  We examine a nonlocal interaction that results from expressing the QCD
Hamiltonian entirely in terms of gauge-invariant quark and gluon fields. The
interaction couples one quark color-charge density to another, much as electric
charge densities are coupled to each other by the Coulomb interaction in QED.
In QCD, this nonlocal interaction also couples quark color-charge densities to
gluonic color. We show how the leading part of the interaction between quark
color-charge densities vanishes when the participating quarks are in a color
singlet configuration, and that, for singlet configurations, the residual
interaction weakens as the size of a packet of quarks shrinks. Because of this
effect, color-singlet packets of quarks should experience final state
interactions that increase in strength as these packets expand in size. For the
case of an SU(2) model of QCD based on the {\em ansatz} that the
gauge-invariant gauge field is a hedgehog configuration, we show how the
infinite series that represents the nonlocal interaction between quark
color-charge densities can be evaluated nonperturbatively, without expanding it
term-by-term. We discuss the implications of this model for QCD with SU(3)
color and a gauge-invariant gauge field determined by QCD dynamics.
