Kohn’s theorem, Larmor’s equivalence principle and the Newton–Hooke group

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–İnönü 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 Brdička–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.

► We show that non-relativistic electrons moving in a magnetic field with trapping potential admits as relativity group the Newton–Hooke group.
► We use this fact to give a group theoretic interpretation of Kohn’s theorem and to obtain the spectrum.
► We obtain the lightlike lift of the system exhibiting showing it coincides with the Nappi–Witten spacetime.