On the L\'evy-Leblond-Newton equation and its symmetries: a geometric view

The L'evy-Leblond-Newton (LLN) equation for non-relativistic fermions with a gravitational self-interaction is reformulated within the framework of a Bargmann structure over a $(n+1)$-dimensional Newton-Cartan (NC) spacetime. The Schr"odinger-Newton (SN) group, introduced in [21] as the maximal group of invariance of the SN equation, turns out to be also the group of conformal Bargmann automorphisms preserving the coupled L'evy-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 4-component spinors is also presented. In particular, when restricted to dilations, the value of the dynamical exponent $z=(n+2)\mathrm{/}3$ is recovered as previously derived in [21] for the SN equation. Subsequently, conserved quantities associated to the (generalized) LLN equation are also exhibited.