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Description
Landau problem and harmonic oscillator in thé plane sbare à Hilbert space which carries a structure of Dirac's remarkable so(2,3) representation.
We show that the orthosymplectic algebra osp(1|4) is the spectrum generating algebra for Landau problem and hence for 2D isotropic harmonic oscillator. The 2D harmonic oscillator is in duality with 2D quantum Coulomb-Kepler system with osp(1|4)-symmetry broken down to the conformal symmetry so(2,3).The even so(2,3)-submodule generated from the ground state of zero angular momentum is identified with Hilbert space of 2D Hydrogen atom. An odd element of the super algebra osp(1|4) creates from the vacuum a pseudo-vacuum with intrinsic angular momentum 1/2. Thé so(2,3) submodule build upon the pseudo vacuum is the Hibert space of a magnetized 2D hydrogen atom: a quantum system of a dyon and an electron.Hence the Hilbert space of Landau problem is a direct sum of a two massless unitary so(2,3) représentation.
