Gauge invariant coefficients in perturbative quantum gravity

Perturbative quantum gravity can be studied in many ways. A traditional approach is to apply covariant quantization schemes to the Einstein-Hilbert action and use heat kernel methods, as pioneered by DeWitt. An alternative approach is to consider the graviton as arising from the first quantization of particle actions, following the same methods used in string theory. An interesting model to describe the graviton is based on the so-called N = 4 spinning particle, which has been used recently to study perturbative properties of quantum gravity, allowing in particular for the calculation of certain gauge-invariant coefficients. The latter are related to the counterterms that renormalize the one-loop effective action of pure quantum gravity with a cosmological constant. Such coefficients have already been tested in D = 4 dimensions. Here we study the general case of arbitrary D. We derive the gauge-invariant coefficients —the simplest one being the number of physical degrees of freedom of the graviton—using the traditional heat kernel method. We compare them with the ones obtained by using the N = 4 spinning particle and discover that the latter fails to reproduce some of those coefficients at arbitrary dimension, suggesting the need of improving that first quantized model. This constitutes a first original result of this thesis. In the second part, we try to find an alternative worldline path integral treatment of the heat kernel, extending a previous worldline construction that was tailored to 4 dimensions only. We succeed in finding suitable worldline actions for the gauge-fixed graviton fluctuations and related ghosts. The action for the graviton fluctuations that we construct reproduces the expected Hamiltonian but does not seem to admit a perturbative path integral treatment.