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 the multi-time probability distributions estimated from the results of sequential measurements of quantum observables; no presuppositions about the underlying mathematical structures are permitted. In the concluding Part II of the paper, we carry out the deduction of the formalism from the phenomenological inputs described in Part I. We show that the resulting formalism exhibits an affinity with Hilbert spaces, and we derive an explicit representation in terms of those mathematical structures. Analogues of the obtained elementary building blocks—such as projection operators—are readily identifiable within the standard formalism. However, once these building blocks are assembled according to the blueprint of the deduced bi-trajectory formalism, it becomes evident that the new and the standard formalisms differ substantially at the conceptual level. These differences do not negate the fact that both formalisms are in perfect agreement with respect to empirically testable predictions. Rather, the emergence of a novel, non-standard formulation should be seen as a relatively rare opportunity to reassess, from a fresh perspective, some of the long-standing foundational issues in the theory. The hope is that the new approach may prove more successful in addressing problems that have resisted resolution within the established theoretical framework.
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 the multi-time probability distributions estimated from the results of sequential measurements of quantum observables; no presuppositions about the underlying mathematical structures are permitted. In the concluding Part II of the paper, we carry out the deduction of the formalism from the phenomenological inputs described in Part I. We show that the resulting formalism exhibits an affinity with Hilbert spaces, and we derive an explicit representation in terms of those mathematical structures. Analogues of the obtained elementary building blocks—such as projection operators—are readily identifiable within the standard formalism. However, once these building blocks are assembled according to the blueprint of the deduced bi-trajectory formalism, it becomes evident that the new and the standard formalisms differ substantially at the conceptual level. These differences do not negate the fact that both formalisms are in perfect agreement with respect to empirically testable predictions. Rather, the emergence of a novel, non-standard formulation should be seen as a relatively rare opportunity to reassess, from a fresh perspective, some of the long-standing foundational issues in the theory. The hope is that the new approach may prove more successful in addressing problems that have resisted resolution within the established theoretical framework.
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.
In order to solve the problem of which is the real logic of Quantum Mechanics (QM), we suggest to study the mutual relationships between the physical theory (as a whole, that is both its experimental part and its theoretical part), its mathematics and its logic. That results to be a more general problem than what is considered by traditional studies on Quantum Logic (QL). These studies focus attention on the problem of the relationship between the experimental part of the physical theory and only its mathematics, from which logic is then derived. The new approach, that we suggest, generalises the interpretation that one of us (A. D.) successfully tested on the foundations of classical physical theories. The foundations of a theory are interpreted in terms of two options on the kind of mathematics, either classical one or constructive one, and the kind of logic, either classical logic or intuitionistic logic. It is remarkable that each theory making use of intuitionistic logic – i. e. L. Carnot’s classical mechanics, S. Carnot’s thermodynamics, Lobacevskij’s non-Euclidean geometry is organised in order to solve an universal problem (expressed by a double negated sentence), rather than in an entirely deductive way; hence, a non-classical logic does not belong exclusively to QM. From an historical and foundational analysis about QM, we conclude that in QM it is the law of double negation and not modular law (introduced by Birkhoff and von Neumann without giving physical evidence to it) which fails, i. e. which represents the true difference between Classical Logic (CL) and QL. We have looked for previous works on QL supporting our suggestion; in the past Destouches-Fevrier (1948), Fine (1974), Bell-Hallett (1981), Gauthier (1983), Adelman-Corbett (1995) already defended this thesis on the relevance of double negation law for QL. Under this light, QL results to have the same logic of a classical theory organised in a problematic manner. Hence, QM has to be referred to intuitionistic logic, in the same way every theory which is developed around a fundamental problem, expressed by a double negated sentence, does. In particular, by following the founder of intuitionism, Brouwer, we reject the identification of intuitionistic logic with an entirely deductive system; rather, the logical system of QL, as a theory, is organised in such a way to put a crucial problem to be solved. In agreement with this kind of organisation, among the several QM formulations we choose Heisenberg’s formulation, which was ignored by previous authors on QL. This formulation is aimed to solve the problem of quantum measurement, that originates the uncertainty methodological principle. Moreover, it is remarkable that it makes use of an operational mathematics (discrete matrices only). We interpret its characteristic arguing about its principal problem through double negated sentences; it results to be a moving from arguments belonging to non-classical logic expressed by a double negated sentence to the recognition of a suitable mathematics for QM formulation. In sum, QL shows how it is possible to state the relationship between QM and mathematics. This point has been almost ignored by the most traditional approaches to QL, which rather drew it from an a priori mathematics of QM (commonly, Hilbert space). As a verification of our approach, we show that even both Lobachevskij’s geometry and S. Carnot’s thermodynamics argued in order to achieve mathematical formulas expressing the content of the basic double negated sentences of respective theory. In the same way the recent book on QM by T. F. Jordan formalised his matrix mechanics. 1. Logic in classical physics and in Quantum Mechanics An historical analysis of Quantum Mechanics (QM) shows that there were at the origin of the theory two distinct formulations: Matrix Mechanics (Heisenberg, 1925) and Wave Mechanics (Schroedinger, 1926). Although Dirac subsumed these two formulations in a comprehensive mathematical framework (Hilbert space), we think that the organisation of the principles is different in Matrix Mechanics from that in Wave Mechanics, and this difference implies different logics. So we think that in order to characterise the logic of QM it is necessary to distinguish between this two formulations. In Matrix Mechanics, Heisenberg’s basic idea was to explain atomic spectra not in terms of atomic orbits (as Old Quantum Theory tried to make), but only by means of the study of experimental data; i. e., frequencies and intensities of light both emitted and absorbed by matter. Through a
TitleStructural Observability: A Framework for Quantum Measurement and Probability This paper proposes the Structural Observability (SO) framework for interpreting quantum measurement and probability. The framework distinguishes between a Complete Relational Configuration (CRC), representing the full relational structure of reality, and the Limited Observational Reference Frame (L-ORF), representing the restricted observational perspective available to observers embedded within spacetime. Within this structure, definite outcomes arise when components of a system’s state become associated with distinguishable physical records in spacetime, a condition termed Structural Observability. Quantum probabilities then arise from the structural limitations imposed by the L-ORF description rather than from fundamental randomness in the underlying configuration. The framework preserves the standard mathematical formalism of quantum mechanics—Hilbert space, unitary evolution, and the Born rule—while providing a structural account of why these elements appear in the observational description. Version 2 — What's new: This version represents a substantial development of the framework presented in Version 1. Major additions include: a formal mathematical structure section with five axioms for the accessibility map Λ, a causal context space (Σ, ≤) making the temporal structure of the framework explicit, and a three-part Structural Observability condition stated as a formal definition with a proof that decoherence is necessary but not sufficient for SO. Six worked examples are included, replacing the purely conceptual treatment of Version 1. Two novel experimental predictions are introduced: a first prediction concerning a critical stability threshold τ below which raw unpost-selected interference is directly observable, and a second prediction concerning a retrievability timescale τR* identified with the quantum information scrambling time, above which interference may again become observable. Together, these define a three-regime non-monotonic structure of interference visibility with no analogue in standard quantum mechanics. A full Statement on AI Assistance has been added, documenting the role of AI language models in the development of the work.* Keywords Quantum foundationsQuantum measurementBorn ruleDecoherenceRelational physics Structural observabilityQuantum interpretation
This work re-examines the long-standing interpretation problems of quantum mechanics—measurement, collapse, probability, irreversibility, the observer, and entanglement—not by proposing a new physical theory, but by relocating the structure of the questionsthat have framed these problems. The paper argues that many interpretational tensions arise not from the formalism itself,but from how concepts such as measurement, state, time, and probability have beenpositioned within the explanatory structure. Measurement is treated not as a privilegedphysical event, but as a structural condition associated with record formation.Collapse is interpreted as the unavoidable fixation of perspective once such records exist,rather than as a dynamical process added to the theory. Time is relocated from a global background parameter to a local phenomenon that appearsonly while interactions remain unresolved. Irreversibility is accordingly understoodnot as a fundamental temporal direction, but as the closure of a perspective.Probability is reinterpreted not as future uncertainty or intrinsic randomness,but as a structural summary of how multiple interactions close in the present. The observer is not treated as an external subject or agent, but as a boundary positionat which correlations between system and environment become fixed.Entanglement is understood not as nonlocal influence, but as the manifestation ofan indivisible whole structure that cannot be decomposed into independent parts. This work does not modify the mathematical formalism or predictive success of quantum mechanics.Instead, it constrains the interpretational space by showing how different questionsabout the same formal results generate persistent disagreement.The contribution is therefore conceptual rather than predictive,and is intended as a foundation for further discussion in quantum foundationsand phase-based theoretical frameworks.
We derive the Hilbert space formalism of quantum mechanics from epistemic principles. A key assumption is that a physical theory that relies on entities or distinctions that are unknowable in principle gives rise to wrong predictions. An epistemic formalism is developed, where concepts like individual and collective knowledge are used, and knowledge may be actual or potential. The physical state S corresponds to the collective potential knowledge. The state S is a subset of a state space S={Z}, such that S always contains several elements Z, which correspond to unattainable states of complete potential knowledge of the world. The evolution of S cannot be determined in terms of the individual evolution of the elements Z, unlike the evolution of an ensemble in classical phase space. The evolution of S is described in terms of sequential time n∈N, which is updated according to n→n+1 each time potential knowledge changes. In certain experimental contexts C, there is initial knowledge at time n that a given series of properties P,P′,… will be observed within a given time frame, meaning that a series of values p,p′,… of these properties will become known. At time n, it is just known that these values belong to predefined, finite sets {p},{p′},…. In such a context C, it is possible to define a complex Hilbert space HC on top of S, in which the elements are contextual state vectors SˉC. Born's rule to calculate the probabilities to find the values p,p′,… is derived as the only generally applicable such rule. Also, we can associate a self-adjoint operator Pˉ with eigenvalues {p} to each property P observed within C. These operators obey [Pˉ,Pˉ′]=0 if and only if the precise values of P and P′ are simultaneoulsy knowable.