We show that the equal-time commutation relations (ETCRs) among the time derivatives of the metric tensor identically vanish in the higher-derivative de Donder gauge as well as the conventional de Donder gauge (or harmonic gauge) for general coordinate invariance in the manifestly covariant canonical operator formalism of quadratic gravity. These ETCRs provide us with the vanishing four-dimensional commutation relation, which implies that the metric tensor behaves as if it were not a quantum operator but a classical field. In this case, the micro-causality is valid at least for the metric tensor in an obvious manner. This fact might be a manifestation of renormalizability of quadratic gravity in case of the canonical operator formalism.
Ichiro Oda (Mon,) studied this question.