Quantizing time: Interacting clocks and systems
1Department of Physics and Astronomy, Dartmouth College, Hanover, New Hampshire 03755, USA
2Department of Mathematics and Computer Science, Santa Clara University, Santa Clara, California 95053, USA
| Published: | 2019-07-08, volume 3, page 160 |
| Eprint: | arXiv:1712.00081v3 |
| Doi: | https://doi.org/10.22331/q-2019-07-08-160 |
| Citation: | Quantum 3, 160 (2019). |
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Abstract
This article generalizes the conditional probability interpretation of time in which time evolution is realized through entanglement between a clock and a system of interest. This formalism is based upon conditioning a solution to the Wheeler-DeWitt equation on a subsystem of the Universe, serving as a clock, being in a state corresponding to a time $t$. Doing so assigns a conditional state to the rest of the Universe $|\psi_S(t)\rangle$, referred to as the system. We demonstrate that when the total Hamiltonian appearing in the Wheeler-DeWitt equation contains an interaction term coupling the clock and system, the conditional state $|\psi_S(t)\rangle$ satisfies a time-nonlocal Schrödinger equation in which the system Hamiltonian is replaced with a self-adjoint integral operator. This time-nonlocal Schrödinger equation is solved perturbatively and three examples of clock-system interactions are examined. One example considered supposes that the clock and system interact via Newtonian gravity, which leads to the system's Hamiltonian developing corrections on the order of $G/c^4$ and inversely proportional to the distance between the clock and system.
Popular summary
Such a reformulation of quantum theory was put forward by Don Page and William Wootters in 1982, which begins by partitioning the Universe into two parts: a clock and a system comprised of everything else. Then any statement we would ordinarily make regarding the time dependence of the system takes the form, “The quantum state of the system is $\psi(t)$, if a measurement of the clock reads the time $t$”. The crucial difference in this formulation of quantum theory is that it makes no reference to an external background time; instead, time appears as an operational concept defined in terms of the measurement of a physical clock. An entangled state is assigned to the clock and system and the usual dynamics of quantum theory, as described by the Schrödinger equation, emerge from quantum correlations between the clock and system.
This article explores the effects of when the Universe cannot be partitioned cleanly into two parts, that is, when the clock and system interact. In some situations we should expect such an interaction given that gravity couples to everything! It is shown that such clock-system interactions lead to a more general quantum dynamics described by a time-nonlocal modification to the Schrödinger equation. This temporal nonlocality implies that the way in which the system changes in time depends not just on its current state, but also on its state in the future and the past. Three different clock-system interactions are examined to explore the effects of this temporal nonlocality. The first interaction considered leads to the system’s Hamiltonian developing a particular time dependence. The second example supposes that the clock and system interact via Newtonian gravity, which leads to corrections to the system’s Hamiltonian at order $G / c^4$ and is inversely proportional to the distance separating the clock and system. The third interaction gives an example of a qubit clock interacting with another qubit.
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► References
[1] C. J. Isham, Canonical Quantum Gravity and the Problem of Time, edited by L. A. Ibort, M. A. Rodríguez, 157 (1993).
https://doi.org/10.1007/978-94-011-1980-1_6
[2] K. V. Kuchař, Proceedings of the 4th Canadian Conference on General Relativity and Relativistic Astrophysics, edited by G. Kunstatter, D. Vincent and J. Williams (World Scientific, Singapore, 1992).
http://inspirehep.net/record/967799?ln=en
[3] D. N. Page and W. K. Wootters, Phys. Rev. D 27, 2885 (1983).
https://doi.org/10.1103/PhysRevD.27.2885
[4] W. K. Wootters, Int. J. Theor. Phys. 23, 701 (1984).
https://doi.org/10.1007/BF02214098
[5] D. N. Page, ITP preprint NSF-ITP-89-18 (1989).
http://inspirehep.net/record/25248?ln=en
[6] D. N. Page, in Physical Origins of Time Asymmetry, edited by J. J. Halliwell, J. Pérez-Mercader, and W. H. Zurek, 287 (Cambridge University Press, Cambridge, 1994).
https://www.cambridge.org/us/academic/subjects/physics/history-philosophy-and-foundations-physics/physical-origins-time-asymmetry?format=PB
[7] C. E. Dolby, arXiv:gr-qc/0406034.
arXiv:gr-qc/0406034
[8] V. Giovannetti, S. Lloyd, and L. Maccone, Phys. Rev. D 79, 945933 (2015).
https://doi.org/10.1103/PhysRevD.92.045033
[9] R. Gambini, R. A. Porto, and J. Pullin,.
[10] R. Gambini, R. A. Porto, and J. Pullin, New J. Phys. 6, 45 (2004).
https://doi.org/10.1088/1367-2630/6/1/045
[11] R. Gambini, R. A. Porto, S. Torterolo, and J. Pullin, Phys. Rev. D 79, 041501 (2009).
https://doi.org/10.1103/PhysRevD.79.041501
[12] R. Gambini, R. A. Porto, and J. Pullin, Phys. Rev. Lett. 93, 24041 (2004).
https://doi.org/10.1103/PhysRevLett.93.240401
[13] J. Leon and L. Maccone, L. Found. Phys. 47, 1597 (2017).
https://doi.org/10.1007/s10701-017-0115-2
[14] T. Martinelli and D. O. Soares-Pinto, Phys. Rev. A 99, 042124 (2019).
https://doi.org/10.1103/PhysRevA.99.042124
[15] L. R. S. Mendes and D. O. Soares-Pinto, arXiv:1806.09669.
arXiv:1806.09669
[16] A. H. R. Smith and M. Ahmadi, arXiv:1904.12390.
arXiv:1904.12390
[17] V. Corbin and N. J. Cornish, Found. Phys. 39, 474 (2009).
https://doi.org/10.1007/s10701-009-9298-5
[18] E. Moreva, G. Brida, M. Gramegna, V. Giovannetti, L. Maccone, and M. Genovese, Phys. Rev. A 89, 052122 (2014).
https://doi.org/10.1103/PhysRevA.89.052122
[19] C. Marletto and V. Vedral, Phys. Rev. D 95, 043510 (2017).
https://doi.org/10.1103/PhysRevD.95.043510
[20] A. Boette, R. Rossignoli, N. Gigena, and M. Cerezo, Phys. Rev. A 93, 062127 (2016).
https://doi.org/10.1103/PhysRevA.93.062127
[21] K. L. H. Bryan and A. J. M. Medved, Found. Phys. 48, 48 (2018).
https://doi.org/10.1007/s10701-017-0128-x
[22] C. Rovelli, Quantum Gravity, (Cambridge University Press, Cambridge, 2004).
https://www.cambridge.org/us/academic/subjects/physics/cosmology-relativity-and-gravitation/quantum-gravity?format=PB#ohIMJrBHQlmhXwL1.97
[23] C. Kiefer, Quantum Gravity, (Oxford University Press, Oxford, 2012).
https://global.oup.com/academic/product/quantum-gravity-9780199585205?cc=us&lang=en&
[24] P. A. M. Dirac, Lectures on Quantum Mechanics, (Dover Publications, New York, 1964).
https://store.doverpublications.com/0486417131.html
[25] P. A. M. Dirac, Proc. Camb. Philos. Soc. 29, 389 (1933).
https://doi.org/10.1017/S0305004100016431
[26] A. S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory, (North-Holland, Amsterdam, 1982).
https://doi.org/10.1007/978-88-7642-378-9
[27] S. L. Braunstein and C. M. Caves, Phys. Rev. Lett. 22, 3439 (1994).
https://doi.org/10.1103/PhysRevLett.72.3439
[28] P. Busch, M. Grabowski, and P. J. Lahti, Operational Quantum Physics, (Springer-Verlag, Berlin, Heidelberg, 1995).
https://doi.org/10.1007/978-3-540-49239-9
https://www.springer.com/gp/book/9783662140345
[29] S. L. Braunstein, C. M. Caves, and G. J. Milburn, Ann. Phys. 247, 135 (1996).
https://doi.org/10.1006/aphy.1996.0040
[30] B. C. Hall, Quantum Theory for Mathematicians, (Springer-Verlag, New York, 2013).
https://doi.org/10.1007/978-1-4614-7116-5
[31] W. G. Unruh and R. M. Wald, Phys. Rev. D 40, 2598 (1989).
https://doi.org/10.1103/PhysRevD.40.2598
[32] B. S. DeWitt, On Recent Developments in Theoretical and Experimental General Relativity, Gravitation, and Relativistic Field Theories, edited by T. Piran, 6, (World Scientific, Singapore, 1999).
https://doi.org/10.1142/4044
[33] K. Yosida, Functional Analysis, (Springer-Verlag, Berlin, Heidelberg, 1980).
https://doi.org/10.1007/978-3-642-61859-8
[34] P. Busch, Math. Phys., Anal. and Geom. 2, 83 (1999).
https://doi.org/10.1023/A:1009822315406
[35] S. D. Bartlett, T. Rudolph, and R. W. Spekkens, Rev. Mod. Phys. 79, 555 (2007).
https://doi.org/10.1103/RevModPhys.79.555
[36] D. Šafránek, M. Ahmadi, and I. Fuentes, New J. Phys. 17, 033012 (2015).
https://doi.org/10.1088/1367-2630/17/3/033012
[37] A. Peres, Quantum Theory: Concepts and Methods, (Kluwer Academic Publishers, Springer, Dordrecht, 1995).
https://doi.org/10.1007/0-306-47120-5
[38] S. Massar and P. Spindel, Phys. Rev. Lett. 100, 190401 (2008).
https://doi.org/10.1103/PhysRevLett.100.190401
[39] M. P. Woods, R. Silva, and J. Oppenheim, J. Ann. Henri Poincaré 20, 125 (2018).
https://doi.org/10.1007/s00023-018-0736-9
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[2] Ismael L. Paiva, Amit Te’eni, Bar Y. Peled, Eliahu Cohen, and Yakir Aharonov, "Non-inertial quantum clock frames lead to non-Hermitian dynamics", Communications Physics 5 1, 298 (2022).
[3] David Edward Bruschi and Andreas Wolfgang Schell, "Gravitational Redshift Induces Quantum Interference", Annalen der Physik 535 1, 2200468 (2023).
[4] Ovidiu Racorean, "The Born rule in a timeless universe", arXiv:2203.07074, (2022).
[5] Christophe Goeller, Philipp A. Hoehn, and Josh Kirklin, "Diffeomorphism-invariant observables and dynamical frames in gravity: reconciling bulk locality with general covariance", arXiv:2206.01193, (2022).
[6] Philipp A Hoehn and Augustin Vanrietvelde, "How to switch between relational quantum clocks", arXiv:1810.04153, (2018).
[7] Philipp A. Hoehn, Isha Kotecha, and Fabio M. Mele, "Quantum Frame Relativity of Subsystems, Correlations and Thermodynamics", arXiv:2308.09131, (2023).
[8] Marcello Rotondo and Yasusada Nambu, "Clock Time in Quantum Cosmology", Universe 5 2, 66 (2019).
[9] Lorenzo Maccone, "A Fundamental Problem in Quantizing General Relativity", Foundations of Physics 49 12, 1394 (2019).
[10] K. L. H. Bryan and A. J. M. Medved, "Requiem for an ideal clock", arXiv:1803.02045, (2018).
[11] Jürg Fröhlich and Alessandro Pizzo, "The Time-Evolution of States in Quantum Mechanics according to the ETH-Approach", Communications in Mathematical Physics 389 3, 1673 (2022).
[12] K. L. H. Bryan and A. J. M. Medved, "No time for isolated clocks", Journal of Physics Conference Series 1275 1, 012037 (2019).
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