The Schrödinger-Newton equation and its foundations

• Published in: New journal of physics Vol. 16; no. 11; pp. 115007 - 17
• Main Authors: Bahrami, Mohammad, Großardt, André, Donadi, Sandro, Bassi, Angelo
• Format: Journal Article
• Language: English
• Published: Bristol IOP Publishing 07.11.2014
• Subjects: 03.65.-w; 03.65.Ta; 04.60.-m; Collapse; collapse of the wave-function; Gravitation; Gravitational collapse; Gravitational fields; Mathematical analysis; Mathematical models; Nonlinearity; Physics; Quantum gravity; Schrodinger equation; Schroedinger equation; Schrödinger-Newton equation; semi-classical gravity; Signaling; Signalling
• ISSN: 1367-2630; 1367-2630
• DOI: 10.1088/1367-2630/16/11/115007

The necessity of quantising the gravitational field is still subject to an open debate. In this paper we compare the approach of quantum gravity, with that of a fundamentally semi-classical theory of gravity, in the weak-field non-relativistic limit. We show that, while in the former case the Schrödinger equation stays linear, in the latter case one ends up with the so-called Schrödinger-Newton equation, which involves a nonlinear, non-local gravitational contribution. We further discuss that the Schrödinger-Newton equation does not describe the collapse of the wave-function, although it was initially proposed for exactly this purpose. Together with the standard collapse postulate, fundamentally semi-classical gravity gives rise to superluminal signalling. A consistent fundamentally semi-classical theory of gravity can therefore only be achieved together with a suitable prescription of the wave-function collapse. We further discuss, how collapse models avoid such superluminal signalling and compare the nonlinearities appearing in these models with those in the Schrödinger-Newton equation.