QUANTUM-MECHANICS OF THE GENERAL SPATIALLY HOMOGENEOUS GEOMETRY COUPLED TO A SCALAR FIELD

The canonical quantization of the general spatially homogeneous geometry minimally coupled to a scalar field is considered. The classical reduction of the full Einstein plus matter action is performed in a way that is independent of both the coordinate system and the dimension of the isometry group acting on each hypersurface. Exact solutions of the Wheeler-DeWitt equation are given for all but one Bianchi types in the case of two-dimensional minisuperspace, and for some Bianchi types in the case of three-dimensional minisuperspace. The results of the quantization suggest that the probability of the classically collapsed configurations to occur as quantum states is zero.