Linear conformal quantum gravity

This paper is an investigation of field theories that satisfy the following two criteria. (1) Among the propagating modes is a pair of massless particles with helicities ±2. (2) The canonical communtation relations are conformally invariant. This study of "linear conformal gravity" is motivated by the belief that conformal invariance may be the key to a future theory of quantum gravity. Our first conclusion is that the fields of linear conformal gravity include a tensor field of rank 3 and mixed symmetry, and a symmetric tensor field of rank 2, tentatively interpreted as a torsion field and a metric field. The free quantum field operator is constructed explicitly, and the propagator is calculated. The Fourier transform is of dimension $p^{-4\mathrm{}}$, which is encouraging for renormalizability. The field inevitably carries along a nonunitary ghost, similar to the one that turns up in linearized Weyl gravity. Our main result is that the ghost can be exorcised by imposing constraints on the external sources and boundary conditions on the physical states.