Kohn's Theorem, Larmor's Equivalence Principle and the Newton-Hooke Group

TL;DR

This paper explores the symmetry groups of non-relativistic charged particles in magnetic fields, linking Kohn's theorem and Larmor's theorem to the Newton-Hooke group, and studies their geometric and group-theoretic properties.

Contribution

It introduces a group-theoretic framework connecting Kohn's theorem, Larmor's theorem, and the Newton-Hooke group through deformations of the Galilei group, and analyzes the system's geometric lift as a pp-wave.

Findings

The system admits a Newton-Hooke symmetry group as a deformation of the Galilei group.

Larmor's theorem shows the deformations are all isomorphic.

The Eisenhart lift of the system corresponds to a known pp-wave solution.

Abstract

We consider non-relativistic electrons, each of the same charge to mass ratio, moving in an external magnetic field with an interaction potential depending only on the mutual separations, possibly confined by a harmonic trapping potential. We show that the system admits a "relativity group" which is a one-parameter family of deformations of the standard Galilei group to the Newton-Hooke group which is a Wigner-Inonu contraction of the de Sitter group. This allows a group-theoretic interpretation of Kohn's theorem and related results. Larmor's Theorem is used to show that the one-parameter family of deformations are all isomorphic. We study the "Eisenhart" or "lightlike" lift of the system, exhibiting it as a pp-wave. In the planar case, the Eisenhart lift is the Brdicka-Eardley-Nappi-Witten pp-wave solution of Einstein-Maxwell theory, which may also be regarded as a bi-invariant metric on the Cangemi-Jackiw group.