they span the quantum states of the geometry. The quantum transition amplitude is given by $$W_{\triangle}(j_{l})=\mathcal{N}_{\triangle}\sum_{j_{f}}\prod_{f}A_{j_{f}}\prod_{v}A_{v}(j_{f}),\label{eq:transitionamplitudegeneral}$$ where $\mathcal{N}_{\triangle}$ is a normalization constant that can depend on the discretization, and there is an amplitude $A_{j_f}$ for each face of $\triangle$\* and an amplitude $A_v(j_f)$ for each vertex, both in 3D and 4D. Equivalently, in 3D, the amplitude $A_{j_f}$ can be defined on the segments of $\triangle$ and $A_v(j_f)$ on the tetrahedrons[^1]. Similarly, in 4D the amplitude $A_{j_f}$ can be defined on the triangles of $\triangle$ and $A_v(j_f)$ on the 4-simplexes[^2]. The partition function ($Z_{\triangle}$) in the situation with the triangulation without boundaries is the same as eq. (\[eq:transitionamplitudegeneral\]) but summing also over the boundary representations.

From now on we consider the 3-dimensional spin foam model, but it was necessary to point out in the preceding review, that the step from 3D to 4D is not a big one. In the spin network basis we have that the amplitudes $A_{j_{f}}$ are given by the dimension of the representation, $d_{j}$, and the vertex amplitude $A_{v}(j_{f})$, which implements the $SU(2)$ symmetry, is a function that takes the 6 spin quantum numbers of the representations on the 6 edges of the tetrahedron around a vertex 