The Schrödinger-Newton equations beyond Newton

Abstract

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 ($G$, $\hslash$, 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.

To appear in General Relativity and Gravitation

References in corpus (5)

• The Schrödinger-Newton equation as non-relativistic limit of self-gravitating Klein-Gordon and Dirac fields
• Problems with the Newton-Schrödinger Equations
• Centre-of-mass motion in multi-particle Schroedinger-Newton dynamics
• Lagrangian approach to the semi-relativistic electron dynamics in the mean-field approximation
• Variational approach to the time-dependent Schrödinger-Newton equations

Cited by in corpus (6)

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• Gravitoelectromagnetic knot fields
• Hybrid Classical-Quantum Newtonian Gravity with stable vacuum
• Phase space analysis of Rényi Holographic dark energy model