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arXiv:2610.07254

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

Markus Frembs, Andreas Döring

quant-phSubmitted Oct 5, 2026• Announced Oct 7, 2026• 0 citations
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Abstract

The 'unreasonable effectiveness of mathematics in the natural sciences' may at times give the impression that natural science, especially modern theoretical physics, is all but a formal apparatus. However, a key challenge of physics remains to align everyday notions of experience -- be it directly or mediated through instruments of arbitrary sophistication -- with their mathematical formalisation. This applies in particular to quantum theory, arguably the most successful theory of physics ever invented. One of its core assumptions postulates: $Observables\ correspond\ with\ self-adjoint\ operators.$ Here, we argue that already this most basic premise about the quantum formalism falls apart under careful scrutiny, at least when understood for all observables at once. In other words, we challenge the role of the noncommutative algebra of bounded operators as an adequate formalisation of the 'algebra of observables', understood as (equivalence classes of) measurement procedures that admit a coherent empirical interpretation.

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20 pages, 1 figure; comments welcome

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The 'unreasonable effectiveness of mathematics in the natural sciences' may at times give the impression that natural science, especially modern theoretical physics, is all but a formal apparatus. However, a key challenge of physics remains to align everyday notions of experience -- be it directly or mediated through instruments of arbitrary sophistication -- with their mathematical formalisation. This applies in particular to quantum theory, arguably the most successful theory of physics ever invented. One of its core assumptions postulates: Observables correspond with self−adjoint operators.Observables\ correspond\ with\ self-adjoint\ operators.Observables correspond with self−adjoint operators. Here, we argue that already this most basic premise about the quantum formalism falls apart under careful scrutiny, at least when understood for all observables at once. In other words, we challenge the role of the noncommutative algebra of bounded operators as an adequate formalisation of the 'algebra of observables', understood as (equivalence classes of) measurement procedures that admit a coherent empirical interpretation.
Quantum physics consists of a formal core that is universally agreed upon (basically being a piece of mathematics with suggestive names for concepts matching related concepts of experimental culture) and an interpretational halo (trying to make this relation more precise) that remains highly disputed even after more than 90 years of modern quantum physics. The latter is the subject of the interpretation of quantum mechanics (see Part II), where many interpretations coexist and compete for the attention of those interested in what quantum physics really means. In this chapter, an axiomatic introduction to the undisputed formal core of quantum physics is given. The axioms described present nonrelativistic quantum statistical mechanics in the Schrödinger picture. The relativistic case is outside the scope of these axioms, as it must be treated by quantum field theory; see Chapter 7. As in any axiomatic setting (necessary for a formal discipline), there are a number of different but equivalent sets of axioms or postulates that can be used to define formal quantum physics. Since they are equivalent, their choice is a matter of convenience. The choice presented here is a formulation featuring three aspects: It emphasises the similarity of quantum mechanics and classical mechanics, gives most direct access to statistical mechanics, and is free from allusions to measurement. The reason for the first is that this similarity serves as a guide to meaning. From the axiomatic setting for quantum mechanics presented below, classical mechanics (in the Koopman setting discussed in Section 7.10) is obtained by simply restricting the operators to be diagonal, so that all operations happen pointwise on the diagonal elements. Thus, multiplication is commutative, and one can identify operators and functions. In particular, the density operator degenerates into a probability density. Therefore, the quantum case appears as a generalization of the classical case allowing for noncommutativity, so that both q-observables and the density operator are (usually infinite-dimensional) operators. The reason for the second is that statistical mechanics is the main tool for applications of quantum physics to the macroscopic systems we are familiar with. The reason for the third is that realistic measurements constitute a complex process involving macroscopic detectors, hence should be explained by quantum statistical mechanics rather than be part of the axiomatic foundations themselves. (This is in marked contrast to other foundations, and distinguishes the present system of axioms.)
Quantum mechanics, despite its remarkable empirical success, has been subject to longstanding interpretational challenges. This work introduces the Prescribed Observation Problem (POP) framework, which derives the complete formulation of quantum theory from the empirical basis of measurement outcomes, treated as an optimization constraint. Rather than postulating quantum principles axiomatically, the POP framework formulates quantum mechanics as the solution to an entropy maximization problem subject to this constraint, resulting in quantum theory's axioms emerging as theorems – the optimal, least biased probability measure consistent with the given measurement data. Notably, the POP framework is sufficiently powerful to reduce the foundation of quantum mechanics to a single axiom. Conventionally, interpretations of quantum mechanics need only align with the theory's postulated axioms, allowing for a plurality of viable interpretations that cannot definitively rule each other out. In contrast, the POP framework derives the axioms as theorems directly from empirical first principles, grounding them in observational constraints from the outset. This substantially restricts the allowable theoretical landscape, imposing stringent requirements for the viability of interpretations and invalidating all those that contradict this genesis. By engendering quantum theory from a formal empirical footing, the POP framework offers a compelling resolution to the enduring interpretational puzzles that have long plagued the field, effectively eliminating all but one interpretation consistent with both the axioms and their empirical genesis.
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Some of the problems connected with the interpretation of quantum mechanics are enumerated, in particular those related to some well known paradoxes and, above all, to the measurement process. We then show how the so called "Physics Laboratory Assumption" introduced in [1], which considers as "observables'' only the self-adjoint operators corresponding to existing measuring instruments, can propose a new perspective on the aforementioned problems and can replace the wavefunction collapse postulate. [1] Luigi E. Picasso, "On the Concept of State in Quantum Mechanics: Another Way to Decoherence?'' Int. J. Theor. Phys. 62 (2), (2023)
It presents the evolution of the quantum observables from their classical counterpart and distinctly analyses the meaning of observable in context to both the classical and the quantum realms, with an emphasis on W. Heisenberg's realization of observable and unobservable [1925]. It is stated that quantum observables are, in the first place, indeed distinctive to classical observables since each appears unobservable to the other; however, to associate physical essence as possessed by the latter, the expectation value of the former is determined. Further, the possibility of mapping quantum information to the classical ones through the action of the operator may be realized after re-interpreting analogies between the quantum commutator and Poisson brackets.