  We study some properties of the canonical transformations in classical
mechanics and quantum field theory and give a number of practical formulas
concerning their generating functions. First, we give a diagrammatic formula
for the perturbative expansion of the composition law around the identity map.
Then, we propose a standard way to express the generating function of a
canonical transformation by means of a certain "componential" map, which obeys
the Baker-Campbell-Hausdorff formula. We derive the diagrammatic interpretation
of the componential map, work out its relation with the solution of the
Hamilton-Jacobi equation and derive its time-ordered version. Finally, we
generalize the results to the Batalin-Vilkovisky formalism, where the conjugate
variables may have both bosonic and fermionic statistics, and describe
applications to quantum field theory.
