Jordan algebra conformal toolbox, ICMS Seminar Talk by Todor Popov

We employ the Jordan algebras for a succinct description of the dynamical conformal symmetries of integrable models. Given an Euclidean Jordan algebra J via Tits-Kantor-Kocher construction we obtain a representation of the conformal (Mobius) group Co(J). Since the seminal work of Ger-
hard Mack and Ivan Todorov [see attached files] on irreducible minimal conformal group U(2, 2)-representations it is known that the orbital wavefunctions of the
hydrogen atom transform in a minimal U(2, 2)-representation.

Given the Jordan algebra of hermitian 2×2 matrices (Pauli matrices)[see attached files] we recover the hydrogen spectrum U(2, 2)-representation from the TKK construction. A reality condition imposed on the Jordan algebra of Pauli matrices yields the Jordan algebra of real symmetric matrices
and reduces the 3D H-atom to a 2D system. The Majorana reduction of the 4D Dirac spinor transforming under SU(2, 2) yields the dynamical con formal symmetry of the quantum motion of an electron in magnetic field (Landau problem). Different Landau levels turn out to be packed into a single conformal spinorial representation of SO(3, 2) which is identified with the Dirac’s ”Remarkable representation of the 3+2 de Sitter group”[3]. We finally speculate on higher Jordan algebras and their relevance to the mass spectrum of elementary particles [4].