The phase of the gravitational path integral

TL;DR

This work develops a streamlined quadratic-quantum-gravity framework to extract the phase of the gravitational path integral on Einstein spaces. By decomposing metric fluctuations into tensor, vector, and scalar sectors and carefully accounting for gauge fixing, zero/mode exclusion, and one-loop determinants, the authors derive a general phase formula controlled by tensor negative modes and a scalar CKV/constant-mode count. Applied to $S^D$, $S^2\times S^{D-2}$, and general products of spheres, the phase is shown to be real and, in many cases, positive, supporting a norm/state-counting interpretation over a decay-rate interpretation for de Sitter spacetimes with these geometries. The analysis extends to charged and rotating black holes (via Kaluza–Klein reductions) and provides bounds on the Lichnerowicz spectrum, offering a geometric lens on stability and phase structure across a broad class of Einstein manifolds. The results bear on Maldacena’s proposals and invite further exploration of observer effects, AdS generalizations, and inclusion of fermions.

Abstract

The gravitational path integral on $S^2 \times S^2$ can be interpreted either as evaluating a contribution to the norm of the Hartle-Hawking wavefunction conditional on spatial $S^1 \times S^2$ topology, or the pair creation rate of black holes in de Sitter. Both interpretations are distinguished at the quantum level. The former requires the path integral to be real and the latter to be imaginary. We develop a formalism to efficiently compute the phase of the gravitational path integral on Einstein spaces. We apply it to a broad class of spacetimes and in particular $S^2\times S^{D-2}$, finding it to be real and positive. We generalize some of the analysis to cases with charge and rotation.