Properties of 4D spinfoam quantum geometry: Results from next-to-leading order spinfoam large-$j$ asymptotics of 1-5 Pachner move

TL;DR

This paper investigates the geometric properties influencing spinfoam amplitudes in 4D quantum geometry, revealing how volume variations and normal directions affect amplitude magnitudes at different orders.

Contribution

It introduces criteria to analyze 4D geometry effects on spinfoam amplitudes and provides numerical evidence on how these geometric factors influence amplitude behavior at next-to-leading order.

Findings

Large volume standard deviation correlates with large amplitudes.

Small 4-simplex volume enhances amplitude magnitude.

Normals close to null direction significantly impact next-to-leading order amplitudes.

Abstract

This paper proposes several criteria to probe the non-trivialities of 4-dimensional geometry that impact spinfoam amplitude. These criteria include the standard deviation of 4-volumes of the constituting 4-simplices, the smallest 4-simplex volume, and whether the directions of tetrahedron 4-normals are close to the null direction. By numerically computing and analyzing the spinfoam amplitudes up to the next-to-leading order of 1-5 Pachner move samples with the same boundary 4-simplex, we reveal the relationship between 4-dimensional geometry and spinfoam amplitudes, as large standard deviation of 4-simplex volumes and small 4-simplex volume can result in both leading order and next-to-leading order amplitudes being large. Furthermore, the numerical result indicates that the distance from tetrahedron 4-normals to the null direction has a greater impact on increasing the next-to-leading order amplitude than on the leading order amplitude, making it primarily a quantum effect.