Peter Lehleiter

This paper develops a temporal interpretation of quantum mechanics from the guiding thesis that every change possesses a process-relative temporal structure. Chronomorphic reduction specifies when the event latency of a physical process admits a controlled local amplitude–phase representation, termed a chromogent. The wave function represents event latency, understood as the temporally organized possibilities of manifestation. The criteria require process and context dependence, controlled approximation errors, parameter reproduction, and phase closure where applicable. Chronomorphic reconstruction determines an observation equivalence class from a fixed approximation family whenever a compatible process exists. De Broglie’s connection between proper periodicity and relativistic phase waves motivates the approach; phase consistency links this temporal structure to quantization. Semiclassical analysis provides controlled realizations through eikonal and transport equations, symbol calculus, and phase closure. A general stability estimate shows that a controlled initial approximation retains its norm accuracy under the same unitary evolution, with uniform bounds on event probability errors, without requiring a semiclassical limit. A spectral construction for the harmonic oscillator realizes the criteria at fixed Planck constant, including error bounds and a specified convention for phase closure. A manifest event establishes a new process boundary; projection is interpreted as conditioned chronomorphic re-reduction. Quantum-likeness is characterized by the controlled representability of temporally organized event latency. The tangent is recovered as a locally flat representation wherever its errors remain within the specified observational tolerance. The proposal retains the established quantum dynamics and probability rules while reinterpreting the ontological role of their mathematical representation.

Preface The point of departure for this work was a simple question that would not let me go: What exactly does the linear Hilbert-space formalism describe? For decades, it had been treated as an ontological framework, as if the mathematical structure itself revealed the architecture of physical reality. Yet the more closely I examined its formation, the clearer it became that linearity was introduced as a methodological convenience rather than as a faithful representation of physical processes. That realization fundamentally altered how I understood the interpretational problems of quantum mechanics: many of them arise not from nature but from the assumptions embedded in the formalism itself. This insight led me back to the origins of quantum theory and to the historical moment in which Heisenberg’s Umdeutung program transformed an incomplete physical intui-tion into a fully linear representation. The transformation was brilliant, but it reshaped the conceptual landscape: temporal and dynamical structures receded behind algebraic invariants, and the very notion of an “object state” began to reflect the constraints of linearity rather than the ontology of physical systems. As the inquiry progressed, it became necessary to analyze not only this historical devel-opment but the structural logic of linear methods more generally. Linearity imposes categories that physical processes do not necessarily obey; it creates apparent unities where there are only approximations, and it conceals the temporal complexity that early quantum theorists still perceived. Understanding these limitations opened the door to a broader perspective: if the Hilbert-space formalism is a local linearization, what lies beyond it? The present book is an attempt to answer that question by tracing the conceptual path from the origins of the linear method to its present-day interpretational tensions and by exploring how a richer temporal and process-based description may restore intelligibil-ity. It does not offer a final theory, but it seeks to clarify the landscape in which such a theory would have to be built. Abstract This work examines the conceptual foundations of quantum theory and argues that the Hilbert-space formalism constitutes a linearized representation of fundamentally nonlin-ear and temporally structured processes. Part I analyzes the historical formation and categorical scope of the linear method, showing that many characteristic features of quantum mechanics—superposition, object states, and spectral decomposition—arise from one-dimensional invariance imposed by linearization rather than from physical ontology. The reconstruction of Heisenberg’s Umdeutung program reveals how the shift from process descriptions to linear operators generated an internally coherent cal-culus while obscuring the dynamical structure of physical systems. Parts II and III form a coherent unit, which is why the chapter numbering continues consecutively across both parts. Starting from the formalism of quantum mechanics, Part II develops the case for viewing quantum mechanics as an event-oriented theory of time. Part III subjects the textbook axioms to a critical analysis and proposes experi-ments aimed at testing whether quantum theory involves an intrinsic temporal structure beyond externally parametrized time. Part IV proposes reopening the question of electron trajectories. Part V discusses the results of a specific mesoscopic model, which once again demon-strates that quantum mechanics is not a matter of smallness, but of the available degrees of freedom. Part VI develops the corresponding positive account. Once temporal dynamics is treat-ed as intrinsic rather than parametric, complex energy parameters decompose naturally into circular and radial process modes. Circular modes encode temporal embedding, while radial modes encode activation and transitions. Their non-commuting generators yield a regime structure in which persistent circular evolution is intermittently interrupt-ed by radial episodes, giving rise to intelligible process trajectories. Ground-state stabil-ity follows not from spectral postulates but from minimal temporal rotation: the lowest configuration is the unique mode in which radial activation cannot operate. Within this framework, electron trajectories reappear—not as mechanical orbits, but as process-space curves determined by alternating temporal modes. Across all six parts, the work shows that the interpretational tensions of quantum mechanics arise from treating a local linearization as a global ontology. A coherent pic-ture emerges in which nonlinear temporal processes underlie the familiar linear struc-tures of Hilbert-space quantum mechanics, and in which dynamical intelligibility is re-stored without abandoning empirical adequacy.

2 Abstract The replacement of classical electron orbits by transition quantities was central to the construction of quantum mechanics. Its success raises a further question: which restrictions belong to the adopted representation, and which justify conclusions about the processes represented? This article examines that distinction through selected historical decisions and explicit mathematical examples. Scalar complex evolution separates phase rotation from amplitude change; finite-dimensional operator dynamics identifies exactly what norm conservation excludes in a fixed inner product. A coupled model then shows how a positive metric can restore unitarity to a matrix that is non-Hermitian in the original metric. Conditional quantum dynamics separates loss of an unnormalised norm from energy loss and from the occurrence of an event. These results do not establish electron trajectories, but they locate the assumptions needed to discuss them. A spatial process requires an evolution law and a connection to measurement records; stationary densities alone do not determine its paths. The resulting interpretation keeps the trajectory question open while retaining the restrictions of Bell-type results. The Dirac/Hestenes rotor provides a concrete example of non-scalar rotational structure that remains mathematically well defined without being identified with a microscopic particle trajectory. Chronomorphic reduction is proposed as a framework for investigating temporal process structure beyond a particular representation, without identifying mathematical possibility with physical confirmation. Keywords: quantum foundations; trajectories; temporal structure; Hermiticity; measurement; chronomorphic reduction

This note argues that mathematical formalism functions in physics as an epistemic instrument, analogous in important respects to a measuring device. Just as recorded signals may contain artefacts introduced by an apparatus, formal structures may partly reflect the architecture of a representation rather than the underlying physical process. The note therefore calls for a systematic distinction between target structure, instrumental mediation, and formal representation, especially in fundamental physics.

This paper distinguishes the predictive use of the linear Hilbert-space formalism from claims about its ontological completeness. A reconstruction of the bounded periodic sector of Heisenberg’s Umdeutung identifies the role of a uniform affine classical action rule in his quantum condition on transition amplitudes. Semiclassical refinements clarify the limits of that starting rule. The derivation examined does not establish universally exact physical linearity; spectral equivalence between matrix and wave mechanics within their common domain supplies no independent justification of this stronger claim. The analysis also distinguishes quantum-state representation from its reification as the complete physical condition of an enduring object. Measurement is examined through the different roles of unitary correlations, conditional projection, and definite events. Fourier representations and Bohm’s concept of metamorphosis motivate interpreting the evolving phase structure as an organization of event possibilities. On this reading, formal superposition does not describe simultaneously actual, mutually incompatible macroscopic properties of an object. The proposal retains the predictive formalism while reframing the measurement problem in terms of event actualization. It does not derive individual outcomes or the Born rule from a deeper dynamics. Keywords: Quantum foundations; Hilbert space; Linearity; Measurement problem; Reification; Event ontology.

Early quantum mechanics can be read, in a precise sense, as a theory of time. A spatially homogeneous proper periodicity gives rise, under a Lorentz transformation, to a spatiotemporal phase wave; de Broglie’s relation between rest energy and proper frequency then yields λ=h/p, and global phase consistency leads to the quantization of closed structures. This derivation motivates event latency: the wave function describes not the instantaneous ontic state of a persisting object, but the organization of possible events in terms of amplitude and phase. This paper explicitly addresses the previously omitted transition from the physical process to the wave function and formulates it as chronomorphic reduction. Chronomorphic reduction is the controlled local representation of a process by a chromogent χ=A exp⁡(iS/ℏ). Semiclassical analysis provides a controlled formal realization of this construction: the eikonal and transport equations determine phase and amplitude, the symbol calculus reproduces physical parameters, and the quasimode argument establishes the concentration of the reduced representative in the corresponding spectral region. A manifest event establishes a new process boundary; projection is interpreted as a conditioned chronomorphic re-reduction. Quantum-likeness thus denotes not a class of objects defined by size, but the controlled representability of temporally organized event latency. Keywords: time wave; event latency; chronomorphic reduction; chromogent; semiclassical analysis; phase closure; operators; projection

Early quantum mechanics can, in a precise sense, be read as a theory of time. A Lorentz transformation turns spatially homogeneous proper periodicity into a spacetime phase wave; de Broglie’s relation between rest energy and proper frequency then yields λ=h/p, while global phase consistency leads to the quantization of closed structures. This derivation motivates the concept of event latency: the wave function describes not the instantaneous ontic state of a persisting object, but the organization of possible events in terms of amplitude and phase. This paper explicitly addresses the previously omitted passage from physical process to wave function and formulates it as chronomorphic reduction. Chronomorphic reduction is the controlled local representation of a process by a chromogent χ=A exp⁡" " (iS/ℏ). Semiclassical analysis provides a controlled formal realization of this scheme: the eikonal and transport equations determine phase and amplitude, symbol calculus reproduces physical parameters, and the quasimode argument establishes the concentration of the reduced representative within the corresponding spectral region. A manifest event establishes a new process boundary; projection is interpreted as conditioned chronomorphic re-reduction. Quantum-like character thus denotes the capacity for a controlled representation of temporally organized event latency, rather than a class of objects distinguished by size. Keywords: temporal wave; event latency; chronomorphic reduction; chromogent; semiclassical analysis; phase closure; operators; projection

My original topic is time. Since I wanted to discuss this topic in a setting where the foundational questions of quantum mechanics are taken seriously, I decided to present my ideas at QIP26 in Växjö. My reason for entering quantum mechanics was not that I wanted to add another technical interpreta-tion to an already crowded field. My real topic was, and remains, time. Quantum mechanics became rele-vant to me because it is precisely there that the inherit-ed concept of time appears to become too narrow. The standard formalism still carries a strong at-tachment to classical thinking. This is visible, for exam-ple, in the language of states. A state is easily under-stood as something a system has at a time, much as in classical mechanics. But quantum mechanics repeated-ly undermines this picture. It confronts us with phase, latency, possible events, measurement conditions, and actual outcomes. These are not naturally described as ordinary classical states. Even more restrictive is the usual reduction of time to clock time: time as what clocks display. This notion is indispensable, of course. Every registered event must have a time of occurrence. But for a rich physical pro-cess, clock time alone may not be sufficient. The inner structure of a process is not exhausted by the number of seconds it lasts. What matters may also be the re-gimes, thresholds, transitions, and structural stages through which the process passes. A physical example is provided by dissipative struc-tures. In such systems, the decisive feature is not merely how long a regime remains stable, but which transition occurs when a control parameter is changed. The temporal structure of the process is then given not only by duration, but also by thresholds, regimes, and transformations. The same distinction can be illustrated by a simple image from literature or film. A screenplay can be printed in different formats, with different page num-bers, fonts, or line spacings. The external numbering changes, but the inner rhythm of the story does not. What matters dramatically is the sequence of scenes: conflict, crisis, turning point, apparent failure, and pos-sible resolution. In this analogy, the page numbers correspond to clock time. They are useful for orientation, but they do not capture the inner temporal form of the process. The dramatic sequence corresponds more closely to a structured temporal order: not merely duration, but ordered transformation. This is the perspective from which I approached quantum mechanics. I suspected that its conceptual difficulties might not arise merely from state-space ontology, but even more fundamentally from a too narrow concept of time. If time is reduced to external clock time, quantum mechanics is forced back into a classical framework that it constantly exceeds. It may then be more natural to read it not primarily as a theo-ry of states evolving in time, but as a theory of possible events, phase-conditioned latencies, and their actual-ization in time. One might object that this is only another descrip-tion of dynamics. But that objection would remain too superficial. It already assumes that dynamics can be understood within the classical framework of external clock time and localized change. My skepticism toward the usual description of dy-namics begins precisely here. As David Bohm already suggested, in quantum mechanics what we call dy-namics often looks less like the motion of an object through a pre-given coordinate space and more like a metamorphosis of the whole process. A metamorpho-sis is not adequately captured by a Cartesian–Newtonian picture in which reality is decomposed into localized processes described by coordinates. Of course, one can always localize aspects of a process. But if everything is localized in this way, the coherence of the transformation itself is easily lost. From this perspective, quantum dynamics is not merely change taking place in time. It is an inseparable unity of temporal rhythm, structural order, regimes, thresholds, and possible event formation. This also gives the proposal a holistic aspect. The relevant whole is not merely spatial. It is also temporal and structural. A process is not exhausted by its local components at a given clock time; it is organized by the relations, phases, thresholds, and possible transitions through which events can become actual. In this sense, the time-event reading points toward a holism along space, along time, and across levels of organization. In doing so, I made an important strategic decision. In my three previous books, I had already discussed many of the underlying ideas, including a pilot project based on mesoscopic random event data. However, these ideas and results are not easily accessible to readers who are not already familiar with the historical and conceptual background. I therefore decided not to begin by presenting a fully developed alternative on-tology. Instead, I wanted first to create a motivation for rethinking the foundations. For this reason, the talk begins with a structural de-construction of the established reading of quantum mechanics. I do not do this by entering the usual inter-pretive debate and simply adding another interpreta-tion to the existing list. Rather, I return to the origins of quantum mechanics and ask how the basic formal structures emerged: what Heisenberg and Schrödinger actually did, which classical and semiclassical assump-tions entered their constructions, and how these as-sumptions were later stabilized within the Hilbert-space formalism. The purpose of this deconstruction is not polemical. It is a structural analysis of the actual historical and mathematical steps by which quantum mechanics ac-quired its present form. Only if these steps are made visible can one ask responsibly whether the standard formalism should be read as a complete ontology, or whether it may instead be the first successful linear closure of a deeper structure. For that reason, I prepared the talk with particular care. The list of possible questions and objections be-came extensive, and it already covered many pages before the conference. At that stage, however, the material was still fragmentary. Some answers were worked out in detail, others existed only as sketches or alternative formulations. The first priority was the talk itself. The final form presented in this book emerged only after the talk and after many fruitful discussions at the conference. Several points became clearer to me there. One of them was that the connection between quan-tum mechanics, time, and events can be motivated more directly than I had initially formulated it. At the practical level, quantum mechanics is largely concerned with probabilities. But probabilities are probabilities for possible events. Without events, probabilities would not measure anything physically meaningful. Events, in turn, require a time of occur-rence. An event that never occurs at any time is not an event, but only a latency. In this simple chain — probability, event, time — one already sees why time and events should be cen-tral concepts in quantum mechanics. What is surprising is not that one can read quantum mechanics in this way. What is surprising is that this connection has been so easily obscured by the state-centered lan-guage of the standard formalism. The present book therefore has a deliberately two-fold structure. The talk gives the compact version of the argument. The question catalogue then opens the argument in different directions, addresses possible objections, and clarifies the methodological and onto-logical assumptions behind the proposal. It is not merely an appendix, but part of the argument itself. I have deliberately kept this structure compact. The aim is not to overload the reader with a complete trea-tise, but to make the central line of the argument visi-ble and then to clarify the main objections step by step. For a more detailed historical analysis of the origins of quantum mechanics, I recommend the books by Marco Giliberti and Luisa Lovisetti, especially their work on old quantum theory, early quantum mechanics, de Broglie’s phase-wave idea, Heisenberg’s 1925 Um-deutung paper, and Schrödinger’s wave mechan-ics [1,2].

We argue that the linear Hilbert-space formalism should be distinguished from its ontological interpretation. Heisenberg’s Umdeutung reorganized relations inherited from classical periodic motion into a calculus of quantum transitions, while Schrödinger’s wave mechanics expressed the Planck–de Broglie phase relations in linear differential form. Their equivalence establishes the coherence and empirical reach of a common linear closure, but does not by itself establish that physical reality possesses a global vector-space structure. Semiclassical refinements and the regime-dependent representations of quantum field theory reinforce the need to distinguish a successful representation from the ontology it is taken to represent. Beyond its diagnostic argument, the paper shows that iterating the reconstructed closure formally yields a self-consistent nonlinear Schrödinger-type equation. Whether this equation defines an empirically distinct dynamics remains open, and no physical collapse mechanism is proposed. A corresponding event-based reconstruction of quantum mechanics has been developed separately. The completed paper will use this result to distinguish formal superposition from the ontological superposition of objects and to reconsider the measurement problem. Keywords: Quantum foundations; Structural interpretation; Tangent linearization; Asymptotic instability; Nonlinear eigenvalue problems; Hilbert space formalism.

Separating de Broglie’s wave construction from physical identifications, we show that a homogeneous proper periodicity becomes a spatiotemporal phase wave under Lorentz transformation. We interpret this as a representation of a temporal structure. Applying his rest-energy–proper-frequency correspondence to wavelength yields λ=h/p. This motivates a reinterpretation of quantum mechanics: the wave function represents not a persisting object’s state but the phase-structured organization of latent event possibilities. Measurement is event manifestation, not wave-function collapse. No outcome-selecting mechanism is postulated; replacing state ontology with a time–event interpretation recasts individual-event unpredictability as a predictive limitation rather than as evidence of missing dynamics.