  Probability distribution of non-Abelian parallel transporters on the group
manifold and the corresponding amplitude are investigated for quantum
Yang-Mills fields. It is shown that when the Wilson area law and the Casimir
scaling hold for the quantum gauge field, this amplitude can be obtained as the
solution of the free Schroedinger equation on the gauge group. Solution of this
equation is written in terms of the path integral and the corresponding action
term is interpreted geometrically. We also note that the partition function of
two-dimensional pure Yang-Mills theory living on the surface spanned on the
loop solves the obtained equation.
