Ghosts of Critical Gravity

TL;DR

This paper analyzes the quantization of linear fluctuations in critical higher-derivative gravity theories in Anti de Sitter space, revealing their non-unitary logarithmic structure and boundary conditions.

Contribution

It provides a detailed quantization of fluctuations in critical gravity, identifying the scalar product structure and demonstrating the non-unitary nature of logarithmic modes.

Findings

Scalar product between Einstein modes vanishes

Scalar product of Einstein and logarithmic modes is nonzero

Logarithmic modes are neither unitary nor unitarizable

Abstract

Recently proposed "critical" higher-derivative gravities in $AdS_D$ $D>3$ are expected to carry logarithmic representation of the Anti de Sitter isometry group. In this note, we quantize linear fluctuations of these critical gravities, which are known to be either identical with linear fluctuations of Einstein's gravity or else satisfy logarithmic boundary conditions at spacial infinity. We identify the scalar product uniquely defined by the symplectic structure implied by the classical action, and show that it does not posses null vectors. Instead, we show that the scalar product between any two Einstein modes vanishes, while the scalar product of an Einstein mode with a logarithmic mode is generically nonzero. This is the basic property of logarithmic representation that makes them neither unitary nor unitarizable.