The Schr\"odinger-Newton equation and its foundations

TL;DR

This paper compares quantum gravity and semi-classical gravity in the non-relativistic limit, focusing on the Schr"odinger-Newton equation's nonlinear gravitational effects and its implications for wave-function collapse and superluminal signaling.

Contribution

It clarifies the differences between quantum and semi-classical gravity approaches and analyzes the Schr"odinger-Newton equation's role and limitations in wave-function collapse theories.

Findings

Schr"odinger-Newton equation involves nonlinear, non-local gravity effects.

Semi-classical gravity with collapse models avoids superluminal signaling.

The Schr"odinger-Newton equation does not describe wave-function collapse.

Abstract

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 Schr\"odinger equation stays linear, in the latter case one ends up with the so-called Schr\"odinger-Newton equation, which involves a nonlinear, non-local gravitational contribution. We further discuss that the Schr\"odinger-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 Schr\"odinger-Newton equation.