Conformal bridge transformation and PT symmetry

The conformal bridge transformation (CBT) is reviewed in the light of the $\mathcal{PT}$ symmetry. Originally, the CBT was presented as a non-unitary transformation (a complex canonical transformation in the classical case) that relates two different forms of dynamics in the sense of Dirac. Namely, it maps the asymptotically free form into the harmonically confined form of dynamics associated with the $\mathfrak{so}(2,1)\cong \mathfrak{sl}(2,{\mathbb R})$ conformal symmetry. However, as the transformation relates the non-Hermitian operator $i\hat{D}$, where $\hat{D}$ is the generator of dilations, with the compact Hermitian generator $\hat{\mathcal{J}}_0$ of the $\mathfrak{sl}(2,{\mathbb R})$ algebra, the CBT generator can be associated with a $\mathcal{PT}$-symmetric metric. In this work we review the applications of this transformation for one- and two-dimensional systems, as well as for systems on a cosmic string background, and for a conformally extended charged particle in the field of Dirac monopole. We also compare and unify the CBT with the Darboux transformation. The latter is used to construct $\mathcal{PT}$-symmetric solutions of the equations of the KdV hierarchy with the properties of extreme waves. As a new result, by using a modified CBT we relate the one-dimensional $\mathcal{PT}$-regularized asymptotically free conformal mechanics model with the $\mathcal{PT}$-regularized version of the de Alfaro, Fubini and Furlan system.