Conformal bridge between asymptotic freedom and confinement

TL;DR

This paper introduces a nonunitary transformation linking asymptotically free conformal quantum systems with their confined, harmonic oscillator counterparts, revealing new algebraic and physical insights.

Contribution

It constructs a novel nonunitary automorphism of the conformal algebra connecting free and confined quantum systems, including explicit mappings of eigenstates and states.

Findings

Mapped zero eigenstates to coherent states

Revealed relation between free particle and Landau problem

Explored algebraic structure of the transformation

Abstract

We construct a nonunitary transformation that relates a given "asymptotically free" conformal quantum mechanical system $H_f$ with its confined, harmonically trapped version $H_c$. In our construction, Jordan states corresponding to the zero eigenvalue of $H_f$, as well as its eigenstates and Gaussian packets are mapped into the eigenstates, coherent states and squeezed states of $H_c$, respectively. The transformation is an automorphism of the conformal $\mathfrak{sl}(2,{\mathbb R})$ algebra of the nature of the fourth-order root of the identity transformation, to which a complex canonical transformation corresponds on the classical level being the fourth-order root of the spatial reflection. We investigate the one- and two-dimensional examples that reveal, in particular, a curious relation between the two-dimensional free particle and the Landau problem.