  Recently, using a local action satisfying the Wess-Zumino condition as a
kinetic term of the conformal mode, we formulated a four-dimensional quantum
geometry (4DQG). The conformal mode can be treated exactly, and it was shown
that the part of the effective action related to this mode is given by the
scale-invariant non-local Riegert action. As for the traceless mode, we
introduce dimensionless coupling, which is a unique gravitational coupling of
this theory satisfying the conditions of renormalizability and asymptotic
freedom. Although this theory is asymptotically free, the physical states are
non-trivial, which should be described as composite fields, like the spectrum
of 2DQG. The possibility that the physical state conditions representing
background-metric independence conceal ghosts is pointed out. The usual
graviton state would be realized when the physical state condition breaks down
dynamically.
