<i>Mathematica</i> program for numerically computing real and complex critical points in four-dimensional Lorentzian spinfoam amplitudes

This work develops a comprehensive algorithm and a Mathematica program to construct boundary data and compute real and complex critical points in spin foam amplitudes. Our approach covers both spacelike tetrahedra and triangles in the Engle-Pereira-Rovelli-Livine model and timelike tetrahedra and triangles in the Conrady-Hnybida extension, aiming at addressing a wide range of physical scenarios such as cosmology and black holes. Starting with a single 4-simplex, we explain how to numerically construct boundary data and corresponding real critical points from any nondegenerate 4-simplex geometry. Extending this to the simplicial complex, we demonstrate the algorithm for constructing boundary data and critical points using examples with two 4-simplices sharing an internal tetrahedron. By revisiting the ${\mathrm{\ensuremath{\Delta}}}_{3}$ triangulation with curved geometry, we demonstrate the numerical computation of the real critical point corresponding to the flat geometry and the deformation to the complex critical points. Additionally, the program evaluates the spin foam action at the critical points and compare to the Regge action.