Newton-Schr\"odinger Equations are not derivable from General Relativity + Quantum Field Theory

Abstract

In this note we show that Newton-Schr\"odinger Equations (NSEs) [arXiv:1210.0457 and references therein] do not follow from general relativity (GR) and quantum field theory (QFT) by way of two considerations: 1) Taking the nonrelativistic limit of the semiclassical Einstein equation, the central equation of relativistic semiclassical gravity, a fully covariant theory based on GR+QFT with self-consistent backreaction of quantum matter on the spacetime dynamics; 2) Working out a model [see C. Anastopoulos and B. L. Hu, Class. Quant. Grav. 30, 165007 (2013), arXiv:1305.5231] with a matter scalar field interacting with weak gravity, in procedures analogous to the derivation of the nonrelativistic limit of quantum electrodynamics. We conclude that the coupling of classical gravity with quantum matter can only be via mean fields, there are no $N$-particle NSEs and theories based on Newton-Schr\"odinger equations assume unknown physics.