  In this paper we apply a variant of Heisenberg's quantization method for
strongly interacting, non-linear fields, to solutions of the classical
Yang-Mills field equations which have bad asymptotic behavior. After
quantization we find that the bad features (i.e. divergent fields and energy
densities) of these solutions are moderated. From these results we argue that
in general the n-point Green's functions for Yang-Mills theories can have
non-perturbative pieces which can not be represented as the sum of Feynman
diagrams. A formalism for dealing with these non-Feynman pieces via
nonassociative field operators is suggested. These methods may also find some
application in dealing with high-$T_c$ superconductors.
