  Following a previous calculation of quark scattering in eikonal
approximation, this paper presents a new, analytic and rigorous approach to the
calculation of QCD phenomena. In this formulation a basic distinction between
the conventional "idealistic" description of QCD and a more "realistic"
description is brought into focus by a non-perturbative, gauge-invariant
evaluation of the Schwinger solution for the QCD generating functional in terms
of the exact Fradkin representations of the Green's functional and the vacuum
functional. Because quarks exist asymptotically only in bound states, their
transverse coordinates can never be measured with arbitrary precision; the
non-perturbative neglect of this statement leads to obstructions that are
easily corrected by invoking in the basic Lagrangian a probability amplitude
which describes such transverse imprecision.
  The second result of this non-perturbative analysis is the appearance of a
new and simplifying output called "Effective Locality", in which the
interactions between quarks by the exchange of a "gluon bundle" - which
"bundle" contains an infinite number of gluons, including cubic and quartic
gluon interactions - display an exact locality property that reduces the
several functional integrals of the formulation down to a set of ordinary
integrals. It should be emphasized that "non-perturbative" here refers to the
effective summation of all gluons between a pair of quark lines, but does not
(yet) include a summation over all closed-quark loops which are tied by
gluon-bundle exchange to the rest of the "Bundle Diagram". As an example of the
power of these methods we offer as a first analytic calculation the
quark-antiquark binding potential of a pion, and the corresponding three-quark
binding potential of a nucleon, obtained in a simple way from relevant eikonal
scattering approximations.
