Quantum polyhedra in loop quantum gravity

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Interwiners describe quanta of space in loop quantum gravity. In this talk I show that the Hilbert space of SU(2) intertwiners has as semiclassical limit the phase space of a classical system originally considered by Minkowski: convex polyhedra with N facets of given areas and normals. This result sharpens Penrose spin-geometry theorem. The knowledge of the classical system associated to intertwiner space can be fruitfully used: I show that many properties of the spectrum of the volume operator can be derived via Bohr-Sommerfeld quantization of the volume of a classical polyhedron.
Moreover, a recent derivation of the entropy of a Black Hole involves the calculation of the dimension of the associated SU(2) intertwiner space. I describe a semiclassical version of this calculation: the microstates counted are shapes of a tessellated horizon having facets of given areas and normals. The calculation reproduces the area law, together with the logarithmic corrections to the entropy.