[Paper Review] Quantum gravity from the point of view of locally covariant quantum field theory

Romeo Brunetti, Klaus Fredenhagen+1 more|arXiv (Cornell University)|Jun 5, 2013

Black Holes and Theoretical Physics52 references18 citations

TL;DR

This paper presents a generally covariant, background-independent formulation of perturbative quantum gravity using the locally covariant quantum field theory framework and the renormalized Batalin-Vilkovisky formalism. It establishes a diffeomorphism-covariant observable algebra through relative observables, resolving the issue of non-invariant local functionals and enabling a consistent perturbative quantization of gravity as an effective field theory.

ABSTRACT

We construct perturbative quantum gravity in a generally covariant way. In particular our construction is background independent. It is based on the locally covariant approach to quantum field theory and the renormalized Batalin-Vilkovisky formalism. We do not touch the problem of nonrenormalizability and interpret the theory as an effective theory at large length scales.

Motivation & Objective

• To develop a generally covariant, background-independent formulation of perturbative quantum gravity.
• To resolve the problem of diffeomorphism-invariant local observables in quantum gravity by introducing relative observables.
• To apply the locally covariant quantum field theory framework to gravity, ensuring covariance across different spacetime backgrounds.
• To construct a consistent perturbative quantum theory of gravity using the renormalized Batalin-Vilkovisky formalism.
• To interpret the resulting theory as an effective field theory valid at large length scales, avoiding issues of nonrenormalizability.

Proposed method

• Adopt the locally covariant quantum field theory approach, formulating all structures in terms of local, spacetime-dependent functionals.
• Use the Batalin-Vilkovisky (BV) formalism to handle gauge symmetries and anomalies in the quantum theory of gravity.
• Define a Peierls bracket on the space of functionals with compact support, ensuring compatibility with the dynamics and diffeomorphism invariance.
• Introduce relative observables—diffeomorphism-covariant functionals whose relations are invariant—enabling a physical interpretation without a fixed background.
• Construct the BRST charge and current from a gauge-fixed version of the Einstein-Hilbert action, ensuring nilpotency and gauge invariance of the quantum theory.
• Apply causal perturbation theory and microlocal analysis to handle singularities and renormalization without regularization, ensuring mathematical consistency.

Experimental results

Research questions

• RQ1 How can quantum gravity be formulated in a background-independent and generally covariant way within the framework of quantum field theory?
• RQ2 What replaces the notion of local, diffeomorphism-invariant observables in quantum gravity, given that such functionals do not exist in the standard sense?
• RQ3 How can the S-matrix and particle interpretation be recovered or replaced in a generally covariant quantum field theory of gravity?
• RQ4 Can the perturbative nonrenormalizability of gravity be managed by treating the theory as an effective field theory with finitely many relevant parameters?
• RQ5 How can the BRST and BV formalisms be consistently applied to gravity to handle gauge symmetries and anomalies in a locally covariant setting?

Key findings

• The space of functionals with compact support and smooth first derivatives forms a Poisson algebra under the Peierls bracket, ensuring a consistent classical structure.
• The Peierls bracket is well-defined and closed on this space, even in the presence of gauge symmetries, due to the use of microlocal analysis and wavefront set conditions.
• The BRST charge is constructed via a gauge-fixed action, and its conservation is ensured by the existence of a suitable 1-form on spacetimes with compact Cauchy surfaces.
• The theory is formulated as an effective field theory, where only finitely many counterterms are needed below a fixed energy scale, circumventing the issue of nonrenormalizability.
• Relative observables provide a diffeomorphism-covariant, physically meaningful interpretation of quantum gravity, replacing the need for a fixed background or S-matrix.
• The locally covariant approach ensures that all structures—algebra of observables, dynamics, and renormalization—are defined independently of a specific spacetime geometry, preserving general covariance.