Remark on quantum mechanics with conformal Galilean symmetry

Conformal Galilei algebra contains so(1,2) subalgebra which is the conformal algebra in one dimension. In this note, we generalize methods previously developed for one-dimensional many-body systems and construct a unitary map relating a quantum mechanics invariant under the conformal Galilean transformations to a set of decoupled particles for which the same symmetry is realized in a nonlocal way. Possible applications of the map are discussed.