Classical and Quantum Behaviour of Multidimensional Integrable Cosmologies

Multidimensional cosmological models with $n(n>1)$ Einstein spaces are discussed classically and with respect to canonical quantization. These models are integrable in the case of Ricci flat internal spaces. For negative curvature of the external space we find exact classical solutions modelling dynamical as well as spontaneous compactification of the internal spaces. Spontaneous compactification turns out to be an attractor solution. Solutions of the quantum Wheeler-DeWitt equation are also obtained. Some of them describe the tunneling process to be interpreted as the birth of the universe from ‘‘nothing’’.