On the Lévy-Leblond–Newton equation and its symmetries: a geometric view

Abstract

The Lévy-Leblond–Newton (LLN) equation for non-relativistic fermions with a gravitational self-interaction is reformulated within the framework of a Bargmann structure over a -dimensional Newton–Cartan (NC) spacetime. The Schrödinger–Newton (SN) group introduced in Duval and Lazzarini (2015 Class. Quantum Grav. 32 175006) as the maximal group of invariance of the SN equation, turns out to be also the group of conformal Bargmann automorphisms preserving the coupled Lévy-Leblond and NC gravitational field equations. Within the Bargmann geometry a generalization of the LLN equation is provided as well. The canonical projective unitary representation of the SN group on four-component spinors is also presented. In particular, when restricted to dilations, the value of the dynamical exponent is recovered as previously derived in Duval and Lazzarini (2015 Class. Quantum Grav. 32 175006) for the SN equation. Subsequently, conserved quantities associated to the (generalized) LLN equation are also exhibited.