Polyhedra in loop quantum gravity

TL;DR

The paper builds a geometric bridge between loop quantum gravity intertwiners and classical polyhedra by showing that intertwiners quantize the Kapovich–Millson phase space and, via Minkowski's theorem, correspond to quantum polyhedra characterized by face areas and normals. It provides explicit reconstruction procedures to obtain edge lengths and volume from area–normal data, introduces a volume operator with correct semiclassical behavior, and connects coherent intertwiners to semiclassical polyhedra within the twisted geometry framework. The work clarifies how non-simplicial graphs can realize Regge-like discretizations and discusses the implications for spinfoams, suggesting that shape matching and polyhedral variables are key to understanding semiclassical limits on general graphs. Together, these results offer a concrete, polyhedral picture of quantum geometry in LQG and a path toward semiclassical dynamics on arbitrary cellular decompositions.

Abstract

Interwiners 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 Euclidean space: 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 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 spinfoams with non-simplicial graphs.

Figures (10)

• Figure 1: A polygon with side vectors $A_i n_i$ and the $(F-3)$ independent diagonals. The space of possible polygons in ${\mathbbm R}^3$ 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 non-coplanar normals, the same data defines 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 separe the dominant ones and span measure-zero subspaces.