Path integral on a group manifold and the lattice gauge theory Hamiltonian

For a quantum mechanical system living on the manifold of a compact simple Lie group we present explicit formulae for the quantum corrections, both in the Hamiltonian and, for the most common time discretization, in the path integral. As a special application of this rather general procedure, we compare, for lattice gauge theories, the path integral corresponding to the Kogut-Susskind Hamiltonian and the Wilson action. The latter is shown to correspond to a very special but elegant way of discretizing the time variable.