Quantal dynamics on non-compact groups

Free quantal motion on group manifolds, compact and non-compact, is considered. An integral representation for the evolution operator is obtained by using the Green function that corresponds to the Laplace-Beltrami operator on the group manifold. The alternative series expressions for the evolution operator - the spectral expansion (when exists) and the sum over classical paths - are reconstructed from the same integral representation. The semi-classical expression is found to be exact for non-compact semi-simple real groups, as well as for compact groups (as was known previously), provided that all the classical trajectories are taken into account properly.