  We consider the background-free quantum gravity based on conformal gravity
with the Riegert-Wess-Zumino action, which is formulated in terms of a
conformal field theory. Employing the $R \times S^3$ background in practice, we
construct the nilpotent BRST operator imposing diffeomorphism invariance.
Physical fields and states are analyzed, which are given only by real primary
scalars with a definite conformal weight. With attention to the presence of
background charges, various significant properties, such as the state-operator
correspondence and the norm structure, are clarified with some examples.
