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

Almost ideal clocks in quantum cosmology: a brief derivation of time

Marolf, D M

Published Dec 5, 1994• 56 citations
OpenAlex

Abstract

A formalism for quantizing time reparametrization invariant dynamics is considered and applied to systems which contain an `almost ideal clock.' Previously, this formalism was successfully applied to the Bianchi models and, while it contains no fundamental notion of `time' or `evolution,' the approach does contain a notion of correlations. Using correlations with the almost ideal clock to introduce a notion of time, the work below derives the complete formalism of external time quantum mechanics. The limit of an ideal clock is found to be closely associated with the Klein-Gordon inner product and the Newton-Wigner formalism and, in addition, this limit is shown to fail for a clock that measures metric-defined proper time near a singularity in Bianchi models.

Related papers

Almost ideal clocks in quantum cosmology: a brief derivation of time

Donald M. Marolf

A formalism for quantizing time reparametrization invariant dynamics is considered and applied to systems which contain an `almost ideal clock'. Previously, this formalism was successfully applied to the Bianchi models and, while it contains no fundamental notion of `time' or `evolution', the approach does contain a notion of correlations. Using correlations with the almost ideal clock to introduce a notion of time, we derive the complete formalism of external time quantum mechanics. The explicit relation between the physical Hilbert space and the usual Hilbert space for a system with an external time is also derived and the limit of an ideal clock is found to be closely associated with the Klein - Gordon inner product and the Newton - Wigner formalism.

arXiv:gr-qc/9412016•Classical and Quantum Gravity•Oct 1, 1995•77 citations

Interpretation of time-reparametrization-invariant quantum mechanics: An exactly soluble model

Ian D. Lawrie, Richard J. Epp

The classical and quantum dynamics of simple time-reparametrization-invariant models containing two degrees of freedom are studied in detail. Elimination of one ``clock'' variable through the Hamiltonian constraint leads to a description of time evolution for the remaining variable which is essentially equivalent to the standard quantum mechanics of an unconstrained system. In contrast with a similar proposal of Rovelli, evolution is with respect to a geometrical proper time, and the Heisenberg equation of motion is exact. The physical phase space contains no coordinate associated with the eliminated ``clock'' variable. Therefore, time evolution is not with respect to the observable readings of a physical clock. Rather, the eliminated variable can be construed as intrinsic to a particular observer, and, hence, as unobservable, in principle, by this observer. The possibility of a ``test clock,'' which would reveal time evolution while contributing negligibly to the Hamiltonian constraint, is examined, and found to be viable in the semiclassical limit of large quantum numbers. \textcopyright{} 1996 The American Physical Society.

arXiv:gr-qc/9509047•Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Jun 15, 1996•11 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

Quantum kinematics of spacetime. II. A model quantum cosmology with real clocks

James B. Hartle

Nonrelativistic model quantum cosmologies are studied in which the basic time variable is the position of a clock indicator and the time parameter of the Schr\"odinger equation is an unobservable label. Familiar Schr\"odinger-Heisenberg quantum mechanics emerges if the clock is ideal---arbitrarily accurate for arbitrarily long times. More realistically, however, the usual formulation emerges only as an approximation appropriate to states of this model universe in which part of the system functions approximately as an ideal clock. It is suggested that the quantum kinematics of spacetime theories such as general relativity may be analogous to those of this model. In particular it is suggested that our familiar notion of time in quantum mechanics is not an inevitable property of a general quantum framework but an approximate feature of specific initial conditions.

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Nov 15, 1988•74 citations

Measuring time with physical clocks

Esteban Castro-Ruiz, Flaminia Giacomini, Časlav Brukner

In general relativity, the picture of spacetime assigns an ideal clock to each spacetime point. Being ideal, gravitational effects due to these clocks are ignored and the flow of time according to one clock is not affected by the presence of surrounding clocks. However, if time is defined operationally, as a pointer position of a physical clock that obeys the laws of quantum mechanics and general relativity, such a picture is at most a convenient fiction. We show that the mass-energy equivalence implies gravitational interaction between the clocks, while the superposition of energy eigenstates leads to a non-fixed metric background. Based only on the assumption that both quantum mechanics and general relativity are valid in this situation, we show that the clocks necessarily get entangled through time dilation effect, which eventually leads to a loss of coherence of a single clock. Hence, the time as measured by a single clock is not well-defined. However, the general relativistic notion of time is recovered in the classical limit of clocks.

•Jul 7, 2015•1 citation

Finite-size clocks in quantum field theory and the twin paradox

Matheus H. Zambianco, T. Rick Perche

Vacuum fluctuations in quantum field theory impose fundamental limitations on our ability to measure time at arbitrarily short scales. To investigate the impact of universal quantum field theory effects on observer-dependent time measurements, we introduce a clock model based on the vacuum decay probability of a finite-sized quantum system. This model defines an effective notion of proper time that depends on the microscopic properties of the clock and on how it samples vacuum fluctuations along its trajectory. We show that, in the long-time regime, this notion of time reduces to the usual proper time of special relativity. However, by studying a microscopic twin-paradox scenario, we find that, in general, time is not determined solely by the trajectory connecting two events, but also by how vacuum fluctuations interact with the internal structure of the clocks.

quant-pharXiv:2512.06076•Dec 9, 2025
A formalism for quantizing time reparametrization invariant dynamics is considered and applied to systems which contain an almost ideal clock.' Previously, this formalism was successfully applied to the Bianchi models and, while it contains no fundamental notion of time' or `evolution,' the approach does contain a notion of correlations. Using correlations with the almost ideal clock to introduce a notion of time, the work below derives the complete formalism of external time quantum mechanics. The limit of an ideal clock is found to be closely associated with the Klein-Gordon inner product and the Newton-Wigner formalism and, in addition, this limit is shown to fail for a clock that measures metric-defined proper time near a singularity in Bianchi models.
A formalism for quantizing time reparametrization invariant dynamics is considered and applied to systems which contain an `almost ideal clock'. Previously, this formalism was successfully applied to the Bianchi models and, while it contains no fundamental notion of `time' or `evolution', the approach does contain a notion of correlations. Using correlations with the almost ideal clock to introduce a notion of time, we derive the complete formalism of external time quantum mechanics. The explicit relation between the physical Hilbert space and the usual Hilbert space for a system with an external time is also derived and the limit of an ideal clock is found to be closely associated with the Klein - Gordon inner product and the Newton - Wigner formalism.
The classical and quantum dynamics of simple time-reparametrization-invariant models containing two degrees of freedom are studied in detail. Elimination of one ``clock'' variable through the Hamiltonian constraint leads to a description of time evolution for the remaining variable which is essentially equivalent to the standard quantum mechanics of an unconstrained system. In contrast with a similar proposal of Rovelli, evolution is with respect to a geometrical proper time, and the Heisenberg equation of motion is exact. The physical phase space contains no coordinate associated with the eliminated ``clock'' variable. Therefore, time evolution is not with respect to the observable readings of a physical clock. Rather, the eliminated variable can be construed as intrinsic to a particular observer, and, hence, as unobservable, in principle, by this observer. The possibility of a ``test clock,'' which would reveal time evolution while contributing negligibly to the Hamiltonian constraint, is examined, and found to be viable in the semiclassical limit of large quantum numbers. \textcopyright{} 1996 The American Physical Society.
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.
Nonrelativistic model quantum cosmologies are studied in which the basic time variable is the position of a clock indicator and the time parameter of the Schr\"odinger equation is an unobservable label. Familiar Schr\"odinger-Heisenberg quantum mechanics emerges if the clock is ideal---arbitrarily accurate for arbitrarily long times. More realistically, however, the usual formulation emerges only as an approximation appropriate to states of this model universe in which part of the system functions approximately as an ideal clock. It is suggested that the quantum kinematics of spacetime theories such as general relativity may be analogous to those of this model. In particular it is suggested that our familiar notion of time in quantum mechanics is not an inevitable property of a general quantum framework but an approximate feature of specific initial conditions.
In general relativity, the picture of spacetime assigns an ideal clock to each spacetime point. Being ideal, gravitational effects due to these clocks are ignored and the flow of time according to one clock is not affected by the presence of surrounding clocks. However, if time is defined operationally, as a pointer position of a physical clock that obeys the laws of quantum mechanics and general relativity, such a picture is at most a convenient fiction. We show that the mass-energy equivalence implies gravitational interaction between the clocks, while the superposition of energy eigenstates leads to a non-fixed metric background. Based only on the assumption that both quantum mechanics and general relativity are valid in this situation, we show that the clocks necessarily get entangled through time dilation effect, which eventually leads to a loss of coherence of a single clock. Hence, the time as measured by a single clock is not well-defined. However, the general relativistic notion of time is recovered in the classical limit of clocks.
Vacuum fluctuations in quantum field theory impose fundamental limitations on our ability to measure time at arbitrarily short scales. To investigate the impact of universal quantum field theory effects on observer-dependent time measurements, we introduce a clock model based on the vacuum decay probability of a finite-sized quantum system. This model defines an effective notion of proper time that depends on the microscopic properties of the clock and on how it samples vacuum fluctuations along its trajectory. We show that, in the long-time regime, this notion of time reduces to the usual proper time of special relativity. However, by studying a microscopic twin-paradox scenario, we find that, in general, time is not determined solely by the trajectory connecting two events, but also by how vacuum fluctuations interact with the internal structure of the clocks.