  Retrieval of classical behaviour in quantum cosmology is usually discussed in
the framework of {\em midi}superspace models in the presence of scalar fields
and the inhomogeneous modes corresponding either to gravitational or scalar
fields. In this work, we propose an alternative model to study the decoherence
of homogeneous and isotropic geometries where the scalar field is replaced by a
massive vector field with a global internal symmetry. We study here the cases
with $U(1)$ and $SO(3)$ global internal symmetries. The presence of a mass term
breaks the conformal invariance and allows for the longitudinal modes of the
spin-1 field to be present in the Wheeler-DeWitt equation. In the case of the
U(1) global internal symmetry, we have only one single ``classical'' degree of
freedom while in the case of the SO(3) global symmetry, we are led to consider
a simple two-dimensional minisuperspace model. These minisuperspaces are shown
to be equivalent to a set of coupled harmonic oscillators where the kinetic
term of the longitudinal modes has a coefficient proportional to the inverse of
the scale factor. The conditions for a suitable decoherence process and
correlations between coordinate and momenta are established. The validity of
the semi-classical Einstein equations when massive vector fields (Abelian and
non-Abelian) are present is also discussed.
