New spinfoam vertex for quantum gravity

We introduce a new spinfoam vertex to be used in models of 4d quantum gravity based on SU(2) and SO(4) BF theory plus constraints. It can be seen as the conventional vertex of SU(2) BF theory, the $15j$ symbol, in a particular basis constructed using SU(2) coherent states. This basis makes the geometric interpretation of the variables transparent: they are the vectors normal to the triangles within each tetrahedron. We study the condition under which these states can be considered semiclassical, and we show that the semiclassical ones dominate the evaluation of quantum correlations. Finally, we describe how the constraints reducing BF to gravity can be directly written in terms of the new variables, and how the semiclassicality of the states might improve understanding the correct way to implement the constraints.

Figure 1

Plots of (41) for the equilateral configuration $j_i=j_0\forall i$. The dots represent the exact numerical evaluation, whereas the line is the Gaussian (42). The left panel shows the case $j_0=10$, whereas the right panel shows the case $j_0=100$. In the small spin case, the Gaussian approximation is already capturing the right behavior, and it becomes very accurate in the large spin case.

Figure 2

The real part of the integrand for $V=4$ as a function of $\gamma \in [0,0.2]$ (symmetric at $\pi /2$) and $\beta \in [0,\pi ]$, for fixed $\alpha$. Left panel: the equilateral case, at $j_i=100$ for all $i$. Right panel: a generic open configuration, with different spins but all of order 100.

Figure 3

Bilogarithmic plots. Left panel: the dots are the numerical evaluation of the exact norm (15) for the equilateral tetrahedron ($j_i=j$ for all $i$), for different values of the spin $j$. The line is the analytic calculation of the leading order (26). Right panel: an open case with $j_i=j$ for all $i$ but the normals not closing. The line is the analytic result (30).