Vanishing Metric Commutation Relation and Higher-derivative De Donder Gauge in Quadratic Gravity

TL;DR

This paper demonstrates that in quadratic gravity, the equal-time commutation relations among metric derivatives vanish, suggesting the metric behaves classically and micro-causality holds, which may relate to the theory's renormalizability.

Contribution

It shows the vanishing of ETCRs in quadratic gravity under specific gauges, indicating classical behavior of the metric and potential implications for renormalizability.

Findings

ETCRs among metric derivatives vanish in quadratic gravity.

The metric behaves as a classical field in this formalism.

Micro-causality is preserved for the metric tensor.

Abstract

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.