  We revisit the issue that the effective potential for the conformal factor of
the metric, which is generated by quantized matter fields, possesses a
non-vanishing vacuum expectation value (VEV) or not. We prove that the
effective potential has a vanishing vacuum expectation value on the basis of a
global $GL(4)$ symmetry. We also account for the reason why there seem to be
two different effective potentials for the conformal factor in a theory, one of
which gives rise to a vanishing VEV for the conformal factor whereas the other
does a non-vanishing VEV.
