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arXiv · 1511.00828

Some reference formulas for the generating functions of canonical transformations

Damiano Anselmi
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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.

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Damiano Anselmi. 2016-04-05. Some reference formulas for the generating functions of canonical transformations. https://doi.org/10.1140/epjc%2Fs10052-015-3874-y

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