Effective Potential for Conformal Factor and GL(4) Symmetry

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.