Noether symmetric classical and quantum scalar field cosmology
  We study the evolution of a two dimensional minisuperspace cosmological model
in classical and quantum levels by the Noether symmetry approach. The phase
space variables turn out to correspond to the scale factor of a
Friedmann-Robertson-Walker (FRW) model and a scalar field with which the action
of the model is augmented. It is shown that the minisuperspace of such a model
is a two dimensional manifold with vanishing Ricci scalar. We present a
coordinate transformation which cast the corresponding minisuper metric to a
Minkowskian or Euclidean one according to the choices of an ordinary or phantom
model for the scalar field. Then, the Noether symmetry of such a cosmological
model is investigated by utilizing the behavior of the corresponding Lagrangian
under the infinitesimal generators of the desired symmetry. We explicitly
calculate the form of the scalar field potential functions for which such
symmetries exist. For these potential functions, the exact classical and
quantum solutions in the cases where the scalar field is an ordinary or a
phantom one, are presented and compared.
