Conformal Bridge Between Asymptotic Freedom and Confinement

We construct a nonunitary transformation that relates a given "asymptotically free" conformal quantum mechanical system Hf with its confined, harmonically trapped version Hc. In our construction, Jordan states corresponding to the zero eigenvalue of Hf, as well as its eigenstates and Gaussian packets, are mapped into the eigenstates, coherent states, and squeezed states of Hc, respectively. The transformation is an automorphism of the conformal sl(2,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.