Polyhedra in loop quantum gravity

TL;DR

This paper establishes a geometric interpretation of intertwiners in loop quantum gravity as quantum polyhedra, providing formulas for their geometry, and explores their semiclassical properties and implications for the theory.

Contribution

It introduces a novel geometric framework linking intertwiners to convex polyhedra and generalizes the quantum tetrahedron concept to arbitrary polyhedra in loop quantum gravity.

Findings

Intertwiners correspond to quantum polyhedra with face areas and normals.

Formulas for edge lengths, volume, and face adjacency of polyhedra are provided.

Coherent intertwiners are peaked on classical polyhedral geometries.

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.