Batalin-Vilkovisky formalism in perturbative algebraic quantum field theory

TL;DR

This paper develops the Batalin-Vilkovisky formalism within perturbative algebraic quantum field theory, establishing a renormalized algebraic framework without relying on path integrals or intermediate regularizations.

Contribution

It constructs the renormalized Batalin-Vilkovisky complex directly in algebraic terms, proving the equivalence of time-ordered and pointwise products and identifying the key anomaly operator.

Findings

Renormalized time-ordered product equals classical pointwise product.

The renormalized BV algebra is expressed via the classical algebra with a modified Laplacian.

The anomaly operator is identified with the Master Ward Identity anomaly.

Abstract

On the basis of a thorough discussion of the Batalin-Vilkovisky formalism for classical field theory presented in our previous publication, we construct in this paper the Batalin-Vilkovisky complex in perturbatively renormalized quantum field theory. The crucial technical ingredient is a proof that the renormalized time-ordered product is equivalent to the pointwise product of classical field theory. The renormalized Batalin-Vilkovisky algebra is then the classical algebra but written in terms of the time-ordered product, together with an operator which replaces the ill defined graded Laplacian of the unrenormalized theory. We identify it with the anomaly term of the anomalous Master Ward Identity of Brennecke and D\"utsch. Contrary to other approaches we do not refer to the path integral formalism and do not need to use regularizations in intermediate steps.