CONFORMAL SYMMETRY AND DUALITY BETWEEN FREE PARTICLE, H ATOM, AND HARMONIC-OSCILLATOR - ART. NO. 066006

We establish a duality between the free massless relativistic particle in d dimensions, the non-relativistic hydrogen atom (1/r potential) i n (d-1) space dimensions, and the harmonic oscillator in (d-2) space d imensions with its mass given as the light cone momentum of an additio nal dimension. The duality is in the sense that the classical action o f these systems are gauge fixed forms of the same worldline gauge theo ry action at the classical level, and they are all described by the sa me unitary representation of the conformal group SO (d,2) at the quant um level. The world line action has a gauge symmetry Sp(2) which treat s canonical variables (x,p) 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 vari ous dual solutions correspond to solutions of the constraints with dif ferent topologies. For example, for the H atom the two timelike dimens ions X-0',X-0 live on a circle. The model provides an example of how r ealistic 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. [S0556-2821(98)00416-0].