Dynamical quantization of the Kepler manifold

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The dynamical quantization of the ’’Kepler manifold’’ in any number of degrees of freedom is constructed. The Kepler manifold is the phase space of the regularized Kepler motion and is shown to be an SO(n,2) -homogeneous symplectic manifold, corresponding to an extremely singular orbit in the co-adjoint representation; the quantization is obtained by ’’approximating’’ this orbit by more regular ones, which are equivalent to homogeneous bounded domains. The most relevant result is that the usual quantum-mechanical ’’hydrogen atom’’ model is recovered in the particular representation introduced by Fock in 1935 [SO(n) -homogeneous integral equation in momentum space].