Conformal symmetry and duality between free particle, H atom, and harmonic oscillator

We establish a duality between the free massless relativistic particle in $d$ dimensions, the non-relativistic hydrogen atom $(1/r$ potential) in $\left(d-1\right)$ space dimensions, and the harmonic oscillator in $\mathrm{}(d-2\mathrm{})\mathrm{}$ space dimensions with its mass given as the light cone momentum of an additional dimension. The duality is in the sense that the classical action of these systems are gauge fixed forms of the same worldline gauge theory action at the classical level, and they are all described by the same unitary representation of the conformal group SO $\mathrm{}(d,2\mathrm{})\mathrm{}$ at the quantum level. The world line action has a gauge symmetry Sp(2) which treats canonical variables $\mathrm{}(x,p)\mathrm{}$ as doublets and exists only with a target spacetime that has $d$ spacelike dimensions and two timelike dimensions. This spacetime is constrained due to the gauge symmetry, and the various dual solutions correspond to solutions of the constraints with different topologies. For example, for the H atom the two timelike dimensions $X^{0^{\prime }}{,X}^0$ live on a circle. The model provides an example of how realistic physics can be viewed as existing in a larger covariant space that includes two timelike coordinates, and how the covariance in the larger space unifies different looking physics into a single system.