Complex critical points in Lorentzian spinfoam quantum gravity: Four-simplex amplitude and effective dynamics on a double-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mi mathvariant="normal">Δ</mml:mi><mml:mn>3</mml:mn></mml:msub></mml:math> complex

Abstract

The complex critical points are analyzed in the four-dimensional Lorentzian Engle-Pereira-Rovelli-Livine spinfoam model in the large-$j$ regime. For the four-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-${\mathrm{\ensuremath{\Delta}}}_{3}$ simplicial complex. We numerically compute the effective action and the solution of the effective equation of motion on the double-${\mathrm{\ensuremath{\Delta}}}_{3}$ complex. The effective theory reproduces the classical Regge gravity when the Barbero-Immirzi parameter $\ensuremath{\gamma}$ is small.