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Non-A Priori Construction of Time III: Physical Realization of Clocks, the Quantum Time Axiom, and Atemporal Quantum Mechanics

Kei Itoh

Published Oct 29, 2026• 0 citations
Publisher

Abstract

AbstractAs the third paper on the non-a-priori construction of time, this paper constructs clocks from mathematical temporal-like structures, defines time in quantum theory axiomatically, and then forgets its temporal interpretation to obtain a formulation of quantum evolution in which physical time is not a primitive variable. Numberless time and temporal numbers from Papers I and II are reintroduced as repetition and recurrence through group actions, and as succession and history through the Peano successor operation, respectively. Physical clocks are defined by equipping their combination with readout and calibration, and an ideal equivalence relation between different types of clocks is examined. A “quantum time axiom” is adopted to define clock time through the energy gap and accumulated relative phase of a two-level quantum clock. The mutual reference between quantum theory and clocks is thereby treated as an axiomatic stopping point grounded in concrete physical realization. A forgetting operation then removes the temporal interpretation while retaining successor operations, group actions, and transformations on Hilbert space as pure operations. A dimensionless unitary group is constructed from a self-adjoint energy operator and the energy scale of a reference clock, and Peano updates, continuous extensions, and invariance under changes of reference clock are examined. The standard Schrödinger equation is recovered through the quantum time axiom, and the dimensionless differential equation is presented as one construction of an atemporal Schrödinger equation. A possible adjunction between free construction and forgetting for time in general is also considered. Finally, comparison with the Page–Wootters mechanism and related approaches distinguishes the nonprimitiveness of a time variable from timelessness in a completely closed system. The paper presents a coherent foundational construction of quantum theory: construct time, realize it as a clock, axiomatize time in quantum theory, and then forget its temporal interpretation. Detailed mathematical derivations and explanations of the constructions presented in this paper are provided in a separate supplementary manuscript. AI Assistance DisclosureGenerative AI was used as an auxiliary tool in drafting, adjusting expressions, and organizing the structure of this paper. The author bears final responsibility for the text, claims, citations, and factual verification. Keywords: non-a-priori construction of time, numberless time, temporal numbers, rhythm time, count time, quantum clocks, quantum time axiom, forgetting the temporal interpretation, atemporal quantum mechanics, group actions, Peano arithmetic, reference-clock invariance

DOI
10.5281/zenodo.23762339

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AbstractAs the third paper on the non-a-priori construction of time, this paper constructs clocks from mathematical temporal-like structures, defines time in quantum theory axiomatically, and then forgets its temporal interpretation to obtain a formulation of quantum evolution in which physical time is not a primitive variable. Numberless time and temporal numbers from Papers I and II are reintroduced as repetition and recurrence through group actions, and as succession and history through the Peano successor operation, respectively. Physical clocks are defined by equipping their combination with readout and calibration, and an ideal equivalence relation between different types of clocks is examined. A “quantum time axiom” is adopted to define clock time through the energy gap and accumulated relative phase of a two-level quantum clock. The mutual reference between quantum theory and clocks is thereby treated as an axiomatic stopping point grounded in concrete physical realization. A forgetting operation then removes the temporal interpretation while retaining successor operations, group actions, and transformations on Hilbert space as pure operations. A dimensionless unitary group is constructed from a self-adjoint energy operator and the energy scale of a reference clock, and Peano updates, continuous extensions, and invariance under changes of reference clock are examined. The standard Schrödinger equation is recovered through the quantum time axiom, and the dimensionless differential equation is presented as one construction of an atemporal Schrödinger equation. A possible adjunction between free construction and forgetting for time in general is also considered. Finally, comparison with the Page–Wootters mechanism and related approaches distinguishes the nonprimitiveness of a time variable from timelessness in a completely closed system. The paper presents a coherent foundational construction of quantum theory: construct time, realize it as a clock, axiomatize time in quantum theory, and then forget its temporal interpretation. Detailed mathematical derivations and explanations of the constructions presented in this paper are provided in a separate supplementary manuscript. AI Assistance DisclosureGenerative AI was used as an auxiliary tool in drafting, adjusting expressions, and organizing the structure of this paper. The author bears final responsibility for the text, claims, citations, and factual verification. Keywords: non-a-priori construction of time, numberless time, temporal numbers, rhythm time, count time, quantum clocks, quantum time axiom, forgetting the temporal interpretation, atemporal quantum mechanics, group actions, Peano arithmetic, reference-clock invariance
This paper extends the temporal concept of Sequential Time Theory (STT) to the problem of metrology in the quantum domain. In STT, a time interval Δt is constructed by a counter n representing the number of realizations of a standard displacement ΔQ_unit, while time t serves as a coordinate label assigned to this sequence. STT explicitly posits that "time is a scaling of physical change" and that "time is constructed solely by clock phenomena subject to the same conditions within a local system." Consequently, in regimes where ΔQ is indistinguishable, time does not "cease to exist" but becomes "unconstructible (undefined)." Based on these definitions and axioms, this paper (i) organizes criteria for distinguishing between science and metaphysics, (ii) identifies conceptual confusions arising from the implicit assumption of an external classical clock in quantum theory, and (iii) proposes a relativity principle regarding differences in observation scale (e.g., macroscopic observation of microscopic systems). We demonstrate that issues such as measurement, classicality, and tunneling time can be unifiedly organized as problems of "choice of clock (ΔQ)" and "recording (event-token) protocols."Furthermore, we provide a mathematical derivation (Appendix A) showing that the intrinsic uncertainty of a physical clock (σ) naturally leads to the suppression of quantum interference terms (e−ω2σ2e^{-\omega^2\sigma^2}e−ω2σ2), thereby identifying the metrological origin of decoherence and the phenomenological arrow of time.Version 4 Update: Major revision including Appendix A (Mathematical Proof of Intrinsic Decoherence) and Appendix B (Definition of Phenomenon Arrow). This version provides the quantitative foundation for the theory.
This essay establishes the foundational claim of the Time as Scalar framework: time is not a constituent of nature but a measuring instrument applied by finite observers. The universe operates through causal structure, geometry, and physical law without requiring a clock. What requires time is the observer—specifically memory-based comparison that allows events to be sequenced and rendered meaningful. The work formally separates three compatible layers: the ontological layer of causal structure (physics), the epistemological layer of temporal measurement (observer-based), and the phenomenological layer of lived temporal experience. Central to the argument is the time-independent Schrödinger equation, interpreted not as a special case but as the natural baseline of a clock-free universe. Time appears only when observers measure deviations from stationary geometry. Memory functions as the anchoring end of the temporal ruler, and forgetting is framed as necessary instrument maintenance. Randomness and surprise emerge as properties of the observer’s measurement gap rather than of nature itself. This document serves as a foundation for further formal development connecting Scope Theory, Hilbert space structure, and quantum measurement regularization.
This paper establishes the Quantum Temporal Non-Constructibility Theorem (QTNT): the external parameter t appearing in the Schrödinger equation carries no operational meaning for an observer embedded within a closed quantum system. Working strictly within the standard Hilbert-space formalism, using only Stone’s theorem, POVM measurement theory, and unitary evolution, the paper proves that all observable quantum data (POVM probabilities, conditional statistics, and multi-time correlators) are invariant under smooth, strictly monotonic reparametrizations t' = f(t), provided the Hamiltonian transforms covariantly as H' = H (dt/dt'). The theorem shows that internal observers, who have access only to reduced states and subsystem POVMs, cannot detect or reconstruct the external Schrödinger-time parameter. Observable histories depend solely on the geometric orbit traced by the state in projective Hilbert space, not on the particular parameter used to traverse it. As a result, Schrödinger-time functions as a gauge label rather than a measurable physical quantity. The analysis provides an operational reinterpretation of Pauli’s theorem, clarifies the correlational nature of quantum clocks, and completes the temporal non-constructibility program alongside its classical and relativistic counterparts. Appendices supply supporting derivations from Stone’s theorem, Pauli’s theorem, quantum clocks, explicit examples, and the path-integral formulation. Note. The corresponding classical and relativistic results are developed in the companion papers The Temporal Non-Constructibility Theorem in Classical Dynamical Systems (Zenodo, DOI: 10.5281/zenodo.17753466) and Temporal Non-Constructibility in General Relativity (Zenodo, DOI: 10.5281/zenodo.17765323) respectively.
Following previous works by E. Prugovečki [Physica A 91A, 202 (1978) and Stochastic Quantum Mechanics and Quantum Space–time (Reidel, Dordrecht, 1986)] on common features of classical and quantum mechanics, we develop a unified mathematical framework for classical and quantum mechanics (based on L2-spaces over classical phase space), in order to investigate to what extent quantum mechanics can be obtained as a simple modification of classical mechanics (on both logical and analytical levels). To obtain this unified framework, we split quantum theory in two parts: (i) general quantum axiomatics (a system is described by a state in a Hilbert space, observables are self-adjoints operators, and so on) and (ii) quantum mechanics proper that specifies the Hilbert space as L2(Rn); the Heisenberg rule [pi,qj]=−iℏδij with p=−iℏ∇, the free Hamiltonian H=−ℏ2Δ/2m and so on. We show that general quantum axiomatics (up to a supplementary “axiom of classicity”) can be used as a nonstandard mathematical ground to formulate physical ideas and equations of ordinary classical statistical mechanics. So, the question of a “true quantization” with “ℏ” must be seen as an independent physical problem not directly related with quantum formalism. At this stage, we show that this nonstandard formulation of classical mechanics exhibits a new kind of operation that has no classical counterpart: this operation is related to the “quantization process,” and we show why quantization physically depends on group theory (the Galilei group). This analytical procedure of quantization replaces the “correspondence principle” (or canonical quantization) and allows us to map classical mechanics into quantum mechanics, giving all operators of quantum dynamics and the Schrödinger equation. The great advantage of this point of view is that quantization is based on concrete physical arguments and not derived from some “pure algebraic rule” (we exhibit also some limit of the correspondence principle). Moreover spins for particles are naturally generated, including an approximation of their interaction with magnetic fields. We also recover by this approach the semi-classical formalism developed by E. Prugovečki [Stochastic Quantum Mechanics and Quantum Space–time (Reidel, Dordrecht, 1986)].
In general relativity, the description of spacetime relies on idealised rods and clocks, which identify a reference frame. In any concrete scenario, reference frames are associated to physical systems, which are ultimately quantum in nature. A relativistic description of the laws of physics hence needs to take into account such quantum reference frames (QRFs), through which spacetime can be given an operational meaning. Here, we introduce the notion of a spacetime quantum reference frame, associated to a quantum particle in spacetime. Such formulation has the advantage of treating space and time on equal footing, and of allowing us to describe the dynamical evolution of a set of quantum systems from the perspective of another quantum system, where the parameter in which the rest of the physical systems evolves coincides with the proper time of the particle taken as the QRF. Crucially, the proper times in two different QRFs are not related by a standard transformation, but they might be in a quantum superposition one with respect to the other.Concretely, we consider a system of N relativistic quantum particles in a weak gravitational field, and introduce a timeless formulation in which the global state of the N particles appears "frozen", but the dynamical evolution is recovered in terms of relational quantities. The position and momentum Hilbert space of the particles is used to fix the QRF via a transformation to the local frame of the particle such that the metric is locally inertial at the origin of the QRF. The internal Hilbert space corresponds to the clock space, which keeps the proper time in the local frame of the particle. Thanks to this fully relational construction we show how the remaining particles evolve dynamically in the relational variables from the perspective of the QRF. The construction proposed here includes the Page-Wootters mechanism for non interacting clocks when the external degrees of freedom are neglected. Finally, we find that a quantum superposition of gravitational redshifts and a quantum superposition of special-relativistic time dilations can be observed in the QRF.
The distinction between classical physics and quantum physics is traditionally analyzed based on concepts such as determinism, the continuous or discrete nature of variables, the non-commutativity of observables, and the superposition of states. This article proposes a more fundamental reinterpretation based on two complementary conceptual ingredients. The first is a minimal relational epistemology, developed from the author's previous work, in which space and time are not a priori frames but emerge from comparisons of movements between systems. The second is an analysis of the role played by the possibility—or impossibility—of defining independent events, a possibility that depends precisely on the prior construction of a spatio-temporal and causal structure. We show that the essential difference between classical physics and quantum physics lies in the rule of probability composition, and not in the features usually invoked. Quantum physics appears to be a coherent description of a physical regime prior to the establishment of the causal independence of events. The two conceptual ingredients are summarized in the form of axioms, broad enough to offer a new perspective on the usual notions of quantum physics, which we review here. Accompanying this text, another article presents more technical developments.