  In many interesting models, including superstring theories, a negative vacuum
energy is predicted. Although this effect is usually regarded as undesirable
from a cosmological point of view, we show that this can be the basis for a new
approach to the cosmology of the early Universe. In the framework of quantum
cosmology (in higher dimensions) when we consider a negative cosmological
constant and matter that could be dust or, alternatively, coherent excitations
of a scalar field, the role of cosmic time can be understood. Then we can
predict the existence of a ``quantum inflationary phase'' for some dimensions
and a simultaneous ``quantum deflationary phase'' for the remaining dimensions.
We discuss how it may be possible to exit from this inflation-compactification
era to a phase with zero cosmological constant which allows a classical
description at late times.
