In order to discriminate speakable and unspeakable in physics of time, the single ion optical atomic clock and the double time-slit experiments are considered. Their description clarifies that temporal quantities such as time and frequency are fundamentally unspeakable, while speakable is only related to ratios between wavelengths and to boundary conditions of stationary light beams. In addition, time intervals lower than the minimum period realized by the experimental apparatus are the results of an extrapolation out of the domain of definition of measurement operations. A toy model is employed to isolate the hidden source of self-recursivity in the definition of clocks and time, which resides in the abuse of language generated by terms such as duration and repetitive event, instead of interval extension and multiplicity. The timeless scenario which emerges from the experimental picture is theoretically supported by fundamental timelessness of the Hamiltonian formalism employed to define dynamics. Clock metrology, employed to parametrize dynamics in experiments, works successfully as it returns a discrete time variable, which approximates the parametrization of the trajectories in the phase space induced by the invariance of the action and of the time independent Hamiltonian.
In order to discriminate speakable and unspeakable in physics of time, the single ion optical atomic clock and the double time-slit experiments are considered. Their description clarifies that temporal quantities such as time and frequency are fundamentally unspeakable, while speakable is only related to ratios between wavelengths and to boundary conditions of stationary light beams. In addition, time intervals lower than the minimum period realized by the experimental apparatus are the results of an extrapolation out of the domain of definition of measurement operations. A toy model is employed to isolate the hidden source of self-recursivity in the definition of clocks and time, which resides in the abuse of language generated by terms such as duration and repetitive event, instead of interval extension and multiplicity. The timeless scenario which emerges from the experimental picture is theoretically supported by fundamental timelessness of the Hamiltonian formalism employed to define dynamics. Clock metrology, employed to parametrize dynamics in experiments, works successfully as it returns a discrete time variable, which approximates the parametrization of the trajectories in the phase space induced by the invariance of the action and of the time independent Hamiltonian.
Time is not a fundamental coordinate but a relation: all physical dynamics are defined on a timeless, global state space, and what we call “time” is the ordered relation between a chosen reference clock and the rest of the system. Formulate the above relational time postulate in an operationally minimal framework aligned with Wheeler–DeWitt constraint dynamics and the Page–Wootters construction, derive a concrete, testable prediction, when the same physical process is parametrized by two physically different “clocks” (e.g. an optical atomic clock and a mechanical phase clock), a small but measurable phase discrepancy can appear after all known effects are calibrated, if time is purely relational, propose a dual-clock interferometric experiment, and present an analytic sensitivity study. Our framework reproduces GR+QM in the ideal clock limit, yet allows for an operational deviation encoded by two dimensionless parameters. A null result yields new upper bounds across disparate frequency bands. A positive, replicated signal would provide the first operational evidence for “time without time”. This document is licensed under CC BY-NC-ND 4.0. The specific measurement protocol described herein, the unique ϵ and χ parametrization, and the associated data analysis algorithms constitute intellectual property. Any experimental realization or software implementation of the mathematical model is subject to the prior written consent of the author.
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
Optical clocks based on atoms and ions probe relativistic effects with unprecedented sensitivity. They resolve time dilation due to atom motion or different positions in the gravitational potential through frequency shifts. However, all measurements of time dilation so far can be explained effectively as the result of dynamics with respect to a classical proper time parameter. Here we show that atomic clocks can probe effects where a classical description of the proper time dynamics is insufficient as superpositions of proper time emerge. We apply a Hamiltonian formalism to derive time dilation effects in harmonically trapped clock atoms and show how second-order Doppler shifts due to the vacuum energy, squeezing, and quantum corrections to the dynamics arise. We also demonstrate that time-dilation-induced entanglement between motion and clock evolution can become observable in state-of-the-art clocks when the motion of the atoms is strongly squeezed, realizing proper time interferometry. Our results show that experiments with trapped ion clocks are within reach of probing relativistic evolution of clocks for which a quantum description of proper time becomes necessary.
A fundamental description of time can be consistent not only with the usual monotonic behavior but also with an oscillating physical clock variable, coupled to the degrees of freedom of a system evolving in time. Generically, one would in fact expect some kind of oscillating motion of a system that is dynamical and interacts with its surroundings, as required for a fundamental clock that can be noticed by any other system. Unitary evolution does not require a monotonic clock variable and can be achieved more generally by formally unwinding the oscillating clock movement, keeping track not only of the value of the clock variable but also of the number of cycles it has gone through at any moment. The specific treatment used here employs concatenations of evolution operators, alternating between forward and backward evolution of the oscillating clock. Because the clock and an evolving system have a common conserved energy, the clock is in different cycles for different energy eigenstates of the system state. As a result, the clock is generically in a quantum state with a superposition of different clock cycles, a key feature that distinguishes oscillating clocks from monotonic time. In addition, coherence could be lost faster than observed, for instance if a system that would be harmonic in isolation is made anharmonic by interactions with a fundamental clock, implying observational bounds on fundamental clocks. Numerical computations show that coherence is maintained over long timescales provided the clock period is much smaller than the system period. A small loss of coherence nevertheless remains and, measured in terms of the relative standard deviation of the system period, is proportional to the ratio of the system period and the clock period. Since the precision of atomic clocks could not be achieved if atomic frequencies would be subject to additional variations from coupling to a fundamental clock, an upper bound on the clock period can be obtained that turns out to be much smaller than currently available direct or indirect measurements of time.
We present an implementation of a recently proposed procedure for defining time, based on the description of the evolving system and its clock as noninteracting, entangled systems, according to the Page and Wootters approach. We study how the quantum dynamics transforms into a classical-like behavior when conditions related to macroscopicity are met by the clock alone, or by both the clock and the evolving system. In the description of this emerging behavior finds its place the classical notion of time, as well as that of phase-space and trajectories on it. This allows us to analyze and discuss the relations that must hold between quantities that characterize the system and clock separately, in order for the resulting overall picture to be that of a physical dynamics as we mean it.
Heisenberg, Bohr and others were forced to renounce on the description of the objective reality as the aim of physics because of the paradoxical quantum phenomena observed on the atomic level. The contemporary quantum mechanics created on the base of their positivism point of view must divide the world into speakable apparatus which amplifies microscopic events to macroscopic consequences and unspeakable quantum system. Examination of the quantum phenomena corroborates the confidence expressed by creators of quantum theory that the renunciation of realism should not apply on our everyday macroscopic world. Nanostructures may be considered for the present as a boundary of realistic description for all phenomena including the quantum one.