Quantum Theory of Weyl Invariant Scalar-tensor Gravity

We perform a manifestly covariant quantization of a Weyl invariant, i.e., a locally scale invariant, scalar-tensor gravity in the extended de Donder gauge condition (or harmonic gauge condition) for general coordinate invariance and a new scalar gauge for Weyl invariance within the framework of BRST formalism. It is shown that choral symmetry, which is a Poincar${\rm{\acute{e}}}$-like $\IOSp(8|8)$ supersymmetry in case of Einstein gravity, is extended to a Poincar${\rm{\acute{e}}}$-like $\IOSp(10|10)$ supersymmetry. We point out that there is a gravitational conformal symmetry in quantum gravity and account for how conventional conformal symmetry in a flat Minkowski space-time is related to the gravitational conformal symmetry. Moreover, we examine the mechanism of spontaneous symmetry breakdown of the choral symmetry, and show that the gravitational conformal symmetry is spontaneously broken to the Poincar\'e symmetry and the corresponding massless Nambu-Goldstone bosons are the graviton and the dilaton. We also prove the unitarity of the physical S-matrix on the basis of the BRST quartet mechanism.