Lie point symmetries and the geodesic approximation for the Schr\"odinger-Newton equations

TL;DR

This paper identifies Lie point symmetries of the Schrödinger-Newton equations and uses these symmetries to analyze approximate solutions where probability lumps behave like particles attracted by an inverse-square law.

Contribution

It provides the first symmetry analysis of the Schrödinger-Newton equations and applies it to derive particle-like dynamics of probability lumps.

Findings

Lumps of probability move as point particles under inverse-square attraction.

Lie point symmetries of the equations are explicitly characterized.

Approximate solutions reveal particle-like behavior in the system.

Abstract

We consider two problems arising in the study of the Schr\"odinger-Newton equations. The first is to find their Lie point symmetries. The second, as an application of the first, is to investigate an approximate solution corresponding to widely separated lumps of probability. The lumps are found to move like point particles under a mutual inverse-square law of attraction.