One-loop quantum gravity from a worldline viewpoint

TL;DR

This work develops a four-dimensional worldline formalism for quantum gravity by recasting the quadratic fluctuations of the metric into two worldline particle models (traceless-tensor and vector ghosts) that reproduce the relevant differential operators. Using a background-field split, covariant gauge fixing, and a Hawking–Gibbons prescription, the authors express the one-loop effective action as heat-kernel traces and compute Seeley–DeWitt coefficients up to $a_2$, confirming the expected divergences and topological structure. The traceless-tensor and vector path integrals are evaluated with dimensional regularization, and their combination with the scalar sector reproduces the standard one-loop divergence structure for pure gravity in $D=4$, including the Gauss–Bonnet term contribution. The results validate the worldline approach as a potentially simpler tool for perturbative quantum gravity amplitudes and set the stage for unified worldline models of gravitons on general backgrounds.

Abstract

We develop a worldline approach to quantum gravity in D=4. Using the background field method we consider the covariantly gauge fixed Einstein-Hilbert action with cosmological constant, and find a worldline representation of the differential operators identified by its quadratic approximation. We test it by computing the correct one-loop divergencies. Alternative worldline methods, such as the use of the O(4) spinning particle that is known to describe correctly the propagation of a massless spin 2 particle in D=4, find obstructions in the coupling to an arbitrary background metric, apparently preventing a more extensive use in perturbative descriptions of quantum gravity. We expect that our model might simplify calculations of one-loop amplitudes with respect to standard quantum field theoretical methods.