# Gupta-Bleuler quantization for linearized gravity in de Sitter spacetime

## TL;DR

This paper develops a Gupta-Bleuler quantization method for linearized gravity in de Sitter space, addressing gauge anomalies and preserving de Sitter covariance at the quantum level.

## Contribution

It introduces a coordinate-independent Gupta-Bleuler formalism for linearized quantum gravity in de Sitter space, ensuring symmetry preservation and consistent gauge treatment.

## Key findings

- Successfully constructs a covariant graviton quantum field on de Sitter space.
- Maintains de Sitter invariance and gauge invariance in the quantum theory.
- Ensures positive energy in physical states despite negative norm states.

## Abstract

In a recent Letter, we have pointed out that the linearized Einstein gravity in de Sitter (dS) spacetime besides the spacetime symmetries generated by the Killing vectors and the evident gauge symmetry also possesses a hitherto `hidden' local (gauge-like) symmetry which becomes anomalous on the quantum level. This gauge-like anomaly makes the theory inconsistent and must be canceled at all costs. In this companion paper, we first review our argument and discuss it in more detail. We argue that the cancelation of this anomaly makes it impossible to preserve dS symmetry in linearized quantum gravity through the usual canonical quantization in a consistent manner. Then, demanding that all the classical symmetries to survive in the quantized theory, we set up a coordinate-independent formalism \`{a} \emph{la} Gupta-Bleuler which allows for preserving the (manifest) dS covariance in the presence of the gauge and the gauge-like invariance of the theory. On this basis, considering a new representation of the canonical commutation relations, we present a graviton quantum field on dS space, transforming correctly under isometries, gauge transformations, and gauge-like transformations, which acts on a state space containing a vacuum invariant under all of them. Despite the appearance of negative norm states in this quantization scheme, the energy operator is positive in all physical states, and vanishes in the vacuum.