Complex critical points and curved geometries in four-dimensional Lorentzian spinfoam quantum gravity

TL;DR

This paper demonstrates through numerical analysis that curved geometries emerge as dominant contributions in the semiclassical limit of 4D Lorentzian spinfoam quantum gravity, resolving the flatness problem and supporting the model's consistency.

Contribution

The work explicitly finds curved Regge geometries from Lorentzian EPRL spinfoam amplitudes, showing their dominance and relation to complex critical points, thus resolving the flatness problem.

Findings

Curved Regge geometries are explicitly obtained from the amplitude.

Dominant contributions are proportional to the exponential of the Regge action.

The results support the semiclassical consistency of the spinfoam model.

Abstract

This paper focuses on the semiclassical behavior of the spinfoam quantum gravity in 4 dimensions. There has been long-standing confusion, known as the flatness problem, about whether the curved geometry exists in the semiclassical regime of the spinfoam amplitude. The confusion is resolved by the present work. By numerical computations, we explicitly find curved Regge geometries from the large-$j$ Lorentzian Engle-Pereira-Rovelli-Livine (EPRL) spinfoam amplitudes on triangulations. These curved geometries are with small deficit angles and relate to the complex critical points of the amplitude. The dominant contribution from the curved geometry to the spinfoam amplitude is proportional to $e^{i \mathcal{I}}$, where $\mathcal{I}$ is the Regge action of the geometry plus corrections of higher order in curvature. As a result, the spinfoam amplitude reduces to an integral over Regge geometries weighted by $e^{i \mathcal{I}}$ in the semiclassical regime. As a byproduct, our result also provides a mechanism to relax the cosine problem in the spinfoam model. Our results provide important evidence supporting the semiclassical consistency of the spinfoam quantum gravity.