Polyhedra in loop quantum gravity

By Eugenio Bianchi, P. Donà, Simone Speziale. 2010

226 citations recorded by the source corpus.

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 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 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.

Subfields: Neurology, Nuclear and High Energy Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/acaed57aa52e6f75e980cac83a7fec86462dfab4

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Citations in the corpus

Emergent Friedmann dynamics with a quantum bounce from quantum gravity condensates (2016, background): “First, quantum states in the kinematical LQG Hilbert space (and in GFT) can be viewed [41–44] as a collection of polyhedra that are glued together across faces with equal areas (although the faces glued together need not have the same shape).”
Zakopane lectures on loop gravity (2011, background): “A good definition of the operator for general n-valent nodes (which reduces to the one on (n− 1)-valent nodes when one of the links has zero spin) is still missing (see [46] for an interesting track).”
The Spin-Foam Approach to Quantum Gravity (2012, background): “For new results on the volume see [72, 71].”