Volume simplicity constraint in the Engle-Livine-Pereira-Rovelli spin foam model

We propose a quantum version of the quadratic volume simplicity constraint for the Engle-Livine-Pereira-Rovelli spin foam model. It relies on a formula for the volume of 4-dimensional polyhedra, depending on its bivectors and the knotting class of its boundary graph. While this leads to no further condition for the 4-simplex, the constraint becomes nontrivial for more complicated boundary graphs. We show that, in the semiclassical limit of the hypercuboidal graph, the constraint turns into the geometricity condition observed recently by several authors.

Figure 1

A 4-simplex, as polytope $P$. The boundary graph of $P$ is the complete graph in five vertices and looks itself like a 4-simplex. The graph can be rearranged to have only one crossing (additionally, we have moved one vertex to the infinitely far point, to make the image clearer).

Figure 2

With orientations, there are two types of crossings $C$ of edge in a graph $\mathrm{\Gamma }$. They receive a crossing number $\sigma (C)=\pm 1$.

Figure 3

A Hopf link, i.e. two closed, linked curves, embedded in $S^3$.

Figure 4

Left: The boundary of a hypercuboid, where every 3-dimensional polyhedron is a cuboid. Right: Its dual graph, where each of the cuboids corresponds to one vertex. Vertex number 8 sits at the infinitely far point in $S^3$.

Figure 5

The six independent loops in the hypercuboid graph $\mathrm{\Gamma }$, color-coded. Each such loop corresponds to one great circle on $S^3$. Two pairs form, respectively, one of the three independent Hopf links $H_1$, $H_2$, $H_3$ in $\mathrm{\Gamma }$. The orientation of the edges in $\mathrm{\Gamma }$ are such that each of the six crossings $C$ has $\sigma (C)=+1$.