  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 BRST operator is also constructed.
