Some reference formulas for the generating functions of canonical transformations

TL;DR

This paper develops practical formulas and diagrammatic methods for generating functions of canonical transformations in classical and quantum theories, including extensions to the Batalin-Vilkovisky formalism.

Contribution

It introduces a diagrammatic expansion for composition laws, a componential map obeying the Baker-Campbell-Hausdorff formula, and extends these results to quantum field theory and BV formalism.

Findings

Diagrammatic formula for composition law expansion

Representation of generating functions via componential map

Extension to quantum field theory and BV formalism

Abstract

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.