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We study some properties of the canonical transformations in classicalmechanics and quantum field theory and give a number of practical formulasconcerning their generating functions. First, we give a diagrammatic formulafor the perturbative expansion of the composition law around the identity map.Then, we propose a standard way to express the generating function of acanonical transformation by means of a certain "componential" map, which obeysthe Baker-Campbell-Hausdorff formula. We derive the diagrammatic interpretationof the componential map, work out its relation with the solution of theHamilton-Jacobi equation and derive its time-ordered version. Finally, wegeneralize the results to the Batalin-Vilkovisky formalism, where the conjugatevariables may have both bosonic and fermionic statistics, and describeapplications to quantum field theory.

Some reference formulas for the generating functions of canonical transformations