Spacetime Quantum Reference Frames and superpositions of proper times

Flaminia Giacomini

Perimeter Institute for Theoretical Physics, 31 Caroline St. N, Waterloo, Ontario, N2L 2Y5, Canada

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.


In general relativity, the description of spacetime relies on idealised rods and clocks, which identify a reference frame. In any concrete scenario, reference frames are associated to physical systems, which are ultimately quantum in nature. A relativistic description of the laws of physics hence needs to take into account such quantum reference frames (QRFs), through which spacetime can be given an operational meaning.

Here, we introduce the notion of a spacetime quantum reference frame, associated to a quantum particle in spacetime. Such formulation has the advantage of treating space and time on equal footing, and of allowing us to describe the dynamical evolution of a set of quantum systems from the perspective of another quantum system, where the parameter in which the rest of the physical systems evolves coincides with the proper time of the particle taken as the QRF. Crucially, the proper times in two different QRFs are not related by a standard transformation, but they might be in a quantum superposition one with respect to the other.

Concretely, we consider a system of $N$ relativistic quantum particles in a weak gravitational field, and introduce a timeless formulation in which the global state of the $N$ particles appears "frozen", but the dynamical evolution is recovered in terms of relational quantities. The position and momentum Hilbert space of the particles is used to fix the QRF via a transformation to the local frame of the particle such that the metric is locally inertial at the origin of the QRF. The internal Hilbert space corresponds to the clock space, which keeps the proper time in the local frame of the particle. Thanks to this fully relational construction we show how the remaining particles evolve dynamically in the relational variables from the perspective of the QRF. The construction proposed here includes the Page-Wootters mechanism for non interacting clocks when the external degrees of freedom are neglected. Finally, we find that a quantum superposition of gravitational redshifts and a quantum superposition of special-relativistic time dilations can be observed in the QRF.

► BibTeX data

► References

[1] Yakir Aharonov and Leonard Susskind. Charge Superselection Rule. Phys. Rev., 155: 1428–1431, 1967a. 10.1103/​PhysRev.155.1428. URL https:/​/​​doi/​10.1103/​PhysRev.155.1428.

[2] Yakir Aharonov and Leonard Susskind. Observability of the sign change of spinors under $2{\pi}$ rotations. Phys. Rev., 158: 1237–1238, 1967b. 10.1103/​PhysRev.158.1237. URL https:/​/​​doi/​10.1103/​PhysRev.158.1237.

[3] Y. Aharonov and T. Kaufherr. Quantum frames of reference. Phys. Rev. D, 30: 368–385, 1984. 10.1103/​PhysRevD.30.368. URL https:/​/​​doi/​10.1103/​PhysRevD.30.368.

[4] Stephen D. Bartlett, Terry Rudolph, and Robert W. Spekkens. Reference frames, superselection rules, and quantum information. Rev. Mod. Phys., 79: 555–609, 2007. 10.1103/​RevModPhys.79.555. URL https:/​/​​doi/​10.1103/​RevModPhys.79.555.

[5] Stephen D. Bartlett, Terry Rudolph, Robert W. Spekkens, and Peter S. Turner. Quantum communication using a bounded-size quantum reference frame. New J. Phys., 11 (6): 063013, 2009. 10.1088/​1367-2630/​11/​6/​063013. URL http:/​/​​1367-2630/​11/​i=6/​a=063013.

[6] Gilad Gour and Robert W Spekkens. The resource theory of quantum reference frames: manipulations and monotones. New J. Phys., 10 (3): 033023, 2008. 10.1088/​1367-2630/​10/​3/​033023. URL http:/​/​​1367-2630/​10/​i=3/​a=033023.

[7] Alexei Kitaev, Dominic Mayers, and John Preskill. Superselection rules and quantum protocols. Phys. Rev. A, 69 (5): 052326, 2004. 10.1103/​PhysRevA.69.052326.

[8] Matthew C. Palmer, Florian Girelli, and Stephen D. Bartlett. Changing quantum reference frames. Phys. Rev. A, 89: 052121, 2014. 10.1103/​PhysRevA.89.052121. URL https:/​/​​doi/​10.1103/​PhysRevA.89.052121.

[9] Stephen D Bartlett, Terry Rudolph, Robert W Spekkens, and Peter S Turner. Degradation of a quantum reference frame. New J. Phys., 8 (4): 58, 2006. 10.1088/​1367-2630/​8/​4/​058. URL http:/​/​​1367-2630/​8/​i=4/​a=058.

[10] Alexander R. H. Smith, Marco Piani, and Robert B. Mann. Quantum reference frames associated with noncompact groups: The case of translations and boosts and the role of mass. Phys. Rev. A, 94: 012333, 2016. 10.1103/​PhysRevA.94.012333. URL https:/​/​​doi/​10.1103/​PhysRevA.94.012333.

[11] David Poulin and Jon Yard. Dynamics of a quantum reference frame. New J. Phys., 9 (5): 156, 2007. 10.1088/​1367-2630/​9/​5/​156. URL http:/​/​​1367-2630/​9/​i=5/​a=156.

[12] Florian Girelli and David Poulin. Quantum reference frames and deformed symmetries. Phys. Rev. D, 77: 104012, 2008. 10.1103/​PhysRevD.77.104012. URL https:/​/​​doi/​10.1103/​PhysRevD.77.104012.

[13] Michael Skotiniotis, Borzu Toloui, Ian T. Durham, and Barry C. Sanders. Quantum Frameness for $CPT$ Symmetry. Phys. Rev. Lett., 111: 020504, 2013. 10.1103/​PhysRevLett.111.020504. URL https:/​/​​doi/​10.1103/​PhysRevLett.111.020504.

[14] David Poulin. Toy model for a relational formulation of quantum theory. Int. J. Theor. Phys., 45 (7): 1189–1215, 2006. 10.1007/​s10773-006-9052-0.

[15] Takayuki Miyadera, Leon Loveridge, and Paul Busch. Approximating relational observables by absolute quantities: a quantum accuracy-size trade-off. J. Phys. A, 49 (18): 185301, 2016. 10.1088/​1751-8113/​49/​18/​185301. URL http:/​/​​1751-8121/​49/​i=18/​a=185301.

[16] L. Loveridge, P. Busch, and T. Miyadera. Relativity of quantum states and observables. EPL (Europhysics Letters), 117 (4): 40004, 2017. 10.1209/​0295-5075/​117/​40004. URL http:/​/​​0295-5075/​117/​i=4/​a=40004.

[17] Leon Loveridge, Takayuki Miyadera, and Paul Busch. Symmetry, reference frames, and relational quantities in quantum mechanics. Found. Phys., 48 (2): 135–198, 2018. 10.1007/​s10701-018-0138-3.

[18] Jacques Pienaar. A relational approach to quantum reference frames for spins. arXiv:1601.07320, 2016.

[19] Renato M Angelo, Nicolas Brunner, Sandu Popescu, Anthony J Short, and Paul Skrzypczyk. Physics within a quantum reference frame. J. Phys. A, 44 (14): 145304, 2011. 10.1088/​1751-8113/​44/​14/​145304. URL https:/​/​​10.1088/​1751-8113/​44/​14/​145304.

[20] R M Angelo and A D Ribeiro. Kinematics and dynamics in noninertial quantum frames of reference. J. Phys. A, 45 (46): 465306, 2012. 10.1088/​1751-8113/​45/​46/​465306. URL https:/​/​​10.1088/​1751-8113/​45/​46/​465306.

[21] S. T. Pereira and R. M. Angelo. Galilei covariance and Einstein's equivalence principle in quantum reference frames. Phys. Rev. A, 91: 022107, 2015. 10.1103/​PhysRevA.91.022107. URL https:/​/​​doi/​10.1103/​PhysRevA.91.022107.

[22] Bryce S DeWitt. Quantum theory of gravity. I. The canonical theory. Phys. Rev., 160 (5): 1113, 1967. 10.1103/​PhysRev.160.1113.

[23] C Rovelli. Quantum reference systems. Class. Quant. Grav., 8 (2): 317, 1991. 10.1088/​0264-9381/​8/​2/​012. URL http:/​/​​0264-9381/​8/​i=2/​a=012.

[24] Carlo Rovelli. Relational quantum mechanics. Int. J. Theor. Phys., 35 (8): 1637–1678, 1996. 10.1007/​BF02302261.

[25] Flaminia Giacomini, Esteban Castro-Ruiz, and Časlav Brukner. Quantum mechanics and the covariance of physical laws in quantum reference frames. Nat. Commun., 10 (1): 494, 2019a. 10.1038/​s41467-018-08155-0. URL https:/​/​​10.1038/​s41467-018-08155-0.

[26] Augustin Vanrietvelde, Philipp A Höhn, Flaminia Giacomini, and Esteban Castro-Ruiz. A change of perspective: switching quantum reference frames via a perspective-neutral framework. Quantum, 4: 225, 2020. 10.22331/​q-2020-01-27-225.

[27] Augustin Vanrietvelde, Philipp A Höhn, and Flaminia Giacomini. Switching quantum reference frames in the N-body problem and the absence of global relational perspectives. arXiv:1809.05093, 2018.

[28] Jianhao M. Yang. Switching Quantum Reference Frames for Quantum Measurement. Quantum, 4: 283, June 2020. 10.22331/​q-2020-06-18-283. URL https:/​/​​10.22331/​q-2020-06-18-283.

[29] Flaminia Giacomini, Esteban Castro-Ruiz, and Časlav Brukner. Relativistic quantum reference frames: the operational meaning of spin. Phys. Rev. Lett., 123 (9): 090404, 2019b. 10.1103/​PhysRevLett.123.090404.

[30] Lucas F. Streiter, Flaminia Giacomini, and Časlav Brukner. Relativistic bell test within quantum reference frames. Phys. Rev. Lett., 126: 230403, Jun 2021. 10.1103/​PhysRevLett.126.230403. URL https:/​/​​doi/​10.1103/​PhysRevLett.126.230403.

[31] Anne-Catherine de la Hamette and Thomas D. Galley. Quantum reference frames for general symmetry groups. Quantum, 4: 367, November 2020. 10.22331/​q-2020-11-30-367. URL https:/​/​​10.22331/​q-2020-11-30-367.

[32] Marius Krumm, Philipp A. Hoehn, and Markus P. Mueller. Quantum reference frame transformations as symmetries and the paradox of the third particle. arXiv:2011.01951, 2020.

[33] Angel Ballesteros, Flaminia Giacomini, and Giulia Gubitosi. The group structure of dynamical transformations between quantum reference frames. Quantum, 5: 470, 2021. 10.22331/​q-2021-06-08-470.

[34] Don N. Page and William K. Wootters. Evolution without evolution: Dynamics described by stationary observables. Phys. Rev. D, 27: 2885, 1983. 10.1103/​PhysRevD.27.2885.

[35] Vittorio Giovannetti, Seth Lloyd, and Lorenzo Maccone. Quantum time. Phys. Rev. D, 92 (4): 045033, 2015. 10.1103/​PhysRevD.92.045033.

[36] Esteban Castro-Ruiz, Flaminia Giacomini, Alessio Belenchia, and Časlav Brukner. Quantum clocks and the temporal localisability of events in the presence of gravitating quantum systems. Nat. Commun., 11 (1): 1–12, 2020. 10.1038/​s41467-020-16013-1.

[37] Flaminia Giacomini and Časlav Brukner. Einstein's Equivalence principle for superpositions of gravitational fields and quantum reference frames. arXiv:2012.13754, 2020.

[38] Philipp A. Höhn and Augustin Vanrietvelde. How to switch between relational quantum clocks. New J. Phys., 22 (12): 123048, 2020. 10.1088/​1367-2630/​abd1ac.

[39] Philipp A Höhn. Switching internal times and a new perspective on the `wave function of the universe'. Universe, 5 (5): 116, 2019. 10.3390/​universe5050116.

[40] Alexander R. H. Smith and Mehdi Ahmadi. Quantizing time: Interacting clocks and systems. Quantum, 3: 160, 2019. 10.22331/​q-2019-07-08-160.

[41] Philipp A. Höhn, Alexander R. H. Smith, and Maximilian P. E. Lock. The Trinity of Relational Quantum Dynamics. arXiv:1912.00033, 2019.

[42] Philipp A. Hoehn, Alexander R. H. Smith, and Maximilian P. E. Lock. Equivalence of Approaches to Relational Quantum Dynamics in Relativistic Settings. Front. in Phys., 9: 181, 2021. 10.3389/​fphy.2021.587083.

[43] Alexander R. H. Smith and Mehdi Ahmadi. Quantum clocks observe classical and quantum time dilation. Nat. Commun., 11 (1): 1–9, 2020. 10.1038/​s41467-020-18264-4.

[44] Lucien Hardy. The Construction Interpretation: Conceptual Roads to Quantum Gravity. arXiv:1807.10980, 2020.

[45] Magdalena Zych, Fabio Costa, and Timothy C Ralph. Relativity of quantum superpositions. arXiv:1809.04999, 2018.

[46] C. J. Isham. Canonical quantum gravity and the problem of time. volume 409, pages 157–287. 1993.

[47] Carlo Rovelli. Quantum Gravity. Cambridge University Press, 2004. 10.1017/​CBO9780511755804.

[48] Karel V Kuchař. Time and interpretations of quantum gravity. Int. J. Mod. Phys. D, 20 (supp01): 3–86, 2011. 10.1142/​S0218271811019347.

[49] Carlo Rovelli. Quantum mechanics without time: a model. Phys. Rev. D, 42 (8): 2638, 1990. 10.1103/​PhysRevD.42.2638.

[50] Michael Reisenberger and Carlo Rovelli. Spacetime states and covariant quantum theory. Phys. Rev. D, 65 (12): 125016, 2002. 10.1103/​PhysRevD.65.125016.

[51] Frank Hellmann, Mauricio Mondragon, Alejandro Perez, and Carlo Rovelli. Multiple-event probability in general-relativistic quantum mechanics. Phys. Rev. D, 75 (8): 084033, 2007. 10.1103/​PhysRevD.75.084033.

[52] Magdalena Zych, Łukasz Rudnicki, and Igor Pikovski. Gravitational mass of composite systems. Phys. Rev. D, 99 (10): 104029, 2019. 10.1103/​PhysRevD.99.104029.

[53] Magdalena Zych, Fabio Costa, Igor Pikovski, and Časlav Brukner. Quantum interferometric visibility as a witness of general relativistic proper time. Nat. Commun., 2: 505, 2011. 10.1038/​ncomms1498.

[54] Igor Pikovski, Magdalena Zych, Fabio Costa, and Časlav Brukner. Time dilation in quantum systems and decoherence. New J. Phys., 19 (2): 025011, 2017. 10.1088/​1367-2630/​aa5d92. URL http:/​/​​1367-2630/​19/​i=2/​a=025011.

[55] Magdalena Zych, Igor Pikovski, Fabio Costa, and Časlav Brukner. General relativistic effects in quantum interference of ``clocks''. In Journal of Physics: Conference Series, volume 723, page 012044. IOP Publishing, 2016. 10.1088/​1742-6596/​723/​1/​012044.

[56] Igor Pikovski, Magdalena Zych, Fabio Costa, and Časlav Brukner. Universal decoherence due to gravitational time dilation. Nat. Phys., 11 (8): 668, 2015. 10.1038/​nphys3366.

[57] A. Ashtekar. Lectures on nonperturbative canonical gravity, volume 6. 1991. 10.1142/​1321.

[58] Donald Marolf. Refined algebraic quantization: Systems with a single constraint. Banach Center Publications, 39, 09 1995. 10.4064/​-39-1-331-344.

[59] James B. Hartle and Donald Marolf. Comparing formulations of generalized quantum mechanics for reparametrization - invariant systems. Phys.Rev., D56: 6247–6257, 1997. 10.1103/​PhysRevD.56.6247.

[60] Achim Kempf and John R Klauder. On the implementation of constraints through projection operators. J Phys A, 34 (5): 1019, 2001. 10.1088/​0305-4470/​34/​5/​307.

[61] Guglielmo M Tino. Testing gravity with cold atom interferometry: Results and prospects. Quantum Science and Technology, 2020. 10.1088/​2058-9565/​abd83e.

[62] Philippe Allard Guérin and Časlav Brukner. Observer-dependent locality of quantum events. New J. Phys., 20 (10): 103031, 2018. 10.1088/​1367-2630/​aae742. URL http:/​/​​1367-2630/​20/​i=10/​a=103031.

[63] Piotr T. Grochowski, Alexander R. H. Smith, Andrzej Dragan, and Kacper Dębski. Quantum time dilation in atomic spectra. Phys. Rev. Research, 3: 023053, Apr 2021. 10.1103/​PhysRevResearch.3.023053. URL https:/​/​​doi/​10.1103/​PhysRevResearch.3.023053.

[64] GM Tino, L Cacciapuoti, S Capozziello, G Lambiase, and F Sorrentino. Precision gravity tests and the Einstein Equivalence Principle. Prog. Part. Nucl. Phys., page 103772, 2020. 10.1016/​j.ppnp.2020.103772.

[65] Esteban Castro Ruiz, Flaminia Giacomini, and Časlav Brukner. Entanglement of quantum clocks through gravity. PNAS, 114 (12): E2303–E2309, 2017. ISSN 0027-8424. 10.1073/​pnas.1616427114. URL https:/​/​​content/​114/​12/​E2303.

[66] Flavio Mercati. Shape dynamics: Relativity and relationalism. Oxford University Press, 2018. 10.1093/​oso/​9780198789475.001.0001.

[67] Julian Barbour, Tim Koslowski, and Flavio Mercati. Identification of a gravitational arrow of time. Phys. Rev. Lett., 113 (18): 181101, 2014. 10.1103/​PhysRevLett.113.181101.

Cited by

[1] Philipp A. Hoehn, Maximilian P. E. Lock, Shadi Ali Ahmad, Alexander R. H. Smith, and Thomas D. Galley, "Quantum Relativity of Subsystems", arXiv:2103.01232.

[2] Philipp A. Hoehn, Marius Krumm, and Markus P. Mueller, "Internal quantum reference frames for finite Abelian groups", arXiv:2107.07545.

The above citations are from SAO/NASA ADS (last updated successfully 2021-08-04 14:31:04). The list may be incomplete as not all publishers provide suitable and complete citation data.

On Crossref's cited-by service no data on citing works was found (last attempt 2021-08-04 14:31:01).