Causal perturbation theory in terms of retarded products, and a proof of the Action Ward Identity

TL;DR

This paper develops a local construction of interacting quantum fields using retarded products, proving the Action Ward Identity and analyzing the renormalization group flow within perturbative algebraic quantum field theory.

Contribution

It provides a local, functional approach to interacting fields, proving the Action Ward Identity and deriving renormalization group flow independently of infrared issues.

Findings

Proof of the Action Ward Identity in this formalism

Explicit computation of renormalization of interaction and fields

Flow of free parameters under renormalization group

Abstract

In the framework of perturbative algebraic quantum field theory a local construction of interacting fields in terms of retarded products is performed, based on earlier work of Steinmann. In our formalism the entries of the retarded products are local functionals of the off shell classical fields, and we prove that the interacting fields depend only on the action and not on terms in the Lagrangian which are total derivatives, thus providing a proof of Stora's 'Action Ward Identity'. The theory depends on free parameters which flow under the renormalization group. This flow can be derived in our local framework independently of the infrared behavior, as was first established by Hollands and Wald. We explicitly compute non-trivial examples for the renormalization of the interaction and the field.