Quantum conformal mechanics emerging from unitary representations of SL(2,R)

The quantum mechanics of one degree of freedom exhibiting the exact conformal SL(2,R) symmetry is presented. The starting point is the classification of the unitary irreducible representations of the SL(2,R) group (or, to some extent, its universal covering). The coordinate representation is defined as the basis diagonalizing the special conformal generator Kˆ. As a particular case one obtains the AFF (de Alfaro et al., 1976) system with positive coupling constant. It is shown that for the negative coupling the conformal quantum mechanics in AFF form can be immersed into well-defined quantum theory with global action of SL(2,R) as a symmetry group. It is indicated how the resulting theory emerges from the canonical/geometric quantization of the Hamiltonian dynamics on the relevant coadjoint orbits.