, unchanging structure. It is a construction that works well enough for practical purposes but breaks down under close scrutiny when we try to understand the full scope of what we are measuring. The authors conclude that we need a new way to think about the relationship between the math and the machine, one that respects the complexity of the experimental context rather than trying to erase it. This is not a failure of the theory, but a call to refine our understanding of what the theory is actually telling us about the nature of reality.

Technical Summary: "The parliament of observers: Against the observable axiom of quantum mechanics"

Problem
The paper challenges the foundational axiom of quantum mechanics, often attributed to von Neumann and Dirac, which posits a bijective and faithful correspondence between physical observables and self-adjoint operators on a Hilbert space (denoted as axiom $(\dagger)$ ). While this axiom serves as the bridge between physical concepts and mathematical formalism, the authors argue that it fails under careful scrutiny when applied to the totality of observables. Specifically, the paper identifies a conflict between the standard mathematical representation of observables (the non-commutative algebra of bounded self-adjoint operators, $N_{sa}$ ) and the operational requirement that observables must be "consistently recordable."

The core issue is that the standard formalism abstracts away the "experimental context" (the specific setup and communication protocol of an observer) in a way that destroys the ability to unambiguously track and communicate observables across different contexts. The authors argue that for a physical theory to be well-posed, an experimenter must be able to record and communicate experimental setups and outcomes in a coherent classical language. The standard observable-operator correspondence, however, leads to a situation where this coherence is mathematically impossible for systems with dimension $d \geq 3$ .

Methodology
The authors employ a rigorous operational and algebraic approach to deconstruct the definition of an observable:

Minimal Formalisation of Observables: The paper defines an observable not merely as a mathematical object, but as a "real-world measurement" described by an experimental context $E$ (a measurable space of outcomes $\Omega_E$ ) and the algebra of bounded measurable functions $A_E$ on that space.
Consistent Recordability: The authors introduce the concept of a "classical coordinatisation." This is a set of bijections mapping the outcomes of various experimental contexts to a shared reference alphabet $\Lambda$ . A space of observables is defined as "consistently recordable" if these mappings can be chosen such that they are compatible across overlapping contexts. This ensures that an experimenter can unambiguously refer to the same observable across different experimental arrangements.
Experimental Context Connections: The paper formalizes the relationships between isomorphic experimental contexts using "context connections" ( $\nabla$ ). A key consistency condition (Proposition 1) requires that the composition of these connections around any closed loop of contexts must return to the identity map. This formalizes the requirement that an observer's record of an observable remains invariant regardless of the path taken through different experimental contexts.
Integration with Quantum Formalism: The authors then apply the standard observable-operator correspondence $(\dagger)$ , which maps observables to self-adjoint operators and experimental contexts to commutative subalgebras of $N_{sa}$ . They analyze whether the structure of $N_{sa}$ supports the consistency conditions derived from the operational definition.

Key Contributions and Results
The central result is Theorem 1, which states that the quantum-mechanical observable-operator correspondence $(\dagger)$ is incompatible with the notion that quantum observables are consistently recordable.

The Incompatibility: When the observable-operator correspondence is imposed, experimental contexts are identified with commutative subalgebras of $N_{sa}$ . The requirement for consistent recordability (Proposition 1) translates into a requirement for a "context connection" on the lattice of these commutative subalgebras that satisfies a specific cycle condition (Eq. 6).
Connection to Kochen-Specker: The authors demonstrate that the existence of such a consistent context connection is mathematically equivalent to the existence of a non-contextual hidden variable assignment. By invoking the reformulation of the Kochen-Specker theorem (specifically Theorem 1 in Ref. [41]), they show that for Hilbert spaces of dimension $d \geq 3$ , no such consistent connection exists.
The Consequence: Consequently, the set of self-adjoint operators $N_{sa}$ cannot serve as a consistently recordable space of observables. If one assumes that self-adjoint operators faithfully represent all observables, one arrives at a contradiction: the same observable (identified by its experimental context) would correspond to different operators depending on the path of context connections, or conversely, the same operator would represent different observables. The "eigenvector-eigenvalue link" is effectively broken in the sense that the mapping from physical observables to operators is not well-defined globally.

Significance and Claims
The paper claims to provide a new, more fundamental reading of the Kochen-Specker theorem and the nature of quantum contextuality:

Reversal of Perspective: Traditionally, the Kochen-Specker theorem is interpreted as a statement about the non-existence of microstates (hidden variables) that assign definite values to observables. The authors argue the perspective should be reversed: the theorem is a constraint on the mathematical formalisation of observables itself. It proves that the standard identification of observables with self-adjoint operators is structurally inadequate for a global, coherent description.
Critique of the "Algebra of Observables": The paper challenges the status of the non-commutative algebra $N_{sa}$ as the "algebra of observables." While the algebra exists as a mathematical object, the authors argue it lacks the necessary structural properties to be interpreted as the set of physically accessible, communicable measurements without losing coherence.
Relation to Bohr: The authors align their findings with Niels Bohr's doctrine of classical concepts, emphasizing that experimental accounts must be expressible in unambiguous classical terms. They argue that the standard formalism fails to maintain this unambiguity when considering the totality of contexts, a failure that is obscured by the probabilistic nature of the Born rule and the focus on specific states rather than the kinematical structure.
Modest Scope: The authors explicitly state they are not challenging the predictive success of quantum mechanics or the utility of operator algebras for specific calculations. Rather, they highlight a foundational inconsistency in the axiomatisation of the theory. They suggest that the "unreasonable effectiveness" of the formalism may mask a deep tension between the mathematical abstraction and the physical reality of measurement contexts.

In summary, the paper argues that the "observable axiom" is not a neutral bridge between physics and math, but a flawed abstraction that collapses under the requirement of global consistency. The solution, they imply, lies in recognizing that the mathematical representation of observables must retain explicit contextual data to remain physically meaningful.

Technical Summary: "The parliament of observers: Against the observable axiom of quantum mechanics"

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