Origin of a classical space in guantum cosmologies

The influence of vector fields on the origin of a classical space in quantum cosmologies and on the possible compactification process in multidimensional gravity is investigated. It is shown that all general features of the transition between classical and quantum regimes of the evolution can be obtained within the simplest Bianchi-I model for arbitrary number of dimensions. It is shown that the classical space appears when the horizon size reaches the smallest of characteristic scales (the characteristic scale of inhomogeneity or a scale associated with vector fields). In multidimensional case the presence of vector fields completely removes the initial stage of the compactification process which takes place in the case of vacuum models.