The Schrödinger-Newton equations beyond Newton

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The scope of this paper is twofold. First, we derive rigorously a low-velocity and Galilei-covariant limit of the gravitoelectromagnetic (GEM) equations. Subsequently, these reduced GEM equations are coupled to the Schrödinger equation with gravitoelectric and gravitomagnetic potentials. The resulting extended Schrödinger-Newton equations constitute a minimal model where the three fundamental constants of nature (G, ℏ, and c) appear naturally. We show that the relativistic correction coming from the gravitomagnetic potential scales as the ratio of the mass of the system to the Planck mass, and that it reinforces the standard Newtonian (gravitoelectric) attraction. The theory is further generalized to many particles through a Wigner function approach.