  The canonical front form Hamiltonian for non-Abelian SU(N) gauge theory in
3+1 dimensions is mapped non-perturbatively on an effective Hamiltonian which
acts only in the Fock space of a quark and an antiquark. Emphasis is put on
dealing with the many-body aspects of gauge field theory, and it is shown
explicitly how the higher Fock-space amplitudes can be retrieved
self-consistently from solutions in the $q\bar q$-space. The approach is based
on the novel method of iterated resolvents and on discretized light-cone
quantization, driven to the continuum limit. It is free of the usual
perturbative Tamm-Dancoff truncations in particle number and coupling constant,
and respects all symmetries of the Lagrangian including covariance and gauge
invariance. It is emphasized that vertex regularization as opposed to
Fock-space regularization allows an explicit renormalization, which yields
linear confinement. The string constant must be determined from experiment.
