Time Observables in a Timeless Universe

Tommaso Favalli; Augusto Smerzi

doi:10.22331/q-2020-10-29-354

Time Observables in a Timeless Universe

Tommaso Favalli^1,2 and Augusto Smerzi^1,1,2

¹QSTAR, INO-CNR and LENS, Largo Enrico Fermi 2, I-50125 Firenze, Italy
²Universitá degli Studi di Napoli Federico II, Via Cinthia 21, I-80126 Napoli, Italy

Published:	2020-10-29, volume 4, page 354
Eprint:	arXiv:2003.09042v3
Doi:	https://doi.org/10.22331/q-2020-10-29-354
Citation:	Quantum 4, 354 (2020).

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Updated after initial publication: This publication was updated to version v3 after the initial publication. The authors left the following comment on the arXiv:

13 pages

Abstract

Time in quantum mechanics is peculiar: it is an observable that cannot be associated to an Hermitian operator. As a consequence it is impossible to explain dynamics in an isolated system without invoking an external classical clock, a fact that becomes particularly problematic in the context of quantum gravity. An unconventional solution was pioneered by Page and Wootters (PaW) in 1983. PaW showed that dynamics can be an emergent property of the entanglement between two subsystems of a static Universe. In this work we first investigate the possibility to introduce in this framework a Hermitian time operator complement of a clock Hamiltonian having an equally-spaced energy spectrum. An Hermitian operator complement of such Hamiltonian was introduced by Pegg in 1998, who named it "Age". We show here that Age, when introduced in the PaW context, can be interpreted as a proper Hermitian time operator conjugate to a "good" clock Hamiltonian. We therefore show that, still following Pegg's formalism, it is possible to introduce in the PaW framework bounded clock Hamiltonians with an unequally-spaced energy spectrum with rational energy ratios. In this case time is described by a POVM and we demonstrate that Pegg's POVM states provide a consistent dynamical evolution of the system even if they are not orthogonal, and therefore partially un-distinguishables.

Note: Equation (35) was corrected post publication – the latest version of this paper is now available at Quantum.

► BibTeX data

@article{Favalli2020timeobservablesin,
  doi = {10.22331/q-2020-10-29-354},
  url = {https://doi.org/10.22331/q-2020-10-29-354},
  title = {Time {O}bservables in a {T}imeless {U}niverse},
  author = {Favalli, Tommaso and Smerzi, Augusto},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {4},
  pages = {354},
  month = oct,
  year = {2020}
}

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Cited by

[1] Tommaso Favalli, "Relativistic Limits on the Discretization and Temporal Resolution of a Quantum Clock", Entropy 27 10, 1068 (2025).

[2] N. L. Diaz, J. M. Matera, and R. Rossignoli, "Spacetime quantum and classical mechanics with dynamical foliation", Physical Review D 109 10, 105008 (2024).

[3] Tommaso Favalli and Augusto Smerzi, "Time Dilation of Quantum Clocks in a Relativistic Gravitational Potential", Entropy 27 5, 489 (2025).

[4] Tommaso Favalli and Augusto Smerzi, "Peaceful coexistence of thermal equilibrium and the emergence of time", Physical Review D 105 2, 023525 (2022).

[5] Leonardo Chataignier, Philipp A Höhn, Maximilian P E Lock, and Fabio M Mele, "Relational dynamics with periodic clocks", New Journal of Physics 28 3, 034504 (2026).

[6] Tommaso Favalli, Springer Theses 89 (2024) ISBN:978-3-031-52351-9.

[7] N.L. Diaz, J.M. Matera, and R. Rossignoli, "Path integrals from spacetime quantum actions", Annals of Physics 479, 170052 (2025).

[8] Tommaso Favalli, Springer Theses 1 (2024) ISBN:978-3-031-52351-9.

[9] Mohammed Basil Altaie, "Quantum Divine Action in Quantum-Time Perspective", Kader 23 1, 1 (2025).

[10] Tommaso Favalli, Springer Theses 69 (2024) ISBN:978-3-031-52351-9.

[11] N. L. Diaz, Paolo Braccia, Martin Larocca, J. M. Matera, R. Rossignoli, and M. Cerezo, "Parallel-in-time quantum simulation via Page and Wootters quantum time", Physical Review Research 7 3, 033294 (2025).

[12] Tommaso Favalli, Springer Theses 135 (2024) ISBN:978-3-031-52351-9.

[13] Philipp A. Höhn, Alexander R. H. Smith, and Maximilian P. E. Lock, "Equivalence of Approaches to Relational Quantum Dynamics in Relativistic Settings", Frontiers in Physics 9, 587083 (2021).

[14] Tommaso Favalli, Springer Theses 47 (2024) ISBN:978-3-031-52351-9.

[15] M. Basil Altaie, Daniel Hodgson, and Almut Beige, "Time and Quantum Clocks: A Review of Recent Developments", Frontiers in Physics 10, 897305 (2022).

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[19] Tommaso Favalli, Springer Theses 13 (2024) ISBN:978-3-031-52351-9.

[20] Tommaso Favalli, Springer Theses 159 (2024) ISBN:978-3-031-52351-9.

[21] T. Favalli and A. Smerzi, "A model of quantum spacetime", AVS Quantum Science 4 4, 044403 (2022).

[22] Leandro R. S. Mendes, Frederico Brito, and Diogo O. Soares-Pinto, "Non-Linear Equation of Motion for Page–Wootters Mechanism with Interaction and Quasi-Ideal Clocks", Universe 11 9, 308 (2025).

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The above citations are from Crossref's cited-by service (last updated successfully 2026-08-08 01:00:13) and SAO/NASA ADS (last updated successfully 2026-08-08 01:00:14). The list may be incomplete as not all publishers provide suitable and complete citation data.

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