The effective action summarizes the content of a field theory in a way which is technically convenient and
physically instructive. It is familiar from perturbation theory as the generating functional of the “1-particle
irreducible” (1-PI) Green’s functions. It can however be given a meaning independent of perturbation
theory and its functional derivatives can be used to reconstruct all the correlation functions of the field
theory under consideration. Since the latter can be viewed as the moments of the functional measure, this
replaces the “cone of measures” (or the space of “Wilsonian effective actions ”) in the Wilsonian setting
by the “space of effective action functionals” as the arena on which the renormalization group
acts.
The standard effective action admits two fruitful generalizations, discussed in Appendices B and C, respectively. The first one is adapted to theories with symmetries, which can be field reparameterization symmetries, gauge symmetries, or both. These effective actions are known as “background effective actions”. In Appendix B.2 we provide a concise summary of these constructions. The background reparameterization and/or gauge invariance also provides an, often crucial, simplification of the renormalization. In Appendix B.3 we describe this for the case of field reparameterization symmetries in Riemannian sigma-models, which also provides some of the renormalization prerequisites for Section 3. In a gravitational context the background field formalism also provides a setting which, despite the name, can reasonably be regarded as “background independent”.
The second generalization of the standard effective action is one where, roughly, the bare action is
replaced by
, where
effectively suppresses field modes with momenta less than
. This
leads to the effective average action reviewed in Appendix C.
For definiteness we assume a Euclidean setting throughout these Appendices.
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