Objective Trajectories In Hybrid Classical-quantum Dynamics

Jonathan Oppenheim¹, Carlo Sparaciari¹, Barbara Šoda^1,2,3, and Zachary Weller-Davies¹

¹Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, United Kingdom
²Dept. of Physics, University of Waterloo, Waterloo, Ontario, Canada
³Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada

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Abstract

Consistent dynamics which couples classical and quantum degrees of freedom exists, provided it is stochastic. This dynamics is linear in the hybrid state, completely positive and trace preserving. One application of this is to study the back-reaction of quantum fields on space-time which does not suffer from the pathologies of the semi-classical equations. Here we introduce several toy models in which to study hybrid classical-quantum evolution, including a qubit coupled to a particle in a potential, and a quantum harmonic oscillator coupled to a classical one. We present an unravelling approach to calculate the dynamics, and provide code to numerically simulate it. Unlike the purely quantum case, the trajectories (or histories) of this unravelling can be unique, conditioned on the classical degrees of freedom for discrete realisations of the dynamics, when different jumps in the classical degrees of freedom are accompanied by the action of unique operators on the quantum system. As a result, the “measurement postulate” of quantum theory is not needed; quantum systems become classical because they interact with a fundamentally classical field.

► BibTeX data

@article{Oppenheim2023objective, doi = {10.22331/q-2023-01-03-891}, url = {https://doi.org/10.22331/q-2023-01-03-891}, title = {Objective trajectories in hybrid classical-quantum dynamics}, author = {Oppenheim, Jonathan and Sparaciari, Carlo and {v{S}}oda, Barbara and Weller-Davies, Zachary}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{“{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {7}, pages = {891}, month = jan, year = {2023}<br /> }

► References

[1] R. Alicki and K. Lendi. Quantum Dynamical Semigroups and Applications. Springer, Berlin, 1987. 10.1007/3-540-70861-8. URL https://doi.org/10.1007/3-540-70861-8.
https://doi.org/10.1007/3-540-70861-8

[2] Robert Alicki and Stanisław Kryszewski. Completely positive bloch-boltzmann equations. Physical Review A, 68 (1): 013809, 2003. 10.1103/PhysRevA.68.013809. URL https://doi.org/10.1103/PhysRevA.68.013809.
https://doi.org/10.1103/PhysRevA.68.013809

[3] Arlen Anderson. Quantum backreaction on” classical” variables. Physical Review Letters, 74 (5): 621, 1995. 10.1103/PhysRevLett.74.621. URL https://doi.org/10.1103/PhysRevLett.74.621.
https://doi.org/10.1103/PhysRevLett.74.621

[4] T. Banks, M. E. Peskin, and L. Susskind. Difficulties for the evolution of pure states into mixed states. Nuclear Physics B, 244: 125–134, September 1984. 10.1016/0550-3213(84)90184-6. URL http://dx.doi.org/10.1016/0550-3213(84)90184-6.
https://doi.org/10.1016/0550-3213(84)90184-6

[5] Carlos Barceló, Raúl Carballo-Rubio, Luis J Garay, and Ricardo Gómez-Escalante. Hybrid classical-quantum formulations ask for hybrid notions. Physical Review A, 86 (4): 042120, 2012. 10.1103/PhysRevA.86.042120. URL https://doi.org/10.1103/PhysRevA.86.042120.
https://doi.org/10.1103/PhysRevA.86.042120

[6] Angelo Bassi, Kinjalk Lochan, Seema Satin, Tejinder P Singh, and Hendrik Ulbricht. Models of wave-function collapse, underlying theories, and experimental tests. Reviews of Modern Physics, 85 (2): 471, 2013. 10.1103/RevModPhys.85.471. URL https://doi.org/10.1103/RevModPhys.85.471.
https://doi.org/10.1103/RevModPhys.85.471

[7] V. P. Belavkin. A stochastic posterior schrödinger equation for counting nondemolition measurement. Letters in Mathematical Physics, (20): 85–89, 1990. 10.1007/BF00398273. URL https://doi.org/10.1007/BF00398273.
https://doi.org/10.1007/BF00398273

[8] Ph. Blanchard and A. Jadczyk. Event-enhanced quantum theory and piecewise deterministic dynamics. Annalen der Physik, 507 (6): 583–599, 1995. https://doi.org/10.1002/andp.19955070605. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/andp.19955070605.
https://doi.org/10.1002/andp.19955070605

[9] Philippe Blanchard and Arkadiusz Jadczyk. On the interaction between classical and quantum systems. Physics Letters A, 175 (3-4): 157–164, 1993. 10.1016/0375-9601(93)90818-K. URL https://doi.org/10.1016/0375-9601(93)90818-K.
https://doi.org/10.1016/0375-9601(93)90818-K

[10] Niels Bohr and Léon Rosenfeld. On the question of the measurability of electromagnetic field quantities. Quantum theory and measurement, pages 478–522, 1933. 10.1007/978-94-009-9349-5_26. URL https://doi.org/10.1007/978-94-009-9349-5_26.
https://doi.org/10.1007/978-94-009-9349-5_26

[11] Denys I Bondar, François Gay-Balmaz, and Cesare Tronci. Koopman wavefunctions and classical–quantum correlation dynamics. Proceedings of the Royal Society A, 475 (2229): 20180879, 2019. 10.1098/rspa.2018.0879. URL https://doi.org/10.1098/rspa.2018.0879.
https://doi.org/10.1098/rspa.2018.0879

[12] Wayne Boucher and Jennie Traschen. Semiclassical physics and quantum fluctuations. Physical Review D, 37 (12): 3522, 1988. 10.1103/PhysRevD.37.3522. URL https://doi.org/10.1103/PhysRevD.37.3522.
https://doi.org/10.1103/PhysRevD.37.3522

[13] H. P. Breuer and F. Petruccione. The Theory of Open Quantum Systems. Oxford University Press, 2002. 10.1093/acprof:oso/9780199213900.001.0001. URL https://doi.org/10.1093/acprof:oso/9780199213900.001.0001.
https://doi.org/10.1093/acprof:oso/9780199213900.001.0001

[14] T.A. Brun, N. Gisin, P.F. O’Mahony, and M. Rigo. From quantum trajectories to classical orbits. Physics Letters A, 229 (5): 267–272, 1997. ISSN 0375-9601. https://doi.org/10.1016/S0375-9601(97)00217-X. URL https://www.sciencedirect.com/science/article/pii/S037596019700217X.
https://doi.org/10.1016/S0375-9601(97)00217-X
https://www.sciencedirect.com/science/article/pii/S037596019700217X

[15] Todd A. Brun. A simple model of quantum trajectories. American Journal of Physics, 70 (7): 719–737, jul 2002. 10.1119/1.1475328. URL https://doi.org/10.1119.
https://doi.org/10.1119/1.1475328

[16] H. Carmichael. An Open Systems Approach to Quantum Optics. Springer, Berlin, 1993. 10.1007/978-3-540-47620-7. URL https://doi.org/10.1007/978-3-540-47620-7.
https://doi.org/10.1007/978-3-540-47620-7

[17] J Caro and LL Salcedo. Impediments to mixing classical and quantum dynamics. Physical Review A, 60 (2): 842, 1999. 10.1103/PhysRevA.60.842. URL https://doi.org/10.1103/PhysRevA.60.842.
https://doi.org/10.1103/PhysRevA.60.842

[18] Jean Dalibard, Yvan Castin, and Klaus Mølmer. Wave-function approach to dissipative processes in quantum optics. Phys. Rev. Lett., 68 (580), 1992. 10.1103/PhysRevLett.68.580. URL https://doi.org/10.1103/PhysRevLett.68.580.
https://doi.org/10.1103/PhysRevLett.68.580

[19] E. B. Davies. Markovian master equations. Commun. Math. Phys., (39): 91–110, 1974. 10.1007/BF01608389. URL https://doi.org/10.1007/BF01608389.
https://doi.org/10.1007/BF01608389

[20] Bryce S DeWitt. Definition of commutators via the uncertainty principle. Journal of Mathematical Physics, 3 (4): 619–624, 1962. 10.1063/1.1724265. URL https://doi.org/10.1063/1.1724265.
https://doi.org/10.1063/1.1724265

[21] Cécile M. DeWitt and Dean Rickles. The role of gravitation in physics: Report from the 1957 Chapel Hill Conference, volume 5. epubli, 2011. 10.34663/9783945561294-00. URL http://doi.org/10.34663/9783945561294-00.
https://doi.org/10.34663/9783945561294-00

[22] Lajos Diosi. A universal master equation for the gravitational violation of quantum mechanics. Physics letters A, 120 (8): 377–381, 1987. 10.1016/0375-9601(87)90681-5. URL https://doi.org/10.1016/0375-9601(87)90681-5.
https://doi.org/10.1016/0375-9601(87)90681-5

[23] Lajos Diósi. Quantum dynamics with two planck constants and the semiclassical limit. arXiv preprint arXiv:quant-ph/9503023, 1995. 10.48550/arXiv.quant-ph/9503023. URL https://doi.org/10.48550/arXiv.quant-ph/9503023.
https://doi.org/10.48550/arXiv.quant-ph/9503023
arXiv:quant-ph/9503023

[24] Lajos Diósi. The gravity-related decoherence master equation from hybrid dynamics. Journal of Physics: Conference Series, 306: 012006, Jul 2011. 10.1088/1742-6596/306/1/012006. URL https://doi.org/10.1088/1742-6596/306/1/012006.
https://doi.org/10.1088/1742-6596/306/1/012006

[25] Lajos Diósi, Nicolas Gisin, Jonathan Halliwell, and Ian C Percival. Decoherent histories and quantum state diffusion. Physical review letters, 74 (2): 203, 1995. 10.1103/PhysRevLett.74.203. URL https://doi.org/10.1103/PhysRevLett.74.203.
https://doi.org/10.1103/PhysRevLett.74.203

[26] Lajos Diósi, Nicolas Gisin, and Walter T Strunz. Quantum approach to coupling classical and quantum dynamics. Physical Review A, 61 (2): 022108, 2000. 10.1103/PhysRevA.61.022108. URL https://doi.org/10.1103/PhysRevA.61.022108.
https://doi.org/10.1103/PhysRevA.61.022108

[27] Fay Dowker and Yousef Ghazi-Tabatabai. Dynamical wavefunction collapse models in quantum measure theory. Journal of Physics A: Mathematical and Theoretical, 41 (20): 205306, 2008. 10.1088/1751-8113/41/20/205306. URL http://doi.org/10.1088/1751-8113/41/20/205306.
https://doi.org/10.1088/1751-8113/41/20/205306

[28] Fay Dowker and Adrian Kent. Properties of consistent histories. Physical Review Letters, 75 (17): 3038, 1995. 10.1103/PhysRevLett.75.3038. URL https://doi.org/10.1103/PhysRevLett.75.3038.
https://doi.org/10.1103/PhysRevLett.75.3038

[29] Kenneth Eppley and Eric Hannah. The necessity of quantizing the gravitational field. Foundations of Physics, 7 (1-2): 51–68, 1977. 10.1007/BF00715241. URL https://doi.org/10.1007/BF00715241.
https://doi.org/10.1007/BF00715241

[30] C. W. Gardiner, A. S. Parkins, and P. Zoller. Wave-function quantum stochastic differential equations and quantum-jump simulation methods. Phys. Rev. A, 46 (7), 1992. 10.1103/PhysRevA.46.4363. URL https://doi.org/10.1103/PhysRevA.46.4363.
https://doi.org/10.1103/PhysRevA.46.4363

[31] Murray Gell-Mann and James B Hartle. Classical equations for quantum systems. Physical Review D, 47 (8): 3345, 1993. 10.1103/PhysRevD.47.3345. URL https://doi.org/10.1103/PhysRevD.47.3345.
https://doi.org/10.1103/PhysRevD.47.3345

[32] G. C. Ghirardi, A. Rimini, and T. Weber. Unified dynamics for microscopic and macroscopic systems. Physical Review D, 34 (2): 470–491, July 1986. 10.1103/PhysRevD.34.470. URL https://doi.org/10.1103/PhysRevD.34.470. Publisher: American Physical Society.
https://doi.org/10.1103/PhysRevD.34.470

[33] Gian Carlo Ghirardi, Philip Pearle, and Alberto Rimini. Markov processes in Hilbert space and continuous spontaneous localization of systems of identical particles. Physical Review A, 42 (1): 78–89, July 1990. 10.1103/PhysRevA.42.78. URL https://doi.org/10.1103/PhysRevA.42.78. Publisher: American Physical Society.
https://doi.org/10.1103/PhysRevA.42.78

[34] N. Gisin. Quantum measurements and stochastic processes. Phys. Rev. Lett., 52: 1657–1660, May 1984. 10.1103/PhysRevLett.52.1657. URL https://doi.org/10.1103/PhysRevLett.52.1657.
https://doi.org/10.1103/PhysRevLett.52.1657

[35] Nicolas Gisin. Stochastic quantum dynamics and relativity. Helv. Phys. Acta, 62 (4): 363–371, 1989.

[36] V. Gorini, A. Kossakowski, and E.C. Sudarsahan. Completely positive semigroups of n-level systems. J. Math. Phys., (17): 821, 1976. 10.1063/1.522979. URL https://doi.org/10.1063/1.522979.
https://doi.org/10.1063/1.522979

[37] Robert B Griffiths. Consistent interpretation of quantum mechanics using quantum trajectories. Physical review letters, 70 (15): 2201, 1993. 10.1103/PhysRevLett.70.2201. URL https://doi.org/10.1103/PhysRevLett.70.2201.
https://doi.org/10.1103/PhysRevLett.70.2201

[38] Michael JW Hall and Marcel Reginatto. Interacting classical and quantum ensembles. Physical Review A, 72 (6): 062109, 2005. 10.1103/PhysRevA.72.062109. URL https://doi.org/10.1103/PhysRevA.72.062109.
https://doi.org/10.1103/PhysRevA.72.062109

[39] D Kafri, J M Taylor, and G J Milburn. 16 (6): 065020, jun 2014. 10.1088/1367-2630/16/6/065020. URL http://doi.org/10.1088/1367-2630/16/6/065020.
https://doi.org/10.1088/1367-2630/16/6/065020

[40] D Kafri, G J Milburn, and J M Taylor. Bounds on quantum communication via newtonian gravity. New Journal of Physics, 17 (1): 015006, jan 2015. 10.1088/1367-2630/17/1/015006. URL http://doi.org/10.1088/1367-2630/17/1/015006.
https://doi.org/10.1088/1367-2630/17/1/015006

[41] Raymond Kapral. Progress in the theory of mixed quantum-classical dynamics. Annu. Rev. Phys. Chem., 57: 129–157, 2006. 10.1146/annurev.physchem.57.032905.104702. URL https://doi.org/10.1146/annurev.physchem.57.032905.104702.
https://doi.org/10.1146/annurev.physchem.57.032905.104702

[42] Raymond Kapral and Giovanni Ciccotti. Mixed quantum-classical dynamics. The Journal of chemical physics, 110 (18): 8919–8929, 1999. 10.1063/1.478811. URL https://doi.org/10.1063/1.478811.
https://doi.org/10.1063/1.478811

[43] Adrian Kent. Quantum histories. Physica Scripta, T76 (1): 78, 1998. ISSN 0031-8949. 10.1238/physica.topical.076a00078. URL http://dx.doi.org/10.1238/Physica.Topical.076a00078.
https://doi.org/10.1238/physica.topical.076a00078

[44] G. Lindblad. On the generators of quantum dynamical semigroups. Commun. Math. Phys., (119): 48, 1976. 10.1007/BF01608499. URL https://doi.org/10.1007/BF01608499.
https://doi.org/10.1007/BF01608499

[45] Chiara Marletto and Vlatko Vedral. Why we need to quantise everything, including gravity. npj Quantum Information, 3 (1): 29, 2017. 10.1038/s41534-017-0028-0. URL https://doi.org/10.1038/s41534-017-0028-0.
https://doi.org/10.1038/s41534-017-0028-0

[46] Klaus Mølmer and Yvan Castin. Monte carlo wavefunctions in quantum optics. Quantum and Semiclassical Optics: Journal of the European Optical Society Part B, 8 (1), 1996. 10.1088/1355-5111/8/1/007. URL http://doi.org/10.1088/1355-5111/8/1/007.
https://doi.org/10.1088/1355-5111/8/1/007

[47] Klaus Mølmer, Yvan Castin, and Jean Dalibard. Monte carlo wave-function method in quantum optics. Journal of the Optical Society of America B, 10 (3): 524–538, 1993. 10.1364/JOSAB.10.000524. URL https://doi.org/10.1364/JOSAB.10.000524.
https://doi.org/10.1364/JOSAB.10.000524

[48] S. Noschese, L. Pasquini, and L. Reichel. Tridiagonal toeplitz matrices: properties and novel applications. Numer. Linear Algebra Appl., 20: 302–326, 2013. 10.1002/nla.1811. URL https://doi.org/10.1002/nla.1811.
https://doi.org/10.1002/nla.1811

[49] Roland Omnes. Consistent interpretations of quantum mechanics. Reviews of Modern Physics, 64 (2): 339, 1992. 10.1103/RevModPhys.64.339. URL https://doi.org/10.1103/RevModPhys.64.339.
https://doi.org/10.1103/RevModPhys.64.339

[50] J. Oppenheim and Z. Weller-Davies. A path integral for completely positive classical-quantum dynamics, 2022a. Manuscript in preparation.

[51] Jonathan Oppenheim. A post-quantum theory of classical gravity?, 2018. URL https://doi.org/10.48550/arXiv.1811.03116.
https://doi.org/10.48550/arXiv.1811.03116

[52] Jonathan Oppenheim and Zachary Weller-Davies. The constraints of post-quantum classical gravity. JHEP, 02: 080, 2022b. 10.1007/JHEP02(2022)080. URL http://doi.org/10.1007/JHEP02(2022)080.
https://doi.org/10.1007/JHEP02(2022)080

[53] Jonathan Oppenheim, Carlo Sparaciari, Barbara Šoda, and Zachary Weller-Davies. The two classes of hybrid classical-quantum dynamics. 3 2022a. 10.48550/arXiv.2203.01332. URL https://doi.org/10.48550/arXiv.2203.01332.
https://doi.org/10.48550/arXiv.2203.01332

[54] Jonathan Oppenheim, Carlo Sparaciari, Barbara Šoda, and Zachary Weller-Davies. Gravitationally induced decoherence vs space-time diffusion: testing the quantum nature of gravity. 3 2022b. 10.48550/arXiv.2203.01982. URL https://doi.org/10.48550/arXiv.2203.01982.
https://doi.org/10.48550/arXiv.2203.01982

[55] Don N Page and CD Geilker. Indirect evidence for quantum gravity. Physical Review Letters, 47 (14): 979, 1981. 10.1103/PhysRevLett.47.979. URL https://doi.org/10.1103/PhysRevLett.47.979.
https://doi.org/10.1103/PhysRevLett.47.979

[56] Juan Pablo Paz and Wojciech Hubert Zurek. Environment-induced decoherence, classicality, and consistency of quantum histories. Physical Review D, 48 (6): 2728, 1993. 10.1103/PhysRevD.48.2728. URL https://doi.org/10.1103/PhysRevD.48.2728.
https://doi.org/10.1103/PhysRevD.48.2728

[57] Philip Pearle. Combining stochastic dynamical state-vector reduction with spontaneous localization. Physical Review A, 39 (5): 2277–2289, March 1989. 10.1103/PhysRevA.39.2277. URL https://doi.org/10.1103/PhysRevA.39.2277. Publisher: American Physical Society.
https://doi.org/10.1103/PhysRevA.39.2277

[58] Roger Penrose. On gravity’s role in quantum state reduction. General relativity and gravitation, 28 (5): 581–600, 1996.

[59] David Poulin, 2017. private communication (result announced in poulinKITP).

[60] David Poulin and John Preskill. Information loss in quantum field theories. Frontiers of Quantum Information Physics, KITP, 2017. URL http://online.kitp.ucsb.edu/online/qinfo-c17/poulin/.
http://online.kitp.ucsb.edu/online/qinfo-c17/poulin/

[61] Oleg V Prezhdo and Vladimir V Kisil. Mixing quantum and classical mechanics. Physical Review A, 56 (1): 162, 1997. 10.1103/PhysRevA.56.162. URL https://doi.org/10.1103/PhysRevA.56.162.
https://doi.org/10.1103/PhysRevA.56.162

[62] Debendranath Sahoo. Mixing quantum and classical mechanics and uniqueness of planck’s constant. Journal of Physics A: Mathematical and General, 37 (3): 997, 2004. 10.1088/0305-4470/37/3/031. URL https://doi.org/10.1088/0305-4470/37/3/031.
https://doi.org/10.1088/0305-4470/37/3/031

[63] LL Salcedo. Absence of classical and quantum mixing. Physical Review A, 54 (4): 3657, 1996. 10.1103/PhysRevA.54.3657. URL https://doi.org/10.1103/PhysRevA.54.3657.
https://doi.org/10.1103/PhysRevA.54.3657

[64] LL Salcedo. Statistical consistency of quantum-classical hybrids. Physical Review A, 85 (2): 022127, 2012. 10.1103/PhysRevA.85.022127. URL https://doi.org/10.1103/PhysRevA.85.022127.
https://doi.org/10.1103/PhysRevA.85.022127

[65] T. N. Sherry and E. C. G. Sudarshan. Interaction between classical and quantum systems: A new approach to quantum measurement.i. Phys. Rev. D, 18: 4580–4589, Dec 1978. 10.1103/PhysRevD.18.4580. URL https://doi.org/10.1103/PhysRevD.18.4580.
https://doi.org/10.1103/PhysRevD.18.4580

[66] T. N. Sherry and E. C. G. Sudarshan. Interaction between classical and quantum systems: A new approach to quantum measurement. ii. theoretical considerations. Phys. Rev. D, 20: 857–868, Aug 1979. 10.1103/PhysRevD.20.857. URL https://doi.org/10.1103/PhysRevD.20.857.
https://doi.org/10.1103/PhysRevD.20.857

[67] Daniel R Terno. Inconsistency of quantum—classical dynamics, and what it implies. Foundations of Physics, 36 (1): 102–111, 2006. 10.1007/s10701-005-9007-y. URL https://doi.org/10.1007/s10701-005-9007-y.
https://doi.org/10.1007/s10701-005-9007-y

[68] Antoine Tilloy and Lajos Diósi. Sourcing semiclassical gravity from spontaneously localized quantum matter. Physical Review D, 93: 024026, 2016. 10.1103/PhysRevD.93.024026. URL http://doi.org/10.1103/PhysRevD.93.024026.
https://doi.org/10.1103/PhysRevD.93.024026

[69] Antoine Tilloy and Lajos Diósi. On gkls dynamics for local operations and classical communication. Open Systems & Information Dynamics, 24: 1740020, 2017. 10.1142/S1230161217400200. URL http://doi.org/10.1142/S1230161217400200.
https://doi.org/10.1142/S1230161217400200

[70] HM Wiseman. Quantum trajectories and quantum measurement theory. Quantum and Semiclassical Optics: Journal of the European Optical Society Part B, 8 (1): 205, 1996. 10.1088/1355-5111/8/1/015. URL http://doi.org/10.1088/1355-5111/8/1/015.
https://doi.org/10.1088/1355-5111/8/1/015

[71] Howard M Wiseman and Gerard J Milburn. Quantum measurement and control. Cambridge university press, 2009. 10.1017/CBO9780511813948. URL https://doi.org/10.1017/CBO9780511813948.
https://doi.org/10.1017/CBO9780511813948

[72] Wojciech H Zurek. Environment-induced superselection rules. Physical review D, 26 (8): 1862, 1982. 10.1103/PhysRevD.26.1862. URL https://doi.org/10.1103/PhysRevD.26.1862.
https://doi.org/10.1103/PhysRevD.26.1862

[73] Wojciech Hubert Zurek. Decoherence and the transition from quantum to classical—revisited. In Quantum Decoherence, pages 1–31. Springer, 2006. 10.1007/978-3-7643-7808-0_1. URL https://doi.org/10.1007/978-3-7643-7808-0_1.
https://doi.org/10.1007/978-3-7643-7808-0_1

Cited by

[1] Jonathan Oppenheim, “A post-quantum theory of classical gravity?”, arXiv:1811.03116.

[2] Jonathan Oppenheim, Carlo Sparaciari, Barbara Šoda, and Zachary Weller-Davies, “Gravitationally induced decoherence vs space-time diffusion: testing the quantum nature of gravity”, arXiv:2203.01982.

[3] Jonathan Oppenheim and Zachary Weller-Davies, “The constraints of post-quantum classical gravity”, arXiv:2011.15112.

[4] Isaac Layton, Jonathan Oppenheim, and Zachary Weller-Davies, “A healthier semi-classical dynamics”, arXiv:2208.11722.

[5] Jonathan Oppenheim, Carlo Sparaciari, Barbara Šoda, and Zachary Weller-Davies, “The two classes of hybrid classical-quantum dynamics”, arXiv:2203.01332.

[6] Viqar Husain and Suprit Singh, “Quantum backreaction on a classical universe”, Physical Review D 104 12, 124048 (2021).

The above citations are from SAO/NASA ADS (last updated successfully 2023-01-03 17:07:27). The list may be incomplete as not all publishers provide suitable and complete citation data.

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