and returns a complex number. The vertex amplitude in this case is given mainly by the Wigner 6j-symbol $\{6j\}=\left\{ \begin{array}{ccc}
j_{1} & j_{2} & j_{3}\\
j_{4} & j_{5} & j_{6}
\end{array}\right\} $[^3], and the transition amplitude is $$W_{\triangle}(j_{l})=\mathcal{N}_{\triangle}\sum_{j_{f}}\prod_{f}(-1)^{j_{f}}d(j_{f})\prod_{v}(-1)^{j_{v}}\{6j\},\label{eq:transitionamplitude3d6j}$$ where $j_{v}=\sum_{a=1}^{6}j_{a}$.

Note that even with the definition of this object on a truncation of the triangulation, still there are non-physical divergences. We see this by first writing explicitly eq. (\[eq:transitionamplitude3d6j\]) for a triangulation made of 1 tetrahedron. The dual is made of a vertex inside the tetrahedron connected to a node on each of the 4 triangular faces. The dual has 6 boundary faces that we label with 6 $SU(2)$ representations (or equivalently there is a dual tetrahedron and the representations are living on its edges) and the transition amplitude is immediately $$W_{\triangle_{1}}(j_{l})=(-1)^{j_{v}}\left\{ \begin{array}{ccc}
j_{1} & j_{2} & j_{3}\\
j_{4} & j_{5} & j_{6}
\end{array}\right\} \label{eq:transitionamplitudedelta1}$$ and is well defined, although for the partition function one should sum over the infinite number of $SU(2)$, $j_{f}$, representations. The problem for transition amplitudes occurs already for the triangulation with only four tetrahedrons, made by inserting a point inside the initial large one and connecting it with the 4 tetrahedron points. The dual is 