Correspondence between a scalar field and an effective perfect fluid

It is widely acknowledged that, for formal purposes, a minimally coupled scalar field is equivalent to an effective perfect fluid with equation of state determined by the scalar potential. This correspondence is not complete because the Lagrangian densities L₁=P and L₂=-ρ, which are equivalent for a perfect fluid, are not equivalent for a minimally coupled scalar field. The exchange L₁↔L₂ amounts to exchanging a canonical scalar field with a phantom scalar field.