Abstract
Consistent coupling of quantum and classical degrees of freedom exists so long as there is both diffusion of the classical degrees of freedom and decoherence of the quantum system. In this paper, we derive the Newtonian limit of such classical-quantum (CQ) theories of gravity. Our results are obtained both via the gauge fixing of the recently proposed path integral theory of CQ general relativity and via the CQ master equation approach. In each case, we find the same weak field dynamics. We find that the New-tonian potential diffuses by an amount lower bounded by the decoherence rate into mass eigenstates. We also present our results as an unraveled system of stochastic differential equations for the trajectory of the hybrid classical-quantum state and provide a series of kernels for constructing figures of merit, which can be used to rule out part of the parameter space of classical-quantum theories of gravity by experimentally testing it via the decoherence-diffusion trade-off. We compare and contrast the weak field limit to previous models of classical Newtonian gravity coupled to quantum systems. Here, we find that the Newtonian potential and quantum state change in lock-step, with the flow of time being stochastic.
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Acknowledgments
We would like to thank Antoine Tilloy for his insightful comments on an earlier draft of this work. We would like to thank Maite Arcos, Lajos Diósi, Gary Horowitz, Jorma Louko, Don Marolf, Emanuele Panella, Andy Svesko, and Bill Unruh for valuable discussions. J.O. is supported by an EPSRC Established Career Fellowship, and a Royal SocietyWolfson Merit Award, A.R. acknowledges financial support from EPSRC and UCL Physics Department, I.L. acknowledges financial support from EPSRC. This research was supported by the National Science Foundation under Grant No. NSF PHY11-25915 and by the Simons Foundation It from Qubit Network. Research at Perimeter Institute is supported in part by the Government of Canada through the Department of Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Economic Development, Job Creation and Trade.
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Layton, I., Oppenheim, J., Russo, A. et al. The weak field limit of quantum matter back-reacting on classical spacetime. J. High Energ. Phys. 2023, 163 (2023). https://doi.org/10.1007/JHEP08(2023)163
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DOI: https://doi.org/10.1007/JHEP08(2023)163