Conformal bridge between asymptotic freedom and confinement
Luis Inzunza, Mikhail S. Plyushchay, Andreas Wipf
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 $\mathfrak{s}\mathfrak{l}(2,\mathbb{R})$ 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.