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.
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Ichiro Oda (2024) studied this question.
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