Further comments on the background field method and gauge invariant effective actions

TL;DR

This paper provides a rigorous proof of the existence of a gauge invariant effective action within the background field method for any gauge theory, clarifying previous arguments and emphasizing its physical significance.

Contribution

It offers a clear and firm proof of background gauge invariance of the effective action, enhancing theoretical understanding of gauge theories.

Findings

Confirmed the gauge invariance of the effective action in the background field method.

Clarified the relationship between the effective action and background gauge equivalence.

Reinforced the physical interpretation of the effective action as a generator of proper vertices.

Abstract

The aim of this paper is to give a firm and clear proof of the existence in the background field framework of a gauge invariant effective action for any gauge theory ({\it background gauge equivalence}). Here by effective action we mean a functional whose Legendre transform restricted to the physical shell generates the matrix elements of the connected $S$-matrix. We resume and clarify a former argument due to Abbott, Grisaru and Schaefer based on the gauge-artifact nature of the background fields and on the identification of the gauge invariant effective action with the generator of the proper, background field, vertices.