Conformal Symmetry and Duality between Free Particle, H-atom and Harmonic Oscillator

TL;DR

This paper reveals a duality linking free particles, hydrogen atoms, and harmonic oscillators through a shared conformal symmetry, suggesting a unified framework in higher-dimensional spacetime with two timelike dimensions.

Contribution

It demonstrates a classical and quantum duality between these systems via a common gauge theory and conformal group representation, introducing a novel perspective on spacetime structure.

Findings

Unified description of different physical systems via conformal symmetry

Classical actions are gauge-fixed forms of a single worldline gauge theory

Quantum states form a common unitary representation of SO(d,2)

Abstract

We establish a duality between the free massless relativistic particle in d dimensions, the non-relativistic hydrogen atom (1/r potential) in (d-1) space dimensions, and the harmonic oscillator in (d-2) space dimensions with its mass given as the lightcone 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(d,2) at the quantum level. The worldline action has a gauge symmetry Sp(2) which treats 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 various dual solutions correspond to solutions of the constraints with different topologies. For example, for the H-atom the two timelike dimensions X^{0'},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.