Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2110.07702 · Physical review. D/Physical review. D.

Relational evolution with oscillating clocks

Martin Bojowald A, Luis Martínez, Garrett Wendel

Published May 24, 2022• 9 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevd.105.106020

Related papers

A relational solution to the problem of time in quantum mechanics and quantum gravity: a fundamental mechanism for quantum decoherence

Rodolfo Gambini, Rafael A. Porto, Jorge A. Pullin

The use of a relational time in quantum mechanics is a framework in which one promotes to quantum operators all variables in a system, and later chooses one of the variables to operate like a 'clock'. Conditional probabilities are computed for variables of the system to take certain values when the 'clock' specifies a certain time. This framework is attractive in contexts where the assumption of usual quantum mechanics of the existence of an external, perfectly classical clock, appears unnatural, as in quantum cosmology. Until recently, there were problems with such constructions in ordinary quantum mechanics with additional difficulties in the context of constrained theories like general relativity. A scheme we recently introduced to consistently discretize general relativity removed such obstacles. Since the clock is now an object subject to quantum fluctuations, the resulting evolution in time is not exactly unitary and pure states decohere into mixed states. Here we work out in detail the type of decoherence generated, and we find it to be of Lindblad type. This is attractive since it implies that one can have loss of coherence without violating the conservation of energy. We apply the framework to a simple cosmological model to illustrate how a quantitative estimate of the effect could be computed. For most quantum systems it appears to be too small to be observed, although certain macroscopic quantum systems could in the future provide a testing ground for experimental observation.

arXiv:gr-qc/0402118•New Journal of Physics•Apr 14, 2004•95 citations

Relational dynamics with periodic clocks

Leonardo Chataignier, Philipp A. Höhn, Maximilian P. E. Lock +1 more

Abstract We discuss a systematic way in which a relational dynamics can be established relative to periodic clocks both in the classical and quantum theories, emphasising the parallels between them. We show that: (1) classical and quantum relational observables that encode the value of a quantity relative to a periodic clock are only invariant along the gauge orbits generated by the Hamiltonian constraint if the quantity itself is periodic, and otherwise the observables are only transiently invariant per clock cycle (this implies, in particular, that counting winding numbers does not lead to invariant observables relative to the periodic clock); (2) the quantum relational observables can be obtained from a partial group averaging procedure over a single clock cycle; (3) there is an equivalence (‘trinity’) between the quantum theories based on the quantum relational observables of the clock-neutral picture of Dirac quantisation, the relational Schrodinger picture of the Page–Wootters(PW) formalism, and the relational Heisenberg picture that follows from quantum deparameterisation, all three taken relative to periodic clocks (implying that the dynamics in all three is necessarily periodic); (4) in the context of periodic clocks, the original PW definition of conditional probabilities fails for systems that have a continuous energy spectrum and, using the equivalence between the PW and the clock-neutral, gauge-invariant formalism, must be suitably updated. Finally, we show how a system evolving periodically with respect to a periodic clock can evolve monotonically with respect to an aperiodic clock, without inconsistency. The presentation is illustrated by several examples, and we conclude with a brief comparison to other approaches in the literature that also deal with relational descriptions of periodic clocks.

arXiv:2409.06479•New Journal of Physics•Feb 17, 2026•4 citations

Requiem for an ideal clock

K. L. H. Bryan, A. J. M. Medved

The `problem of time' remains an unresolved issue in all known physical descriptions of the Universe. One aspect of this problem is the conspicuous absence of time in the Wheeler-Dewitt equation, which is the analogue of the Schrodinger equation for the Universal wavefunction. Page and Wootters famously addressed this problem by providing a mechanism for effectively introducing time evolution into this timeless cosmological picture. Their method, which is sometimes called the conditional probability interpretation (CPI), requires the identification of an internal clock system that is meant to keep time for the remainder of the Universe. Most investigations into this idea employ the idealized limit of a non-interacting clock system, the so-called ideal clock. However, by allowing for interactions, we have found the counter-intuitive result that a non-interacting clock is not necessarily the optimal choice, even if it is `ideal'. In particular, the uncertainty that is associated with the physical measurement of an atomic clock is found to decrease monotonically as the interactions grow stronger. This observation, which is reinforced by a previous study using a semi-classical clock, paves the way to an independent argument that is based on the energy conservation of any isolated system. Our conclusion is that ideal clocks must be prohibited from the CPI when recovering cosmological time evolution. Interactions are necessary for describing time evolution as a strict matter of principle. Lastly, we also consider the implications of this result for the experience of time in the evolution of the Universe.

arXiv:1803.02045•Mar 6, 2018•3 citations

Clocks And Dynamics In Quantum Mechanics

Michael J. York

We argue that (1) our perception of time through change and (2) the gap between reality and our observation of it are at the heart of both quantum mechanics and the dynamical mechanism of physical systems. We suggest that the origin of quantum uncertainty lies with the absence of infinities or infinitesimals in observational data and that our concept of time derives from observing changing data (events). We argue that the fundamentally important content of the Superposition Principle is not the "probability amplitude" of posterior state observation but future state availability conditional only on prior information. Since event detection also implies posterior conditions (e.g. a specific type of detectable event occurred) as well as prior conditions, the probabilities of detected outcomes are also conditional on properties of the posterior properties of the observation. Such posterior conditions cannot affect the prior state availabilities and this implies violation of counter-factual definiteness. A component of a quantum system may be chosen to represent a clock and changes in other components can then be expected to be correlated with clocks with which they are entangled. Instead of traditional time-dependent equations of motion we provide a specific mechanism whereby evolution of data is instead quasi-causally related to the relative \availability\ of states and equations of motion are expressed in terms of quantized clock variables. We also suggest that time-reversal symmetry-breaking in weak interactions is an artifice of a conventional choice of co-ordinate time-function. Analysis of a "free" particle suggests that conventional co-ordinate space-time emerges from how we measure the separation of objects and events.

arXiv:1405.1098•May 5, 2014•1 citation

Relativity of Coherence: A Dual-Clock Framework Completing the Standard Model Through Temporal Geometry and Emergent Standing-Wave Matter

Boyd, Chandler Stanten

This paper introduces a dual-clock framework that unifies quantum mechanics, general relativity, and thermodynamics through a single dynamical equation governing temporal evolution. The model proposes that physical time emerges from interference between two counter-propagating temporal waves: an entropic wave driving phase-space exploration and a Lagrangian wave driving action minimization. Their interaction produces a universal refresh constant τ that regulates coherence across all physical systems. The framework recovers standard quantum and relativistic equations as limiting cases (τ = 0) while predicting measurable deviations for τ ≠ 0. The primary experimental prediction is a differential frequency offset between co-located optical atomic clocks of different species: δR ≈ -7 × 10⁻¹⁸, testable with emerging 10⁻¹⁹-precision clock technology. Numerical simulations demonstrate stability over 3.17 × 10¹⁵ equivalent years, reproducing FRW cosmological expansion, gravitational time dilation, and QED phase behavior. The model provides geometric explanations for quantum measurement collapse, black-hole singularities as temporal degree-of-freedom mergers, and biological quantum coherence in photosynthetic systems. This work presents a falsifiable, minimal extension to the Standard Model requiring no new particles or hidden dimensions, testable with near-term atomic clock experiments.

•Nov 23, 2025•0 citations

Measuring time in a timeless universe

Samuel Kuypers, Simone Rijavec

Physical systems are usually assumed to evolve relative to an external time parameter, which is problematic because in quantum theory that parameter is not a physical observable. Page and Wootters (1984) solved this by proposing that the Universe is in a stationary state, eliminating the need for the external time parameter. Instead, their model contains an isolated subsystem, a clock, with which other subsystems are entangled, making the latter appear to evolve relative to different states of the clock. While this resolves the problem of the time parameter, the assumption that the clock is isolated prevents it from being measured, as this requires an interaction with another system. We prove that the clock can be measured while preserving the core features of the Page-Wootters construction. We also discuss clock synchronization.

arXiv:2406.14642•Physical review. D/Physical review. D.•Aug 24, 2025•6 citations
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.
The use of a relational time in quantum mechanics is a framework in which one promotes to quantum operators all variables in a system, and later chooses one of the variables to operate like a 'clock'. Conditional probabilities are computed for variables of the system to take certain values when the 'clock' specifies a certain time. This framework is attractive in contexts where the assumption of usual quantum mechanics of the existence of an external, perfectly classical clock, appears unnatural, as in quantum cosmology. Until recently, there were problems with such constructions in ordinary quantum mechanics with additional difficulties in the context of constrained theories like general relativity. A scheme we recently introduced to consistently discretize general relativity removed such obstacles. Since the clock is now an object subject to quantum fluctuations, the resulting evolution in time is not exactly unitary and pure states decohere into mixed states. Here we work out in detail the type of decoherence generated, and we find it to be of Lindblad type. This is attractive since it implies that one can have loss of coherence without violating the conservation of energy. We apply the framework to a simple cosmological model to illustrate how a quantitative estimate of the effect could be computed. For most quantum systems it appears to be too small to be observed, although certain macroscopic quantum systems could in the future provide a testing ground for experimental observation.
Abstract We discuss a systematic way in which a relational dynamics can be established relative to periodic clocks both in the classical and quantum theories, emphasising the parallels between them. We show that: (1) classical and quantum relational observables that encode the value of a quantity relative to a periodic clock are only invariant along the gauge orbits generated by the Hamiltonian constraint if the quantity itself is periodic, and otherwise the observables are only transiently invariant per clock cycle (this implies, in particular, that counting winding numbers does not lead to invariant observables relative to the periodic clock); (2) the quantum relational observables can be obtained from a partial group averaging procedure over a single clock cycle; (3) there is an equivalence (‘trinity’) between the quantum theories based on the quantum relational observables of the clock-neutral picture of Dirac quantisation, the relational Schrodinger picture of the Page–Wootters(PW) formalism, and the relational Heisenberg picture that follows from quantum deparameterisation, all three taken relative to periodic clocks (implying that the dynamics in all three is necessarily periodic); (4) in the context of periodic clocks, the original PW definition of conditional probabilities fails for systems that have a continuous energy spectrum and, using the equivalence between the PW and the clock-neutral, gauge-invariant formalism, must be suitably updated. Finally, we show how a system evolving periodically with respect to a periodic clock can evolve monotonically with respect to an aperiodic clock, without inconsistency. The presentation is illustrated by several examples, and we conclude with a brief comparison to other approaches in the literature that also deal with relational descriptions of periodic clocks.
The `problem of time' remains an unresolved issue in all known physical descriptions of the Universe. One aspect of this problem is the conspicuous absence of time in the Wheeler-Dewitt equation, which is the analogue of the Schrodinger equation for the Universal wavefunction. Page and Wootters famously addressed this problem by providing a mechanism for effectively introducing time evolution into this timeless cosmological picture. Their method, which is sometimes called the conditional probability interpretation (CPI), requires the identification of an internal clock system that is meant to keep time for the remainder of the Universe. Most investigations into this idea employ the idealized limit of a non-interacting clock system, the so-called ideal clock. However, by allowing for interactions, we have found the counter-intuitive result that a non-interacting clock is not necessarily the optimal choice, even if it is `ideal'. In particular, the uncertainty that is associated with the physical measurement of an atomic clock is found to decrease monotonically as the interactions grow stronger. This observation, which is reinforced by a previous study using a semi-classical clock, paves the way to an independent argument that is based on the energy conservation of any isolated system. Our conclusion is that ideal clocks must be prohibited from the CPI when recovering cosmological time evolution. Interactions are necessary for describing time evolution as a strict matter of principle. Lastly, we also consider the implications of this result for the experience of time in the evolution of the Universe.
We argue that (1) our perception of time through change and (2) the gap between reality and our observation of it are at the heart of both quantum mechanics and the dynamical mechanism of physical systems. We suggest that the origin of quantum uncertainty lies with the absence of infinities or infinitesimals in observational data and that our concept of time derives from observing changing data (events). We argue that the fundamentally important content of the Superposition Principle is not the "probability amplitude" of posterior state observation but future state availability conditional only on prior information. Since event detection also implies posterior conditions (e.g. a specific type of detectable event occurred) as well as prior conditions, the probabilities of detected outcomes are also conditional on properties of the posterior properties of the observation. Such posterior conditions cannot affect the prior state availabilities and this implies violation of counter-factual definiteness. A component of a quantum system may be chosen to represent a clock and changes in other components can then be expected to be correlated with clocks with which they are entangled. Instead of traditional time-dependent equations of motion we provide a specific mechanism whereby evolution of data is instead quasi-causally related to the relative \availability\ of states and equations of motion are expressed in terms of quantized clock variables. We also suggest that time-reversal symmetry-breaking in weak interactions is an artifice of a conventional choice of co-ordinate time-function. Analysis of a "free" particle suggests that conventional co-ordinate space-time emerges from how we measure the separation of objects and events.
This paper introduces a dual-clock framework that unifies quantum mechanics, general relativity, and thermodynamics through a single dynamical equation governing temporal evolution. The model proposes that physical time emerges from interference between two counter-propagating temporal waves: an entropic wave driving phase-space exploration and a Lagrangian wave driving action minimization. Their interaction produces a universal refresh constant τ that regulates coherence across all physical systems. The framework recovers standard quantum and relativistic equations as limiting cases (τ = 0) while predicting measurable deviations for τ ≠ 0. The primary experimental prediction is a differential frequency offset between co-located optical atomic clocks of different species: δR ≈ -7 × 10⁻¹⁸, testable with emerging 10⁻¹⁹-precision clock technology. Numerical simulations demonstrate stability over 3.17 × 10¹⁵ equivalent years, reproducing FRW cosmological expansion, gravitational time dilation, and QED phase behavior. The model provides geometric explanations for quantum measurement collapse, black-hole singularities as temporal degree-of-freedom mergers, and biological quantum coherence in photosynthetic systems. This work presents a falsifiable, minimal extension to the Standard Model requiring no new particles or hidden dimensions, testable with near-term atomic clock experiments.
Physical systems are usually assumed to evolve relative to an external time parameter, which is problematic because in quantum theory that parameter is not a physical observable. Page and Wootters (1984) solved this by proposing that the Universe is in a stationary state, eliminating the need for the external time parameter. Instead, their model contains an isolated subsystem, a clock, with which other subsystems are entangled, making the latter appear to evolve relative to different states of the clock. While this resolves the problem of the time parameter, the assumption that the clock is isolated prevents it from being measured, as this requires an interaction with another system. We prove that the clock can be measured while preserving the core features of the Page-Wootters construction. We also discuss clock synchronization.