Gravitational Noether-Ward identities for scalar field

TL;DR

This paper investigates the gravitational Noether-Ward identities in the context of Einstein gravity coupled to a quantum scalar field, revealing how these identities hold for each term in the equations of motion and their dependence on metric perturbation definitions.

Contribution

It demonstrates that each term in the gravitational perturbation equations satisfies its own Noether-Ward identity and explores how different metric perturbation definitions affect these identities.

Findings

Each term in the gravitational perturbation equations satisfies its own Noether-Ward identity.

Counterterms for graviton self-energy also satisfy their own Noether identities.

The identities depend on the chosen definition of metric perturbations.

Abstract

We consider the gravitational Noether-Ward identities for the evolution of general metric perturbations on quantum matter backgrounds. In this work we consider Einstein's gravity covariantly coupled to a massive, non-minimally coupled, quantum scalar field in general curved backgrounds. We find that each term in the equation of motion for gravitational perturbations satisfies its own Noether-Ward identity. Even though each term is non-transverse, the whole equation of motion maintains transversality. In particular, each counterterm needed to renormalize the graviton self-energy satisfies its own Noether identity, and we derive the explicit form for each. Finally, in order to understand how the Noether-Ward identities are affected by the definition of the metric perturbation, we consider two inequivalent definitions of metric perturbations and derive the Noether-Ward identities for both definitions. This implies that there are Noether-Ward identities for every definition of the metric perturbation.