On the Schr{\"o}dinger-Newton equation and its symmetries: a geometric view

TL;DR

This paper geometrically reformulates the Schrödinger-Newton equation using Bargmann structures on Newton-Cartan spacetimes, identifying its symmetry group and exploring its invariance properties and representations.

Contribution

It provides a geometric framework for the Schrödinger-Newton equation and characterizes its symmetry group as conformal Bargmann automorphisms, revealing new invariance features.

Findings

Identifies the SN group as conformal Bargmann automorphisms.

Determines the maximal invariance group of the SN equation.

Derives unitary representations showing dilations with specific dynamical exponent.

Abstract

The \SN (SN) equation is recast on purely geometrical grounds, namely in terms of Bargmann structures over $(\d+1)$-dimensional Newton-Cartan (NC) spacetimes. Its maximal group of invariance, which we call the SN group, is determined as the group of conformal Bargmann automorphisms that preserve the coupled Schr\"odinger and NC gravitational field equations. Canonical unitary representations of the SN group are worked out, helping us recover, in particular, a very specific occurrence of dilations with dynamical exponent $z=(\d+2)/3$.