Complex critical points in Lorentzian spinfoam quantum gravity: 4-simplex amplitude and effective dynamics on double-${\Delta }_3$ complex

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.

29 pages, 15 figures

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• (2+1) Lorentzian quantum cosmology from spin-foams: opportunities and obstacles for semi-classicality
• Hessian in the spinfoam models with cosmological constant