Phantom Field from Conformal Invariance

We establish a correspondence between a conformally invariant complex scalar field action (with a conformal self-interaction potential) and the action of a phantom scalar field minimally coupled to gravity (with a cosmological constant). In this correspondence, the module of the complex scalar field is used to relate conformally the metrics of both systems while its phase is identified with the phantom scalar field. At the level of the equations, the correspondence allows to map solution of the conformally non-linear Klein-Gordon equation with vanishing energy-momentum tensor to solution of a phantom scalar field minimally coupled to gravity with cosmological constant satisfying a massless Klein-Gordon equation. The converse is also valid with the advantage that it offers more possibilities owing to the freedom of rewriting a metric as the conformal transformation of another metric. Finally, we provide some examples of this correspondence.