Background-free quantum gravity based on conformal gravity and conformal field theory on $M^4$

We study four-dimensional quantum gravity formulated as a certain conformal field theory at the ultraviolet fixed point, whose dynamics is described by the combined system of Riegert-Wess-Zumino and Weyl actions. Background-free nature comes out as quantum diffeomorphism symmetry by quantizing the conformal factor of the metric field nonperturbatively. In this paper, Minkowski background $M^4$ is employed in practice. The generator of quantum diffeomorphism that forms conformal algebra is constructed. Using it, we study the composite scalar operator that becomes a good conformal field. We find that physical fields are described by such scalar fields with conformal dimension 4. Consequently, tensor fields outside the unitarity bound are excluded. Computations of quantum algebra on $M^4$ are carried out in the coordinate space using operator products of the fields. The nilpotent Becchi-Rouet-Stora-Tyupin operator is also constructed.