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This research proposes a "bi-trajectory formalism" derived strictly from sequential experimental observations, offering an alternative mathematical framework distinct from the standard state-focused formulation of quantum mechanics. By analyzing multi-time probability distributions without presupposing underlying structures, the study confirms that observed phenomena like interference and the Zeno effect necessitate a non-classical theory rooted in Hilbert spaces. This foundational work advances our understanding of how to reconstruct quantum formalism purely through phenomenological data rather than abstract axioms.
Phenomenological quantum mechanics: I. Phenomenology of quantum observables
arXiv:2410.14410v4 Announce Type: replace Abstract: We propose an exercise in which one attempts to deduce the formalism of quantum mechanics solely from phenomenological observations. The only assumed inputs are obtained through sequential probing of quantum systems; no presuppositions about the underlying mathematical structures are permitted. We demonstrate that it is indeed possible to derive, on this basis, a complete and fully functional formalism rooted in the structures of Hilbert spaces. However, the resulting formalism--the bi-trajectory formalism--differs significantly from the standard state-focused formulation. In Part I of the paper, we analyze the outcomes of various experiments involving sequential measurements of quantum observables. These outcomes are quantitatively described by phenomenological multi-time probability distributions, estimated from experimental data. Our first conclusion is that the theory describing these experiments must be non-classical: the measured sequences cannot be interpreted as sampling of a uni-trajectory representing the system's observable. The non-classical nature of the investigated systems manifests in a range of observed phenomena, including quantum interference, the quantum Zeno effect, and uncertainty relations between the measured observables.
Abstract:We propose an exercise in which one attempts to deduce the formalism of quantum mechanics solely from phenomenological observations. The only assumed inputs are obtained through sequential probing of quantum systems; no presuppositions about the underlying mathematical structures are permitted. We demonstrate that it is indeed possible to derive, on this basis, a complete and fully functional formalism rooted in the structures of Hilbert spaces. However, the resulting formalism--the bi-trajectory formalism--differs significantly from the standard state-focused formulation. In Part I of the paper, we analyze the outcomes of various experiments involving sequential measurements of quantum observables. These outcomes are quantitatively described by phenomenological multi-time probability distributions, estimated from experimental data. Our first conclusion is that the theory describing these experiments must be non-classical: the measured sequences cannot be interpreted as sampling of a uni-trajectory representing the system's observable. The non-classical nature of the investigated systems manifests in a range of observed phenomena, including quantum interference, the quantum Zeno effect, and uncertainty relations between the measured observables.
Part I of a two-part series; Part II arXiv:2507.04812. Note that the original submission arXiv:2410.14410v1 has been now split into two parts. Published in J. Phys. A: Math. Theor
Quantum Physics (quant-ph); Mathematical Physics (math-ph); History and Philosophy of Physics (physics.hist-ph)
arXiv:2410.14410 [quant-ph]
(or arXiv:2410.14410v4 [quant-ph] for this version)
J. Phys. A: Math. Theor. (2026)
https://doi.org/10.1088/1751-8121/ae97a0 DOI(s) linking to related resources
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