  A basic distinction, long overlooked, between the conventional, "idealistic"
formulation of QCD, and a more "realistic" formulation is brought into focus by
a rigorous, non-perturbative, gauge-invariant evaluation of the Schwinger
solution for the QCD generating functional in terms of exact Fradkin
representations for the Green's functional $\mathbf{G}_{c}(x,y|A)$ and the
vacuum functional $\mathbf{L}[A]$. The quanta of all (Abelian) quantized fields
may be expected to obey standard quantum-mechanical measurement properties,
perfect position dependence at the cost of unknown momenta, and vice-versa, but
this is impossible for quarks since they always appear asymptotically in bound
states, and their transverse position or momenta can never, in principle, be
exactly measured. Violation of this principle produces an absurdity in the
exact evaluation of each and every QCD amplitude. We here suggest a
phenomenological change in the basic QCD Lagrangian, such that a limitation of
transverse precision is automatically contained in the now "realistic" theory,
with the function essential to quark binding into hadrons appearing in the new
Lagrangian. All absurdities in estimates of all "realistic" QCD amplitudes are
then removed, and one obtains the possibility of hadron formation by
appropriate quark binding potentials, and nucleon scattering and binding by
effective, Yukawa-type potentials; the first of these potentials are
constructed, in detail, in the following paper.
