Polyhedra in loop quantum gravity

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.

32 pages, many figures. v2 minor corrections

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• Discrete Gravity Models and Loop Quantum Gravity: a Short Review
• Flux formulation of loop quantum gravity: Classical framework
• An elementary introduction to loop quantum gravity
• Intertwiner Entanglement on Spin Networks
• Loop Quantum Gravity Phenomenology: Linking Loops to Observational Physics
• Loop expansion and the bosonic representation of loop quantum gravity
• Dynamics of anisotropies close to a cosmological bounce in quantum gravity
• New action for simplicial gravity in four dimensions
• Entropy in the Classical and Quantum Polymer Black Hole Models
• On the structure of a background independent quantum theory: Hamilton function, transition amplitudes, classical limit and continuous limit
• Holonomy-flux spinfoam amplitude
• Coherent 3j-symbol representation for the loop quantum gravity intertwiner space
• A "Helium Atom" of Space: Dynamical Instability of the Isochoric Pentahedron
• Generating Functionals for Spin Foam Amplitudes
• Classical and Quantum Polyhedra
• An algorithm to reconstruct convex polyhedra from their face normals and areas
• Aspects of quantum gravity
• Statistical discrete geometry
• On Geometry and Symmetries in Classical and Quantum Theories of Gauge Gravity
• The Quantum Trihedron
• Semiclassical Length Measure from a Quantum-Gravity Wave Function