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

TL;DR

This paper reformulates the Levy-Leblond-Newton equation within a geometric framework, revealing its symmetries, conserved quantities, and connections to the Schrödinger-Newton group in Newton-Cartan spacetime.

Contribution

It provides a geometric reformulation of the LLN equation using Bargmann structures, identifies its symmetry group, and generalizes the equation within this geometric context.

Findings

The SN group is the maximal invariance group of the LLN equation.

The dynamical exponent z = (n+2)/3 is recovered from symmetry considerations.

Conserved quantities related to the generalized LLN equation are identified.

Abstract

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 $z=(n+2)/3$ is recovered as previously derived in [21] for the SN equation. Subsequently, conserved quantities associated to the (generalized) LLN equation are also exhibited.