A New Type of Conformal Dynamics

TL;DR

This paper explores a conformal particle model with added magnetic or Coulomb interactions, revealing hidden symmetries and analyzing bound states through transformations linking it to known solvable potentials.

Contribution

It introduces a conformal particle model with extended interactions, uncovers hidden symmetries, and connects quantum bound states to the Morse potential.

Findings

Hidden SO(2,1) symmetry in the vortex case

Reduction of phase space to four dimensions

Schrödinger equation maps to Morse potential

Abstract

We consider the Lagrangian particle model introduced in [hep-th/9612017] for zero mass but nonvanishing second central charge of the planar Galilei group. Extended by a magnetic vortex or a Coulomb potential the model exibits conformal symmetry. In the former case we observe an additional SO(2,1) hidden symmetry. By either a canonical transformation with constraints or by freezing scale and special conformal transformations at $t=0$ we reduce the six-dimensional phase-space to the physically required four dimensions. Then we discuss bound states (bounded solutions) in quantum dynamics (classical mechanics). We show that the Schr\"odinger equation for the pure vortex case may be transformed into the Morse potential problem thus providing us with an explanation of the hidden SO(2,1) symmetry.