A remarkable dynamical symmetry of the Landau problem

TL;DR

This paper reveals that the dynamical symmetry group of an electron in a magnetic field is the symplectomorphism group $Sp(4,\mathbb{R})$, connecting the Landau problem to conformal and hydrogen atom symmetries through advanced algebraic constructions.

Contribution

It establishes a novel connection between the Landau problem, conformal symmetries, and the hydrogen atom using symplectic and Jordan algebra frameworks.

Findings

The dynamical group of the Landau problem is $Sp(4,\mathbb{R})$.

The Landau problem's symmetry relates to the conformal group $SO(2,3)$.

A duality between the 2D hydrogen atom and the Landau problem is explained.

Abstract

We show that the dynamical group of an electron in a constant magnetic field is the group of symplectomorphisms $Sp(4,\mathbb{R})$. It is generated by the spinorial realization of the conformal algebra $\mathfrak{so}(2,3)$ considered in Dirac's seminal paper "A Remarkable Representation of the 3 + 2 de Sitter Group". The symplectic group $Sp(4,\mathbb{R})$ is the double covering of the conformal group $SO(2,3)$ of 2+1 dimensional Minkowski spacetime which is in turn the dynamical group of a hydrogen atom in 2 space dimensions. The Newton-Hooke duality between the 2D hydrogen atom and the Landau problem is explained via the Tits-Kantor-Koecher construction of the conformal symmetries of the Jordan algebra of real symmetric $2 \times 2$ matrices. The connection between the Landau problem and the 3D hydrogen atom is elucidated by the reduction of a Dirac spinor to a Majorana one in the Kustaanheimo-Stiefel spinorial regularization.