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Duality Between Hydrogen Atom and Oscillator Systems via Hidden SO(d,2) Symmetry and 2T-physics

The relation between motion in $-1/r$ and $r^{2}$ potentials, known since Newton, can be demonstrated by the substitution $r\rightarrow r^{2}$ in the classical/quantum radial equations of the Kepler/Hydrogen problems versus the harmonic oscillator. This suggests a duality-type relationship between these systems. However, when both radial and angular components of these systems are included the possibility of a true duality seems to be remote. Indeed, investigations that explored and generalized Newton's radial relation, including algebraic approaches based on noncompact groups such as SO(4,2), have never exhibited a full duality consistent with Newton's. On the other hand, 2T-physics predicts a host of dualities between pairs of a huge set of systems that includes Newton's two systems. These dualities take the form of rather complicated canonical transformations that relate the full phase spaces of these respective systems in all directions. In this paper we focus on Newton's case by imposing his radial relation to find an appropriate basis for 2T-physics dualities, and then construct the full duality. Using the techniques of 2T-physics, we discuss the hidden symmetry of the actions (beyond the symmetry of Hamiltonians) for the Hydrogen atom in $D$-dimensions and the harmonic oscillator in $\bar{D}$ dimensions. The symmetries lead us to find the one-to-one relation between the quantum states, including angular degrees of freedom, for specific values of $\left( D,\bar{D}\right) $, and construct the explicit quantum canonical transformation in those special cases. We find that the canonical transformation has itself a hidden gauge symmetry that is crucial for the respective phase spaces to be dual even when $D\neq\bar{D}$. In this way we display the surprising beautiful symmetry of the full duality that generalizes Newton's radial duality.