Complex critical points in Lorentzian spinfoam quantum gravity: 4-simplex amplitude and effective dynamics on double-$\Delta_3$ complex

TL;DR

This paper investigates complex critical points in Lorentzian spinfoam quantum gravity, extending asymptotic analysis to non-Regge data, deriving effective theories for simplicial complexes, and numerically analyzing the double-$ riangle_3$ case.

Contribution

It introduces a general procedure to derive effective Regge geometries from spinfoam amplitudes using complex critical points, applicable to generic simplicial complexes.

Findings

Effective theory reproduces classical Regge gravity for small Barbero-Immirzi parameter.

Numerical computation of effective action on double-$ riangle_3$ complex.

Generalization of large-$j$ asymptotics to non-Regge boundary data.

Abstract

The complex critical points are analyzed in the 4-dimensional Lorentzian Engle-Pereira-Rovelli-Livine (EPRL) spinfoam model in the large-$j$ regime. For the 4-simplex amplitude, taking into account the complex critical point generalizes the large-$j$ asymptotics to the situation with non-Regge boundary data and relates to the twisted geometry. For generic simplicial complexes, we present a general procedure to derive the effective theory of Regge geometries from the spinfoam amplitude in the large-$j$ regime by using the complex critical points. The effective theory is analyzed in detail for the spinfoam amplitude on the double-$\Delta_3$ simplicial complex. We numerically compute the effective action and the solution of the effective equation of motion on the double-$\Delta_3$ complex. The effective theory reproduces the classical Regge gravity when the Barbero-Immirzi parameter $\gamma$ is small.