A proper-time cure for the conformal sickness in quantum gravity

TL;DR

The paper tackles the conformal-factor problem in quantum gravity by formulating a non-perturbative, gauge-fixed path integral in proper-time gauge and performing a non-perturbative Wick rotation on the space of Lorentzian metrics. It shows that the leading conformal divergence can be cancelled by a Faddeev-Popov measure contribution, provided the DeWitt metric parameter satisfies $C< -\frac{2}{d}$, a condition that matches insights from Lorentzian dynamical triangulations. A concrete 3d perturbative calculation illustrates the cancellation explicitly, supporting the view that the conformal mode is non-propagating in the full non-perturbative theory. The results strengthen the bridge between continuum proper-time quantization and discrete Lorentzian gravity models, suggesting a viable non-perturbative definition of quantum gravity, at least in the presence of appropriate regularization and measure choices. Further work is needed to extend these insights to four dimensions and to connect with a full continuum limit.

Abstract

Starting from the space of Lorentzian metrics, we examine the full gravitational path integral in 3 and 4 space-time dimensions. Inspired by recent results obtained in a regularized, dynamically triangulated formulation of Lorentzian gravity, we gauge-fix to proper-time coordinates and perform a non-perturbative ``Wick rotation'' on the physical configuration space. Under certain assumptions about the behaviour of the partition function under renormalization, we find that the divergence due to the conformal modes of the metric is cancelled non-perturbatively by a Faddeev-Popov determinant contributing to the effective measure. We illustrate some of our claims by a 3d perturbative calculation.

Figures (1)

• Figure 1: Penrose diagram of anti-de Sitter space, illustrating the convergence of time-like geodesics.