Ghosts of critical gravity

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 article, we quantize linear fluctuations of these critical gravities, which are known to be either identical with linear fluctuations of Einstein's gravity or 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.