  The canonical front form Hamiltonian for non-Abelian SU(N) gauge theory in
3+1 dimensions is mapped on an effective Hamiltonian which acts only in the
Fock space of one quark and one antiquark. The approach is non-perturbative and
exact. It is based on Discretized Light-Cone Quantization and the Method of
Iterated Resolvents. The method resums the diagrams of perturbation theory to
all orders in the coupling constant and is free of Tamm-Dancoff truncations in
the Fock-space. Emphasis is put on dealing accurately with the many-body
aspects of gauge field theory. Pending future renormalization group analysis
the running coupling is derived to all orders in the bare coupling
constant.~--- The derived effective interaction has an amazingly simple
structure and is gauge invariant and frame independent. It is solvable on a
small computer like a work station. The many-body amplitudes can be retrieved
self-consistently from these solutions, by quadratures without solving another
eigenvalue problem. The structures found allow also for developing simple
phenomenological models consistent with non-Abelian gauge field theory.
