  We construct explicitly the canonical transformation that controls the full
dependence (local and non-local) of the vertex functional of a Yang-Mills
theory on a background field. After showing that the canonical transformation
found is nothing but a direct field-theoretic generalization of the Lie
transform of classical analytical mechanics, we comment on a number of possible
applications, and in particular the non perturbative implementation of the
background field method on the lattice, the background field formulation of the
two particle irreducible formalism, and, finally, the formulation of the
Schwinger-Dyson series in the presence of topologically non-trivial
configurations.
