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  • Open Access

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

Matheus H. Zambianco1,2,3,* and T. Rick Perche4,†

  • *Contact author: mhzambia@uwaterloo.ca
  • †Contact author: rick.perche@su.se
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Phys. Rev. Research 8, 043003 – Published 2 October, 2026

DOI: https://doi.org/10.1103/5yl5-4kdk

Abstract

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.

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References (59)

  1. E. F. Taylor, in Spacetime Physics, edited by J. A. Wheeler (W. H. Freeman, San Francisco, 1966).
  2. J. D. Barrow and J. Levin, Twin paradox in compact spaces, Phys. Rev. A 63, 044104 (2001).
  3. F. L. Markley, Relativity twins in the Kerr metric, Am. J. Phys. 41, 1246 (1973).
  4. L. M. Sokołowski, On the twin paradox in static spacetimes: I. Schwarzschild metric, Gen. Relativ. Gravitation 44, 1267 (2012).
  5. B. Rossi and D. B. Hall, Variation of the rate of decay of mesotrons with momentum, Phys. Rev. 59, 223 (1941).
  6. J. C. Hafele and R. E. Keating, Around-the-world atomic clocks: Predicted relativistic time gains, Science 177, 166 (1972).
  7. B. Botermann, D. Bing, C. Geppert, G. Gwinner, T. W. Hänsch, G. Huber, S. Karpuk, A. Krieger, T. Kühl, W. Nörtershäuser, C. Novotny, S. Reinhardt, R. Sánchez, D. Schwalm, T. Stöhlker, A. Wolf, and G. Saathoff, Test of time dilation using stored Li+ ions as clocks at relativistic speed, Phys. Rev. Lett. 113, 120405 (2014).
  8. C. W. Chou, D. B. Hume, T. Rosenband, and D. J. Wineland, Optical clocks and relativity, Science 329, 1630 (2010).
  9. P. Delva, J. Lodewyck, S. Bilicki, E. Bookjans, G. Vallet, R. Le Targat, P.-E. Pottie, C. Guerlin, F. Meynadier, C. Le Poncin-Lafitte, et al., Test of special relativity using a fiber network of optical clocks, Phys. Rev. Lett. 118, 221102 (2017).
  10. H. Salecker and E. P. Wigner, Quantum limitations of the measurement of space-time distances, Phys. Rev. 109, 571 (1958).
  11. Y. J. NG and H. van DAM, Limitation to quantum measurements of space‐time distances, Ann. N. Y. Acad. Sci. 755, 579 (1995).
  12. W. Pauli, General Principles of Quantum Mechanics (Springer, Berlin, 1980).
  13. W. G. Unruh and R. M. Wald, Time and the interpretation of canonical quantum gravity, Phys. Rev. D 40, 2598 (1989).
  14. Edited by J. G. Muga, R. Sala Mayato, and I. L. Egusquiza, in Time in Quantum Mechanics, 2nd ed., Lecture Notes in Physics, Vol. 734 (Springer, Berlin, 2008).
  15. P. Erker, M. T. Mitchison, R. Silva, M. P. Woods, N. Brunner, and M. Huber, Autonomous quantum clocks: Does thermodynamics limit our ability to measure time? Phys. Rev. X 7, 031022 (2017).
  16. X. He, P. Pakkiam, A. A. Gangat, M. J. Kewming, G. J. Milburn, and A. Fedorov, Effect of measurement backaction on quantum clock precision studied with a superconducting circuit, Phys. Rev. Appl. 20, 034038 (2023).
  17. A. R. H. Smith and M. Ahmadi, Quantum clocks observe classical and quantum time dilation, Nat. Commun. 11, 5360 (2020).
  18. A. D. Ludlow, M. M. Boyd, J. Ye, E. Peik, and P. O. Schmidt, Optical atomic clocks, Rev. Mod. Phys. 87, 637 (2015).
  19. T. L. Nicholson, S. L. Campbell, R. B. Hutson, G. E. Marti, B. J. Bloom, R. L. McNally, W. Zhang, M. D. Barrett, M. S. Safronova, G. F. Strouse, W. L. Tew, and J. Ye, Systematic evaluation of an atomic clock at 2 × 10−18 total uncertainty, Nat. Commun. 6, 6896 (2015).
  20. W. F. McGrew, X. Zhang, R. J. Fasano, S. A. Schäffer, K. Beloy, D. Nicolodi, R. C. Brown, N. Hinkley, G. Milani, M. Schioppo, T. H. Yoon, and A. D. Ludlow, Atomic clock performance enabling geodesy below the centimetre level, Nature (London) 564, 87 (2018).
  21. E. Oelker, R. B. Hutson, C. J. Kennedy, L. Sonderhouse, T. Bothwell, A. Goban, D. Kedar, C. Sanner, J. M. Robinson, G. E. Marti, D. G. Matei, T. Legero, M. Giunta, R. Holzwarth, F. Riehle, U. Sterr, and J. Ye, Demonstration of 4.8 × 10−17 stability at 1 s for two independent optical clocks, Nat. Photonics 13, 714 (2019).
  22. S. M. Brewer, J.-S. Chen, A. M. Hankin, E. R. Clements, C. W. Chou, D. J. Wineland, D. B. Hume, and D. R. Leibrandt, Al+27 quantum-logic clock with a systematic uncertainty below 10−18, Phys. Rev. Lett. 123, 033201 (2019).
  23. S. Loriani, A. Friedrich, C. Ufrecht, F. D. Pumpo, S. Kleinert, S. Abend, N. Gaaloul, C. Meiners, C. Schubert, D. Tell, É. Wodey, M. Zych, W. Ertmer, A. Roura, D. Schlippert, W. P. Schleich, E. M. Rasel, and E. Giese, Interference of clocks: A quantum twin paradox, Sci. Adv. 5, eaax8966 (2019).
  24. W. G. Unruh, Notes on black-hole evaporation, Phys. Rev. D 14, 870 (1976).
  25. L. C. B. Crispino, A. Higuchi, and G. E. A. Matsas, The Unruh effect and its applications, Rev. Mod. Phys. 80, 787 (2008).
  26. S. Takagi, Vacuum noise and stress induced by uniform acceleration: Hawking-Unruh effect in Rindler manifold of arbitrary dimension, Prog. Theor. Phys. Suppl. 88, 1 (1986).
  27. D. E. Bruschi, J. Louko, E. Martín-Martínez, A. Dragan, and I. Fuentes, Unruh effect in quantum information beyond the single-mode approximation, Phys. Rev. A 82, 042332 (2010).
  28. L. Sriramkumar and T. Padmanabhan, Finite-time response of inertial and uniformly accelerated Unruh-DeWitt detectors, Class. Quantum Grav. 13, 2061 (1996).
  29. W. Brenna, R. B. Mann, and E. Martín-Martínez, Anti-Unruh phenomena, Phys. Lett. B 757, 307 (2016).
  30. L. J. Garay, E. Martín-Martínez, and J. de Ramón, Thermalization of particle detectors: The Unruh effect and its reverse, Phys. Rev. D 94, 104048 (2016).
  31. T. R. Perche and E. Martín-Martínez, Role of quantum degrees of freedom of relativistic fields in quantum information protocols, Phys. Rev. A 107, 042612 (2023).
  32. E. Poisson, The motion of point particles in curved spacetime, Living Rev. Relativ. 7, 6 (2004).
  33. S. Schlicht, Considerations on the Unruh effect: Causality and regularization, Class. Quantum Grav. 21, 4647 (2004).
  34. J. Louko and A. Satz, How often does the Unruh-DeWitt detector click? Regularization by a spatial profile, Class. Quantum Grav. 23, 6321 (2006).
  35. E. Martín-Martínez, T. R. Perche, and B. de S. L. Torres, General relativistic quantum optics: Finite-size particle detector models in curved spacetimes, Phys. Rev. D 101, 045017 (2020).
  36. E. Martín-Martínez and P. Rodriguez-Lopez, Relativistic quantum optics: The relativistic invariance of the light-matter interaction models, Phys. Rev. D 97, 105026 (2018).
  37. T. R. Perche, Localized nonrelativistic quantum systems in curved spacetimes: A general characterization of particle detector models, Phys. Rev. D 106, 025018 (2022).
  38. E. Martín-Martínez, T. R. Perche, and B. d. S. L. Torres, Broken covariance of particle detector models in relativistic quantum information, Phys. Rev. D 103, 025007 (2021).
  39. W. G. Unruh and R. M. Wald, What happens when an accelerating observer detects a Rindler particle, Phys. Rev. D 29, 1047 (1984).
  40. T. R. Perche, J. Polo-Gómez, B. d. S. L. Torres, and E. Martín-Martínez, Particle detectors from localized quantum field theories, Phys. Rev. D 109, 045013 (2024).
  41. A. M. Alhambra, A. Kempf, and E. Martín-Martínez, Casimir forces on atoms in optical cavities, Phys. Rev. A 89, 033835 (2014).
  42. A. Pozas-Kerstjens and E. Martín-Martínez, Entanglement harvesting from the electromagnetic vacuum with hydrogenlike atoms, Phys. Rev. D 94, 064074 (2016).
  43. R. Lopp and E. Martín-Martínez, Quantum delocalization, gauge, and quantum optics: Light-matter interaction in relativistic quantum information, Phys. Rev. A 103, 013703 (2021).
  44. R. Shah, E. Martín-Martínez, and T. R. Perche, Relativistic QFT description for the interaction of a spin with a magnetic field, Phys. Rev. D 111, 044075 (2025).
  45. B. d. S. L. Torres, T. Rick Perche, A. G. S. Landulfo, and G. E. A. Matsas, Neutrino flavor oscillations without flavor states, Phys. Rev. D 102, 093003 (2020).
  46. T. R. Perche, C. Lima, and E. Martín-Martínez, Harvesting entanglement from complex scalar and fermionic fields with linearly coupled particle detectors, Phys. Rev. D 105, 065016 (2022).
  47. C. Gooding, S. Biermann, S. Erne, J. Louko, W. G. Unruh, J. Schmiedmayer, and S. Weinfurtner, Interferometric Unruh detectors for Bose-Einstein condensates, Phys. Rev. Lett. 125, 213603 (2020).
  48. T. R. Perche, F. Gozzini, and M. K. Oberthaler, Bose polarons as relativistic Unruh-DeWitt detectors: Entanglement harvesting from Bose-Einstein condensates, arXiv:2512.21381.
  49. B. DeWitt, General Relativity: An Einstein Centenary Survey (Cambridge University Press, Cambridge, 1980).
  50. M. Good, B. A. Juárez-Aubry, D. Moustos, and M. Temirkhan, Unruh-like effects: Effective temperatures along stationary worldlines, J. High Energy Phys. 06 (2020) 059.
  51. Notice that it is not possible to directly access the deexcitation probability using a single clock through a one-shot experiment. Instead, one should repeat the experiment N times under the exact same circumstances. Then, if the clock decays in Nd of those experiments, the elapsed time between events xi and xf as measured by this clock model is given by T(C)≡F−1Nd/N≈F−1PD.
  52. S. A. Fulling, M. Sweeny, and R. M. Wald, Singularity structure of the two-point function in quantum field theory in curved spacetime, Commun. Math. Phys. 63, 257 (1978).
  53. Y. Décanini and A. Folacci, Hadamard renormalization of the stress-energy tensor for a quantized scalar field in a general spacetime of arbitrary dimension, Phys. Rev. D 78, 044025 (2008).
  54. C. J. Fewster and R. Verch, The necessity of the Hadamard condition, Class. Quantum Grav. 30, 235027 (2013).
  55. Notice that TA ends up depending on a and T, as it corresponds to the proper time at which Bob meets Alice, and Bob's trajectory is determined by these parameters. Overall, TA and TB are related by TB=4asinh−1(aTA/4).
  56. K. Lorek, J. Louko, and A. Dragan, Ideal clocks—A convenient fiction, Class. Quantum Grav. 32, 175003 (2015).
  57. S. Reinhardt, G. Saathoff, H. Buhr, L. A. Carlson, A. Wolf, D. Schwalm, S. Karpuk, C. Novotny, G. Huber, M. Zimmermann, et al., Test of relativistic time dilation with fast optical atomic clocks at different velocities, Nat. Phys. 3, 861 (2007).
  58. A. Kempf, Replacing the notion of spacetime distance by the notion of correlation, Front. Phys. 9, 655857 (2021).
  59. T. R. Perche and E. Martín-Martínez, Geometry of spacetime from quantum measurements, Phys. Rev. D 105, 066011 (2022).

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Phys. Rev. Research 8, 043003– Published 2 October, 2026
  • Received 6 January 2026
  • Accepted 31 August 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Funded by Bibsam.

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