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Quantum frames of reference and the relational flow of time

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Abstract

In this short review paper, relative evolution in time and related issues are analyzed within classical and quantum mechanics. We first discuss the basics of quantum frames of reference in both space and time. We then focus on the latter, and more specifically on the “timeless” approach to quantum mechanics due to Page and Wootters. We address time–energy uncertainty relations and the emergence of non-unitarity within this framework. We emphasize relational aspects of quantum time as well as unique features of non-inertial clock frames.

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References

  1. Y. Aharonov, L. Susskind, Charge superselection rule. Phys. Rev. 155, 1428–1431 (1967). https://doi.org/10.1103/PhysRev.155.1428

    Article  ADS  CAS  Google Scholar 

  2. S.D. Bartlett, T. Rudolph, R.W. Spekkens, Reference frames, superselection rules, and quantum information. Rev. Mod. Phys. 79(2), 555 (2007). https://doi.org/10.1103/RevModPhys.79.555

    Article  ADS  MathSciNet  CAS  Google Scholar 

  3. Y. Aharonov, T. Kaufherr, Quantum frames of reference. Phys. Rev. D 30(2), 368 (1984). https://doi.org/10.1103/PhysRevD.30.368

    Article  ADS  MathSciNet  Google Scholar 

  4. C. Rovelli, Quantum reference systems. Class. Quantum Gravity 8(2), 317 (1991). https://doi.org/10.1088/0264-9381/8/2/012

    Article  ADS  MathSciNet  Google Scholar 

  5. F. Giacomini, E. Castro-Ruiz, Č Brukner, Quantum mechanics and the covariance of physical laws in quantum reference frames. Nat. Commun. 10, 494 (2019). https://doi.org/10.1038/s41467-018-08155-0

    Article  ADS  CAS  PubMed  PubMed Central  Google Scholar 

  6. A. Vanrietvelde, P.A. Hoehn, F. Giacomini, E. Castro-Ruiz, A change of perspective: switching quantum reference frames via a perspective-neutral framework. Quantum 4, 225 (2020). https://doi.org/10.22331/q-2020-01-27-225

    Article  Google Scholar 

  7. J.M. Yang, Switching quantum reference frames for quantum measurement. Quantum 4, 283 (2020). https://doi.org/10.22331/q-2020-06-18-283

    Article  Google Scholar 

  8. G.W. Leibniz, G.W. Leibniz, Discourse on Metaphysics: 1686 (Springer, Amsterdam, 1989)

    Book  Google Scholar 

  9. E. Mach, The Science of Mechanics: A Critical and Historical Exposition of Its Principles (Open Court Publishing Company, Illinois, 1893)

    Google Scholar 

  10. J.H. Poincare, La science et l’hypothèse [science and hypothesis] (E. Flammarion, Paris, 1903)

    Google Scholar 

  11. P.-L. Maupertuis, Accord de Différentes Loix de la Nature Qui Avoient Jusqu’ici Paru Incompatibles. Institut de France, Paris (1744)

  12. P.-L. Maupertuis, Les loix du mouvement et du repos déduites d’un principe metaphysique. Histoire de l’academie royale des sciences et des belles-lettres de Berlin [pour l’annee] 1746, 267–294 (1748)

  13. C.G.J. Jacobi, C.W. Borchardt, Vorlesungen Über Dynamik. G. Reimer, 11 (1866)

  14. J.B. Barbour, B. Bertotti, Mach’s principle and the structure of dynamical theories. Proc. R. Soc. Lond. A Math. Phys. Sci. 382(1783), 295–306 (1982)

    ADS  MathSciNet  Google Scholar 

  15. J.B. Barbour, The timelessness of quantum gravity: I. The evidence from the classical theory. Class. Quantum Gravity 11(12), 2853 (1994)

    Article  ADS  MathSciNet  Google Scholar 

  16. J.B. Barbour, The timelessness of quantum gravity: II. The appearance of dynamics in static configurations. Class. Quantum Gravity 11(12), 2875 (1994)

    Article  ADS  MathSciNet  Google Scholar 

  17. C. Rovelli, Group quantization of the Barbour-Bertotti model, in Conceptual Problems of Quantum Gravity. ed. by A. Ashtekar, J. Stachel (Birkhauser, Boston, 1991), pp.292–299

    Google Scholar 

  18. S. Gryb, Jacobi’s principle and the disappearance of time. Phys. Rev. D (2010). https://doi.org/10.1103/physrevd.81.044035

    Article  Google Scholar 

  19. W. Pauli, Die allgemeinen prinzipien der wellenmechanik. In: Quantentheorie (Springer, Berlin, 1933), pp. 83–272. https://doi.org/10.1007/978-3-642-52619-0_2

  20. Y. Aharonov, D. Bohm, Time in the quantum theory and the uncertainty relation for time and energy. Phys. Rev. 122(5), 1649 (1961). https://doi.org/10.1103/PhysRev.122.1649

    Article  ADS  MathSciNet  Google Scholar 

  21. J.C. Garrison, J. Wong, Canonically conjugate pairs, uncertainty relations, and phase operators. J. Math. Phys. 11(8), 2242 (1970). https://doi.org/10.1063/1.1665388

    Article  ADS  MathSciNet  Google Scholar 

  22. B.S. DeWitt, Quantum theory of gravity. I. The canonical theory. Phys. Rev. 160(5), 1113 (1967). https://doi.org/10.1103/PhysRev.160.1113

    Article  ADS  CAS  Google Scholar 

  23. D.N. Page, W.K. Wootters, Evolution without evolution: dynamics described by stationary observables. Phys. Rev. D 27(12), 2885 (1983). https://doi.org/10.1103/PhysRevD.27.2885

    Article  ADS  Google Scholar 

  24. W.K. Wootters, “Time’’ replaced by quantum correlations. Int. J. Theor. Phys. 23(8), 701 (1984). https://doi.org/10.1007/BF02214098

    Article  MathSciNet  Google Scholar 

  25. V. Giovannetti, S. Lloyd, L. Maccone, Quantum time. Phys. Rev. D 92(4), 045033 (2015). https://doi.org/10.1103/PhysRevD.92.045033

    Article  ADS  MathSciNet  CAS  Google Scholar 

  26. C. Marletto, V. Vedral, Evolution without evolution and without ambiguities. Phys. Rev. D 95(4), 043510 (2017). https://doi.org/10.1103/PhysRevD.95.043510

    Article  ADS  Google Scholar 

  27. E. Castro Ruiz, F. Giacomini, Č Brukner, Entanglement of quantum clocks through gravity. Proc. Natl. Acad. Sci. 114(12), 2303 (2017). https://doi.org/10.1073/pnas.1616427114

    Article  ADS  MathSciNet  CAS  Google Scholar 

  28. A.R.H. Smith, M. Ahmadi, Quantizing time: interacting clocks and systems. Quantum 3, 160 (2019). https://doi.org/10.22331/q-2019-07-08-160

  29. F. Giacomini, E. Castro-Ruiz, Č Brukner, Quantum mechanics and the covariance of physical laws in quantum reference frames. Nat. Commun. 10, 494 (2019). https://doi.org/10.1038/s41467-018-08155-0

    Article  ADS  CAS  PubMed  PubMed Central  Google Scholar 

  30. N.L. Diaz, R. Rossignoli, History state formalism for Dirac’s theory. Phys. Rev. D 99(4), 045008 (2019). https://doi.org/10.1103/PhysRevD.99.045008

    Article  ADS  MathSciNet  CAS  Google Scholar 

  31. N.L. Diaz, J.M. Matera, R. Rossignoli, History state formalism for scalar particles. Phys. Rev. D 100(12), 125020 (2019). https://doi.org/10.1103/PhysRevD.100.125020

    Article  ADS  MathSciNet  CAS  Google Scholar 

  32. T. Martinelli, D.O. Soares-Pinto, Quantifying quantum reference frames in composed systems: local, global, and mutual asymmetries. Phys. Rev. A 99(4), 042124 (2019). https://doi.org/10.1103/PhysRevA.99.042124

    Article  ADS  CAS  Google Scholar 

  33. E. Castro-Ruiz, F. Giacomini, A. Belenchia, Č Brukner, Quantum clocks and the temporal localisability of events in the presence of gravitating quantum systems. Nat. Commun. 11, 2672 (2020). https://doi.org/10.1038/s41467-020-16013-1

    Article  ADS  CAS  PubMed  PubMed Central  Google Scholar 

  34. A.R.H. Smith, M. Ahmadi, Quantum clocks observe classical and quantum time dilation. Nat. Commun. 11(1), 5360 (2020). https://doi.org/10.1038/s41467-020-18264-4

    Article  ADS  CAS  PubMed  PubMed Central  Google Scholar 

  35. A. Ballesteros, F. Giacomini, G. Gubitosi, The group structure of dynamical transformations between quantum reference frames. arXiv:2012.15769 (2020)

  36. M. Trassinelli, Conditional probability, quantum time and friends. arXiv:2103.08903 (2021)

  37. R.S. Carmo, D.O. Soares-Pinto, Quantifying resources for the Page-Wootters mechanism: shared asymmetry as relative entropy of entanglement. Phys. Rev. A 103(5), 052420 (2021). https://doi.org/10.1103/PhysRevA.103.052420

    Article  ADS  MathSciNet  CAS  Google Scholar 

  38. I.L. Paiva, M. Nowakowski, E. Cohen, Dynamical nonlocality in quantum time via modular operators. Phys. Rev. A 105(4), 042207 (2022)

    Article  ADS  MathSciNet  CAS  Google Scholar 

  39. V. Baumann, M. Krumm, P.A. Guérin, Č Brukner, Noncausal Page–Wootters circuits. Phys. Rev. Res. 4(1), 013180 (2022)

    Article  CAS  Google Scholar 

  40. I.L. Paiva, A.C. Lobo, E. Cohen, Flow of time during energy measurements and the resulting time-energy uncertainty relations. Quantum 6, 683 (2022)

    Article  Google Scholar 

  41. L.R.S. Mendes, F. Brito, D.O. Soares-Pinto, Non-linear equation of motion for Page-Wootters mechanism with interaction and quasi-ideal clocks. arXiv:2107.11452 (2021)

  42. I.L. Paiva, A. Te’eni, B.Y. Peled, E. Cohen, Y. Aharonov, Non-inertial quantum clock frames lead to non-Hermitian dynamics. Commun. Phys. 5(1), 298 (2022). https://doi.org/10.1038/s42005-022-01081-0

    Article  Google Scholar 

  43. E. Adlam, Watching the clocks: interpreting the Page-Wootters formalism and the internal quantum reference frame programme. Found. Phys. 52(5), 99 (2022)

    Article  ADS  MathSciNet  Google Scholar 

  44. C.J. Isham, Canonical quantum gravity and the problem of time. In: Integrable Systems. Quantum Groups, and Quantum Field Theories (Springer, Berlin, 1993), pp. 157–287

  45. C. Rovelli, Quantum mechanics without time: a model. Phys. Rev. D 42(8), 2638 (1990)

    Article  ADS  CAS  Google Scholar 

  46. C. Rovelli, Time in quantum gravity: an hypothesis. Phys. Rev. D 43(2), 442 (1991)

    Article  ADS  MathSciNet  CAS  Google Scholar 

  47. S.W. Hawking, The unpredictability of quantum gravity. Commun. Math. Phys. 87(3), 395 (1982). https://doi.org/10.1007/BF01206031

    Article  ADS  MathSciNet  Google Scholar 

  48. W.G. Unruh, R.M. Wald, Evolution laws taking pure states to mixed states in quantum field theory. Phys. Rev. D 52(4), 2176 (1995). https://doi.org/10.1103/PhysRevD.52.2176

    Article  ADS  MathSciNet  CAS  Google Scholar 

  49. R. Penrose, On gravity’s role in quantum state reduction. Gen. Relat. Gravity 28(5), 581 (1996). https://doi.org/10.1007/BF02105068

    Article  ADS  MathSciNet  Google Scholar 

  50. I. Newton, Philosophiae Naturalis Principia Mathematica (1687). https://doi.org/10.3931/e-rara-440

  51. P.A.M. Dirac, Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950). https://doi.org/10.4153/CJM-1950-012-1

    Article  MathSciNet  Google Scholar 

  52. F. Mercati, Shape Dynamics: Relativity and Relationalism (Oxford University Press, Oxford, 2018)

    Book  Google Scholar 

  53. P.A.M. Dirac, in Lectures on Quantum Mechanics, Belfer Graduate School of Science (Yeshiva University, New York, 1964). https://store.doverpublications.com/0486417131.html

  54. P.A. Höhn, A.R.H. Smith, M.P.E. Lock, Trinity of relational quantum dynamics. Phys. Rev. D 104, 066001 (2021). https://doi.org/10.1103/PhysRevD.104.066001

    Article  ADS  MathSciNet  Google Scholar 

  55. T. Favalli, A. Smerzi, A model of quantum spacetime. AVS Quantum Sci. 4(4), 044403 (2022)

    Article  ADS  Google Scholar 

  56. F. Giacomini, Spacetime quantum reference frames and superpositions of proper times. Quantum 5, 508 (2021). https://doi.org/10.22331/q-2021-07-22-508

  57. P. Busch, M. Grabowski, P.J. Lahti, in Operational Quantum Physics. Lecture Notes in Physics Monographs, vol. 31 (Springer, Berlin, 1995). https://doi.org/10.1007/978-3-540-49239-9

  58. L. Loveridge, T. Miyadera, Relative quantum time. Found. Phys. 49(6), 549 (2019). https://doi.org/10.1007/s10701-019-00268-w

    Article  ADS  MathSciNet  Google Scholar 

  59. H. Salecker, E.P. Wigner, Quantum limitations of the measurement of space-time distances. Phys. Rev. 109(2), 571 (1958). https://doi.org/10.1103/PhysRev.109.571

    Article  ADS  MathSciNet  Google Scholar 

  60. A. Peres, Measurement of time by quantum clocks. Am. J. Phys. 48(7), 552 (1980). https://doi.org/10.1119/1.12061

    Article  ADS  MathSciNet  Google Scholar 

  61. J.B. Hartle, Quantum kinematics of spacetime. II. A model quantum cosmology with real clocks. Phys. Rev. D 38(10), 2985 (1988). https://doi.org/10.1103/PhysRevD.38.2985

    Article  ADS  MathSciNet  CAS  Google Scholar 

  62. A. Singh, S.M. Carroll, Modeling position and momentum in finite-dimensional Hilbert spaces via generalized Pauli operators. arXiv:1806.10134 (2018)

  63. P.A. Höhn, A. Vanrietvelde, How to switch between relational quantum clocks. New J. Phys. 22(12), 123048 (2020). https://doi.org/10.1088/1367-2630/abd1ac

    Article  ADS  MathSciNet  Google Scholar 

  64. P.A.M. Dirac, Bakerian lecture–The physical interpretation of quantum mechanics. Proc. R. Soc. A 180(980), 1 (1942). https://doi.org/10.1098/rspa.1942.0023

    Article  ADS  CAS  Google Scholar 

  65. W. Pauli, On Dirac’s new method of field quantization. Rev. Mod. Phys. 15(3), 175 (1943). https://doi.org/10.1103/RevModPhys.15.175

    Article  ADS  MathSciNet  Google Scholar 

  66. C.M. Bender, S. Boettcher, Real spectra in non-Hermitian Hamiltonians having pt symmetry. Phys. Rev. Lett. 80(24), 5243 (1998)

    Article  ADS  MathSciNet  CAS  Google Scholar 

  67. C.M. Bender, D.C. Brody, H.F. Jones, B.K. Meister, Faster than Hermitian quantum mechanics. Phys. Rev. Lett. 98(4), 040403 (2007)

    Article  ADS  MathSciNet  PubMed  Google Scholar 

  68. C. Zheng, L. Hao, G.L. Long, Observation of a fast evolution in a parity-time-symmetric system. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. 371(1989), 20120053 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  69. S. Massar, S. Popescu, Measurement of the total energy of an isolated system by an internal observer. Phys. Rev. A 71(4), 042106 (2005). https://doi.org/10.1103/PhysRevA.71.042106

    Article  ADS  CAS  Google Scholar 

  70. R. Gambini, R.A. Porto, J. Pullin, Fundamental decoherence from quantum gravity: a pedagogical review. Gen. Relat. Gravity 39(8), 1143 (2007). https://doi.org/10.1007/s10714-007-0451-1

    Article  ADS  MathSciNet  Google Scholar 

  71. E.C. Ruiz, F. Giacomini, Č Brukner, Entanglement of quantum clocks through gravity. Proc. Natl. Acad. Sci. 114(12), 2303 (2017). https://doi.org/10.1073/pnas.1616427114

    Article  MathSciNet  CAS  Google Scholar 

  72. I. Pikovski, M. Zych, F. Costa, Č Brukner, Universal decoherence due to gravitational time dilation. Nat. Phys. 11(8), 668 (2015). https://doi.org/10.1038/nphys3366

    Article  CAS  Google Scholar 

  73. M. Sonnleitner, S.M. Barnett, Mass-energy and anomalous friction in quantum optics. Phys. Rev. A 98(4), 042106 (2018). https://doi.org/10.1103/PhysRevA.98.042106

    Article  ADS  CAS  Google Scholar 

  74. M. Zych, Ł Rudnicki, I. Pikovski, Gravitational mass of composite systems. Phys. Rev. D 99(10), 104029 (2019). https://doi.org/10.1103/PhysRevD.99.104029

    Article  ADS  MathSciNet  CAS  Google Scholar 

  75. M. Zych, F. Costa, I. Pikovski, Č Brukner, Quantum interferometric visibility as a witness of general relativistic proper time. Nat. Commun. 2(1), 505 (2011). https://doi.org/10.1038/ncomms1498

    Article  ADS  CAS  PubMed  Google Scholar 

  76. C. Rovelli, Quantum Gravity (Cambridge University Press, Cambridge, 2004)

    Book  Google Scholar 

  77. C. Kiefer, Quantum gravity: general introduction and recent developments. Annalen der Physik 518(1–2), 129–148 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  78. E. Anderson, Problem of time in quantum gravity. Annalen der Physik 524(12), 757–786 (2012)

    Article  ADS  MathSciNet  Google Scholar 

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Acknowledgements

We acknowledge insightful discussions and fruitful collaborations with Yakir Aharonov, Bar Peled, Amit Te’eni and above all, with Ismael L. Paiva. We also acknowledge constructive comments provided by an anonymous reviewer. E.C. further wishes to thank Václav Špička for organizing FQMT’22 where some of these results were first presented, as well as the conference participants for helpful conversations.

Funding

E.C. was supported by the Israel Innovation Authority under Project 73795, by the Pazy Foundation (Grant no. 49579), by the Israeli Ministry of Science and Technology (Grant no. 17812), and by the Quantum Science and Technology Program of the Israeli Council of Higher Education.

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  1. Faculty of Engineering and the Institute of Nanotechnology and Advanced Materials, Bar-Ilan University, Max Ve-Anna Webb, Ramat Gan, 5290002, Israel

    Michael Suleymanov & Eliahu Cohen

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  1. Michael Suleymanov
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Suleymanov, M., Cohen, E. Quantum frames of reference and the relational flow of time. Eur. Phys. J. Spec. Top. 232, 3325–3337 (2023). https://doi.org/10.1140/epjs/s11734-023-00973-8

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