The Background Field Method as a Canonical Transformation

TL;DR

This paper explicitly constructs a canonical transformation in Yang-Mills theory that manages the dependence of the vertex functional on background fields, generalizing classical Lie transforms and enabling various advanced applications.

Contribution

It introduces a field-theoretic canonical transformation for background dependence, extending classical Lie transform concepts to quantum gauge theories.

Findings

Explicit construction of the canonical transformation

Connection to Lie transform in classical mechanics

Potential applications in non-perturbative and lattice formulations

Abstract

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.