About possible measures in Quantum Gravity

TL;DR

This paper investigates how the choice of path integral measure in quantum gravity affects quantization, exploring invariant and noninvariant options and their anomaly/counterterm interplay. It derives a model-independent invariant measure and discusses model-dependent g^{00}-weighted measures, comparing with existing literature. It demonstrates that, for quartic/ higher-derivative theories including Stelle gravity, the volume divergences δ^4(0) can cancel between measure factors and fluctuations in certain gauges or extremal configurations, using Liouville-type quantization and Dirac-Pauli methods. The work suggests noncovariant measures may be viable if their anomalies are absorbable by counterterms, while noting that fully general curved-background quantization remains an open challenge with connections to heat-kernel approaches.

Abstract

Possible measures for Quantum Gravity are considered. Choices that are invariant under diffeomorphisms are analyzed, but the possibility of employing non invariant measures is also taken into account. The last possibility may be accepted if the anomaly in the measure is compensated by counter term redefinitions of the model under analysis. Particular attention is paid to some concrete examples of non covariant looking measures, which may be useful for generalizing the Veltmann identities when quantizing around curved space times. The results are specified for the Stelle gravity model [1]-[2], which is known to be renormalizable in flat space, although not known to be so in curved ones.