Path integral quantum cosmology

The effect of the Hamiltonian constraint on the quantization of gravity is explored by application of Feynman path integral formalism to the quantum mechanics of spatially homogeneous cosmologies. The reduction method of Arnowitt, Deser, and Misner (ADM) has been implemented within the path integral by a cononical transformation in the subphasespace of the non-dynamical degree of freedom to variables which are the Hamiltonian constraint and a time coordinate condition. The Jacobian of this transformation introduces an extra factor in the measure. Minisuperspace transition amplitudes have been constructed for several cosmological models. Closed form expressions have been obtained for the Kasner and fluid-filled Kasner cosmologies. The infinitesimal transition amplitude has been found for Robertson-Walker with scalar field and mixmaster near the minimum of the potential