Polyhedra in loop quantum gravity

Intertwiners are the building blocks of spin-network states. The space of intertwiners is the quantization of a classical symplectic manifold introduced by Kapovich and Millson. Here we show that a theorem by Minkowski allows us to interpret generic configurations in this space as bounded convex polyhedra in ${\mathbb{R}}^3$: A polyhedron is uniquely described by the areas and normals to its faces. We provide a reconstruction of the geometry of the polyhedron: We give formulas for the edge lengths, the volume, and the adjacency of its faces. At the quantum level, this correspondence allows us to identify an intertwiner with the state of a quantum polyhedron, thus generalizing the notion of the quantum tetrahedron familiar in the loop quantum gravity literature. Moreover, coherent intertwiners result to be peaked on the classical geometry of polyhedra. We discuss the relevance of this result for loop quantum gravity. In particular, coherent spin-network states with nodes of arbitrary valence represent a collection of semiclassical polyhedra. Furthermore, we introduce an operator that measures the volume of a quantum polyhedron and examine its relation with the standard volume operator of loop quantum gravity. We also comment on the semiclassical limit of spin foams with nonsimplicial graphs.

Figure 1

A polygon with side vectors $A_i{n}_i$ and the ($F-3$) independent diagonals. The space of possible polygons in ${\mathbb{R}}^3$ up to rotations is a ($2F-6$)-dimensional phase space, with action-angle variables the pairs $({\mu }_i,{\theta }_i)$ of the diagonal lengths and dihedral angles. For noncoplanar normals, the same data define also a unique polyhedron thanks to Minkowski’s theorem.

Figure 2

Some examples of Schlegel diagrams. From left to right, a tetrahedron, a pyramid, a cube, and a dodecahedron.

Figure 3

Polyhedra with 5 faces: The two possible classes are the triangular prism (left panel) and the pyramid (right panel). The two classes differ in the polygonal faces and in the number of vertices.

Figure 4

The seven classes of polyhedra with 6 faces, grouped according to the dimensionality of their configurations.

Figure 5

Pictorial representation of the phase space: It can be mapped into regions corresponding to the various dominant classes (two in the example). The subdominant classes separate the dominant ones and span measure-zero subspaces.

Figure 6

Mappings of subspaces of ${\mathcal{S}}_6$ realized by using the reconstruction algorithm of Sec. 3 and Wolfram’s Mathematica. We subdivided the phase space into a regular grid and had Mathematica computing the adjacency matrix of the area-normal configurations lying at the center of the cells. This associates a unique class to each cell of the phase space. The information is color-coded: cuboids in blue, pentagonal wedges in red. With this mapping of finite resolution we have measure-zero probability of hitting a subdominant class; thus, the latter are absent in the figures. The holes are configurations for which our numerical algorithm failed. Concerning the specific values of the example, the areas are taken to be (9, 10, 11, 12, 13, 13). In the left panel, we fixed ${\mu }_1=15$, ${\theta }_1=\frac{7}{10}\pi$, ${\mu }_2=13$, and ${\theta }_2=\frac{13}{10}\pi$ and plotted the remaining pair $({\mu }_3,{\theta }_3)$. In the right panel, we fixed ${\mu }_i=(15,13,17)$ and plotted the three angles ${\theta }_i$.

Figure 7

A polyhedron with $F=100$ drawn with Wolfram’s Mathematica, by using the reconstruction algorithm. The example has all areas equal and normals uniformly distributed on a sphere. Notice that most faces have valence 6 and that triangles are nowhere to be seen.

Figure 8

The geometric meaning of Eq. (35): The 2D angle ${\alpha }_{ij,kl}$ belonging to the shaded triangle can be expressed in terms of 3D angles associated the thick edges of the tetrahedron $k$ or, equivalently, of the tetrahedron $l$.

Figure 9

Some eigenvalues of $\widehat{V}$. For comparison, the curve is the classical volume of an equilateral tetrahedron as a function of the area $A=j$ (units $8\pi \gamma L_P^2=1$). The empty circles are single eigenvalues; the full circles have double degeneracy. The spectrum is gapped and bounded from the above by the classical maximal volume, which provides a large spin asymptote.

Figure 10

Left panel: Some eigenvalues of $\widehat{\tilde{V}}$. For comparison, the curve is the classical volume of an equilateral tetrahedron as a function of the area $A=j$ (units $8\pi \gamma L_P^2=1$). All but the zero eigenvalue have double degeneracies. Right panel: The same region of the spectrum for Barbieri’s operator ${\widehat{V}}_B$. Notice that here the asymptotic curve is the equilateral volume with areas $A=\sqrt{j(j+1)}$.