About possible measures in Quantum Gravity

There exist several different proposals for a measure in Quantum Gravity theories. Although sometimes being labelled as non covariant, the measure derived in [7] for GR has the particularity that, in the extremal, the volume divergences cancel out. The analogous for Quadratic Gravity [1]-[2] was considered in [44]-[45]. However, as far as the author knows, the issue of volume divergences was not considered for this last measure. The present work fills this gap and presents an analysis showing that, in the extremal, these divergences cancel as well. This is up to some subtleties related to superdeterminants. The possibility of employing non invariant measures may be accepted if the anomaly in the measure is compensated by counter term redefinitions of the model under analysis. This makes difficult to disprove, at the present times, some choices of measures. Quadratic Gravity[1]-[2], is known to be renormalizable in flat space, and there are a finite number of counter terms needed in order to renormalize its effective action. However, around a curved space this is not known, and this complicates considerably the analysis. These issues are reviewed in the text, together with an analysis of covariant measures. In particular, it is shown how these measures [47]-[49] can be found if one condition in [7] is relaxed.