BRST formalism of Weyl Invariant Gravity and Confinement of Massive Tensor Ghost

We present the manifestly covariant canonical operator formalism of a Weyl invariant (or equivalently, a locally scale invariant) gravity whose classical action consists of the well-known conformal gravity and Weyl invariant scalar-tensor gravity, on the basis of the Becchi-Rouet-Stora-Tyupin (BRST) formalism. Based on this formalism, we analyze the physical states by expanding the metric around a flat Minkowski background and the scalar field around a constant background. It is shown that under the assumption of no bound states the physical modes are composed of both a massive tensor ghost of five components with indefinite norm and a massless graviton of two components with positive semi-definite norm. The unitarity of the physical S-matrix is violated by the presence of the massive ghost. On the other hand, if the Weyl BRST transformation of the massive ghost has a bound state, the massive ghost is confined in the zero-norm states by the BRST-quartet mechanism so the physical S-matrix becomes unitary. This mechanism of the ghost confinement might pave the way for a long-standing problem of the unitarity violation in quadratic gravity or higher-derivative gravity.