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

Mecanismo de Page-Wootters: o papel da coerência e relógios finitos

Mendes, Leandro Raffhael da Silva

Published Oct 25, 2021• 0 citations
OpenAlex

Abstract

Among the many proposals to approach the concept of time in quantum theory, the Page-Wootters mechanism has attracted much attention in the last few years. Originally, this mechanism explored a stationary pure bipartite non-interacting global system, i.e., a system of interest together with an ancillary clock, to determine how the evolution in time can emerge and an equation of motion can be obtained for a quantum particle conditioned to the measurement of the state of the clock. In this mechanism, time is seen as an inaccessible coordinate and the apparent passage of time arises as a consequence of correlations between the subsystems of a global state. Here we propose a measure that captures the relational character of the mechanism, showing that the internal coherence is the necessary ingredient for the emergence of time in the Page–Wootters model. Connecting it to results in quantum thermodynamics, showing that it is directly related to the extractable work from quantum coherence. In a second step we analyze such a timeless approach to quantum theory but deriving an equation of motion for a mixed state system that evolves according to its gravitationally induced interaction with a non-ideal quantum clock. The interaction considered is known to describe the gravitational decoherence mechanism, and the clock model is the recently proposed quasi-ideal clock, i.e., one constructed to approximate the time-energy canonical commutation relation. As a result of our considerations, we obtained an equation of motion that is non-linear in nature, dependent on the systems initial conditions.

Related papers

Non-linear equation of motion for Page-Wootters mechanism with interaction and quasi-ideal clocks

Mendes, Leandro R. S., Brito, Frederico, Diogo O. Soares-Pinto

Among the many proposals to approach the concept of time in quantum theory, the Page-Wootters mechanism has attracted much attention in the last few years. Originally, such a mechanism explored a stationary bipartite non-interacting global system, i.e., a system of interest together with an ancillary clock, to determine how the evolution in time can emerge as an equation of motion for a quantum particle conditioned to the measurement of the state of the clock. After the seminal proposal, many variations of it were considered, and different aspects of the mechanism were elucidated. Our contribution to these investigations is that we analyze such a timeless approach to quantum theory but deriving an equation of motion for a mixed state system that evolves according to its gravitationally induced interaction with a non-ideal quantum clock. The interaction considered is known to describe the gravitational decoherence mechanism, and the clock model is the recently proposed quasi-ideal clock, i.e., one constructed to approximate the time-energy canonical commutation relation. As a result of our considerations, we obtained an equation of motion that is non-linear in nature, dependent on the system's initial conditions.

arXiv:2107.11452•Jul 23, 2021•0 citations

Non-Linear Equation of Motion for Page–Wootters Mechanism with Interaction and Quasi-Ideal Clocks

Leandro R. S. Mendes, Frederico Brito, Diogo O. Soares-Pinto

We explore a timeless approach to quantum theory, in the form of the Page–Wootters mechanism, in which a gravitational interaction is introduced between the system and a finite-dimensional clock. The clock model used is the recently proposed quasi-ideal clock, a construction that can approximate the time–energy canonical commutation relation. We derive equations of motion for the case in which the system is in a pure and mixed state, obtaining a Schrödinger-like equation that leads to a non-linear equation exhibiting decoherence due to the non-ideal nature of the clock and gravitational coupling. A distinctive feature of this equation is that it exhibits dependence on the system’s initial conditions.

Universe•Sep 10, 2025•1 citation

On the Emergence of Time and Space in Closed Quantum Systems

Tommaso Favalli

Time, space and entanglement are the main characters in this work. Their nature is still a great mystery in physics and we study here the possibility that these three phenomena are closely connected, showing how entanglement can be at the basis of the emergence of time and space within closed quantum systems. We revisit and extend the Page and Wootters theory that was originally introduced in order to describe the emergence of time through entanglement between subsystems in a globally static, quantum Universe. In the book, after providing a complete review of the salient aspects of the theory, we establish a connection with recent research on the foundations of statistical mechanics and we propose a new understanding of the thermalization process. Furthermore, we generalize the framework in order describe the spatial degree of freedom and we provide a model of 3+1 dimensional, quantum spacetime emerging from entanglement among different subsystems in a globally "timeless" and "positionless" Universe. Finally, via the Page and Wootters theory, the evolution of quantum clocks within a gravitational field is treated and a time dilation effect is obtained in agreement with the Schwarzschild solution.

quant-pharXiv:2512.08120•Springer theses•Dec 10, 2025•2 citations

Time as a consequence of internal coherence

Leandro R. S. Mendes, Diogo O. Soares-Pinto

Time has been an elusive concept to grasp. Although we do not yet understand it properly, there have been advances made in regard to how we can explain it. One such advance is the Page-Wootters mechanism. In this mechanism, time is seen as an inaccessible coordinate and the apparent passage of time arises as a consequence of correlations between the subsystems of a global state. Here we propose a measure that captures the relational character of the mechanism, showing that the internal coherence is the necessary ingredient for the emergence of time in the Page-Wootters model. Also, we connect it to results in quantum thermodynamics, showing that it is directly related to the extractable work from quantum coherence.

arXiv:1806.09669•Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences•Nov 1, 2019•24 citations

Robustness of the Page-Wootters construction across different pictures, states of the universe, and system-clock interactions

Simone Rijavec

In quantum theory, the concept of time rests on shaky ground. One way to address this problem is to remove the usual background time parameter as a primitive entity and explain its emergence via correlations between physical systems. This approach was adopted by Page and Wootters [Phys. Rev. D 27, 2885 (1983)], who showed how time can emerge in a stationary quantum universe from the correlations between two of its subsystems, one of them acting as a clock for the other. In this work, I study the robustness of the Page-Wootters construction across different pictures, states of the universe, and clock interactions, clarifying the role and the nature of the correlations between the subsystems of the universe. I start by showing how to formulate the Page-Wootters construction in the Heisenberg picture via a unitary change of basis. I consider both pure and mixed states of the universe and extend the analysis to include interactions between the clock and the other subsystem of the universe. The study reveals what kind of correlations are necessary for the construction to work. Interestingly, entanglement is not required as long as there are no interactions with the clock. The study also shows that these interactions can lead to a nonunitary evolution for some mixed states of the universe. In a simple two-level system, this aspect becomes relevant at scales where one would expect strong relativistic effects. At these scales, I also observe an inversion in the system's direction of time.

arXiv:2204.11740•Physical review. D/Physical review. D.•Sep 11, 2023•9 citations

The time of arrival problem in the Page-Wootters formalism

Niyusha Hosseini, Maximilian P. E. Lock

The time-of-arrival problem asks for the probability distribution for when a quantum particle reaches a specified location. It has been the subject of decades of debate, exemplifying the lack of a self-adjoint time observable in quantum theory. In the Page-Wootters framework, time is a relational quantity, emerging from correlations between a system and a clock induced by a global Hamiltonian constraint. We construct a time-of-arrival distribution by inverting the Page-Wootters approach, asking what time a clock reads given that the particle arrives at some fixed position. The result coincides with a common approach to the time-of-arrival problem, suggesting a potential relational interpretation of the latter. In addition to providing a relational description of the time-of-arrival problem, this gives an application of the Page-Wootters formalism to a concrete physical problem, and reveals some complications with its canonical interpretation as a theory of conditional probabilities.

quant-pharXiv:2604.00092•Apr 2, 2026•0 citations
Among the many proposals to approach the concept of time in quantum theory, the Page-Wootters mechanism has attracted much attention in the last few years. Originally, this mechanism explored a stationary pure bipartite non-interacting global system, i.e., a system of interest together with an ancillary clock, to determine how the evolution in time can emerge and an equation of motion can be obtained for a quantum particle conditioned to the measurement of the state of the clock. In this mechanism, time is seen as an inaccessible coordinate and the apparent passage of time arises as a consequence of correlations between the subsystems of a global state. Here we propose a measure that captures the relational character of the mechanism, showing that the internal coherence is the necessary ingredient for the emergence of time in the Page–Wootters model. Connecting it to results in quantum thermodynamics, showing that it is directly related to the extractable work from quantum coherence. In a second step we analyze such a timeless approach to quantum theory but deriving an equation of motion for a mixed state system that evolves according to its gravitationally induced interaction with a non-ideal quantum clock. The interaction considered is known to describe the gravitational decoherence mechanism, and the clock model is the recently proposed quasi-ideal clock, i.e., one constructed to approximate the time-energy canonical commutation relation. As a result of our considerations, we obtained an equation of motion that is non-linear in nature, dependent on the systems initial conditions.
Among the many proposals to approach the concept of time in quantum theory, the Page-Wootters mechanism has attracted much attention in the last few years. Originally, such a mechanism explored a stationary bipartite non-interacting global system, i.e., a system of interest together with an ancillary clock, to determine how the evolution in time can emerge as an equation of motion for a quantum particle conditioned to the measurement of the state of the clock. After the seminal proposal, many variations of it were considered, and different aspects of the mechanism were elucidated. Our contribution to these investigations is that we analyze such a timeless approach to quantum theory but deriving an equation of motion for a mixed state system that evolves according to its gravitationally induced interaction with a non-ideal quantum clock. The interaction considered is known to describe the gravitational decoherence mechanism, and the clock model is the recently proposed quasi-ideal clock, i.e., one constructed to approximate the time-energy canonical commutation relation. As a result of our considerations, we obtained an equation of motion that is non-linear in nature, dependent on the system's initial conditions.
We explore a timeless approach to quantum theory, in the form of the Page–Wootters mechanism, in which a gravitational interaction is introduced between the system and a finite-dimensional clock. The clock model used is the recently proposed quasi-ideal clock, a construction that can approximate the time–energy canonical commutation relation. We derive equations of motion for the case in which the system is in a pure and mixed state, obtaining a Schrödinger-like equation that leads to a non-linear equation exhibiting decoherence due to the non-ideal nature of the clock and gravitational coupling. A distinctive feature of this equation is that it exhibits dependence on the system’s initial conditions.
Time, space and entanglement are the main characters in this work. Their nature is still a great mystery in physics and we study here the possibility that these three phenomena are closely connected, showing how entanglement can be at the basis of the emergence of time and space within closed quantum systems. We revisit and extend the Page and Wootters theory that was originally introduced in order to describe the emergence of time through entanglement between subsystems in a globally static, quantum Universe. In the book, after providing a complete review of the salient aspects of the theory, we establish a connection with recent research on the foundations of statistical mechanics and we propose a new understanding of the thermalization process. Furthermore, we generalize the framework in order describe the spatial degree of freedom and we provide a model of 3+1 dimensional, quantum spacetime emerging from entanglement among different subsystems in a globally "timeless" and "positionless" Universe. Finally, via the Page and Wootters theory, the evolution of quantum clocks within a gravitational field is treated and a time dilation effect is obtained in agreement with the Schwarzschild solution.
Time has been an elusive concept to grasp. Although we do not yet understand it properly, there have been advances made in regard to how we can explain it. One such advance is the Page-Wootters mechanism. In this mechanism, time is seen as an inaccessible coordinate and the apparent passage of time arises as a consequence of correlations between the subsystems of a global state. Here we propose a measure that captures the relational character of the mechanism, showing that the internal coherence is the necessary ingredient for the emergence of time in the Page-Wootters model. Also, we connect it to results in quantum thermodynamics, showing that it is directly related to the extractable work from quantum coherence.
In quantum theory, the concept of time rests on shaky ground. One way to address this problem is to remove the usual background time parameter as a primitive entity and explain its emergence via correlations between physical systems. This approach was adopted by Page and Wootters [Phys. Rev. D 27, 2885 (1983)], who showed how time can emerge in a stationary quantum universe from the correlations between two of its subsystems, one of them acting as a clock for the other. In this work, I study the robustness of the Page-Wootters construction across different pictures, states of the universe, and clock interactions, clarifying the role and the nature of the correlations between the subsystems of the universe. I start by showing how to formulate the Page-Wootters construction in the Heisenberg picture via a unitary change of basis. I consider both pure and mixed states of the universe and extend the analysis to include interactions between the clock and the other subsystem of the universe. The study reveals what kind of correlations are necessary for the construction to work. Interestingly, entanglement is not required as long as there are no interactions with the clock. The study also shows that these interactions can lead to a nonunitary evolution for some mixed states of the universe. In a simple two-level system, this aspect becomes relevant at scales where one would expect strong relativistic effects. At these scales, I also observe an inversion in the system's direction of time.
The time-of-arrival problem asks for the probability distribution for when a quantum particle reaches a specified location. It has been the subject of decades of debate, exemplifying the lack of a self-adjoint time observable in quantum theory. In the Page-Wootters framework, time is a relational quantity, emerging from correlations between a system and a clock induced by a global Hamiltonian constraint. We construct a time-of-arrival distribution by inverting the Page-Wootters approach, asking what time a clock reads given that the particle arrives at some fixed position. The result coincides with a common approach to the time-of-arrival problem, suggesting a potential relational interpretation of the latter. In addition to providing a relational description of the time-of-arrival problem, this gives an application of the Page-Wootters formalism to a concrete physical problem, and reveals some complications with its canonical interpretation as a theory of conditional probabilities.