Maaneli Derakhshani
The Newtonian Limit Of Stochastic Gravity: Does It Entail The Many-Particle Stochastic Schrödinger-Newton Equations, And Gravitationally-Induced Wavefunction Collapse?
We give a different answer from Bera et al. [1] on how to take the Newtonian limit of the Einstein-Langevin equation in stochastic gravity [2]. We first review how the semiclassical Einstein equation (SEE) can be derived, in a formal sense, as a mean-field approximation to the quantum gravitational path-integral in the closed-time-path formulation. Then we take the Newtonian limit of the SEE. We confirm that the resulting ‘single-body’ Schrödinger-Newton (SN) equation corresponds to the large N limit of the theory of perturbatively quantized gravity coupled to a quantum matter field in the Newtonian regime. We then review how the Einstein-Langevin equation (ELE) corresponds to the next-to-leading order correction to the mean-field SEE, after which we take the Newtonian limit of the ELE. We find that the resulting equation corresponds to the single-body stochastic SN equation obtained in [1], but with a crucially different physical meaning: the resulting ‘single-body’ equation corresponds to the next-to-leading-order contribution to the large N limit of perturbatively quantized gravity coupled to a quantum matter field in the Newtonian regime, rather than a standalone Newtonian description of how quantized matter couples to gravity for single particle states. Finally, we compare the mean-field stochastic SN equation to the stochastic Schrödinger equation of dynamical collapse theories such as Diósi-Penrose (DP). We point out that the mean-field stochastic SN equation does not predict objective state-vector collapse in the position basis (in contrast to DP), only (gravitationally-induced) decoherence in the energy basis.
References:
[1] S. Bera, R. Mohan, and T. P. Singh. Stochastic modification of the schrodinger-newton equation. Phys. Rev. D 92, 025054, 2015, https://arxiv.org/abs/1504.05892.
[2] B. L. Hu and E. Verdaguer. Stochastic gravity: Theory and applications. Living Rev. Relativity, 11:3, 2008, http://arxiv.org/abs/0802.0658.
