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

C Duval, Serge Lazzarini

Published 2015-04-20Version 1

The Schr{\"o}dinger-Newton (SN) equation is recast on purely geometrical grounds, namely in terms of Bargmann structures over (n + 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{\"o}dinger 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 = (n + 2)/3.