The weak field limit of quantum matter back-reacting on classical spacetime

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.

Gravitationally induced decoherence vs space-time diffusion: testing the quantum nature of gravity

“…11 , 39 can be found in ref. 81 . It is in taking this limit where some care should be taken, since one is gauge fixing the full general relativistic theory.…”

“…A higher precision calculation would involve a full simulation of CQ dynamics, for example using unravelling methods 26 , 78 or the path integral as in ref. 81 . However, care should be taken, as we have found that relativistic corrections put constraints on the degree of diffusion even at low energy (J. Oppenheim and A. Russo, manuscript in preparation), and one should bear this in mind when drawing conclusions on the models presented here.…”

“…In this section we motivate the Newtonian limit of gravity used in Section ’Physical constraints on the classicality of gravity’ 121 . A fuller treatment can be found in 81 . We begin with classical General Relativity in the ADM formulation 36 .…”

“…The choice of , is to ensure that π Φ is canonically conjugate to Φ. A more detailed derivation starts from the full weak field metric, and gauge fixing of the shift can be found in 81 . Here, we find the effective action can be written where the Newtonian Hamiltonian is given by with the pure gravity Hamiltonian, and is the interaction Hamiltonian, from which we see that non-relativistic matter couples to the Newtonian potential through its mass density m ( x ).…”

“…This is because 24 allows for non-zero conjugate momentum π Φ with the kinetic energy of a different sign, while in ref. 81 , the momentum is set to zero via the constraint equations and gauge fixing of the lapse. The discontinuity then arises because we are operating in the c → ∞ limit, while in ref.…”