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Gravitational Noether-Ward identities for scalar field

Tomislav Prokopec

TL;DR

The paper studies gravitational Noether-Ward identities for the evolution of general metric perturbations on quantum matter backgrounds within semiclassical gravity, focusing on Einstein gravity with a real, massive, nonminimally coupled scalar field. It demonstrates that every term in the graviton perturbation equation satisfies its own Noether-Ward identity, while the total equation remains covariantly transverse; explicit NW identities are derived for all one-loop counterterms involving $R^2$, ${ m Ric}^2$, ${ m Riem^2}$, Weyl$^2$, and Gauss-Bonnet combinations, for both perturbation definitions Case A and Case B. The de Sitter space example with Chernikov-Tagirov propagators illustrates the identities in a cosmological setting, yielding a renormalized background equation and a perturbative expansion for the Hubble parameter $H$ including quantum corrections. The results provide systematic consistency checks for graviton self-energy computations on curved backgrounds and pave the way for extensions to in-in formalisms and more general matter content in early-universe and black-hole spacetimes.

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.

Gravitational Noether-Ward identities for scalar field

TL;DR

The paper studies gravitational Noether-Ward identities for the evolution of general metric perturbations on quantum matter backgrounds within semiclassical gravity, focusing on Einstein gravity with a real, massive, nonminimally coupled scalar field. It demonstrates that every term in the graviton perturbation equation satisfies its own Noether-Ward identity, while the total equation remains covariantly transverse; explicit NW identities are derived for all one-loop counterterms involving , , , Weyl, and Gauss-Bonnet combinations, for both perturbation definitions Case A and Case B. The de Sitter space example with Chernikov-Tagirov propagators illustrates the identities in a cosmological setting, yielding a renormalized background equation and a perturbative expansion for the Hubble parameter including quantum corrections. The results provide systematic consistency checks for graviton self-energy computations on curved backgrounds and pave the way for extensions to in-in formalisms and more general matter content in early-universe and black-hole spacetimes.

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.
Paper Structure (7 sections, 222 equations, 3 figures)

This paper contains 7 sections, 222 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Effective action
  4. Noether-Ward identities for gravitational perturbations
  5. De Sitter space example
  6. Discussion
  7. Appendices

Figures (3)

  • Figure 1: The three- and four-point vertices. The solid straight lines represent the scalar propagators, the wavy lines correspond to the graviton propagators.
  • Figure 2: The one-loop graviton tadpole diagram contributing to the background equation of motion (\ref{['SC Einstein equation']}). This diagram contributes to the semiclassical Einstein equation (\ref{['SC Einstein equation']}) such that the external graviton leg ought to be amputated.
  • Figure 3: The one-loop diagrams contributing to the graviton self-energy (\ref{['total self-energy']}) induced by scalars. The first two diagrams are the three- and four-point contributions, and the last diagram represents the local counterterms.