- #1

- 24,775

- 792

==quote==

http://arxiv.org/abs/1009.3402

**Polyhedra in loop quantum gravity**

Eugenio Bianchi, Pietro Doná, Simone Speziale

32 pages, many figures

(Submitted on 17 Sep 2010)

"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 a polyhedron. 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."

==endquote==

To put it differently, we have already seen the CDT program get interesting results just by gluing tetrahedra together in different ways (and doing the 4D analog of that, as well.)

You get curvature, using tet building blocks, the same way you get curvature in a 2D surface made of identical equilateral triangles---where you can put more or less than 6 around a given point and putting exactly 6 makes it flat there. CDT introduces quantum rules for gluing the tets, and so it gets an uncertain quantum geometry--one which has turned out to be very interesting.

So by analogy we can ask

**what about LQG?**Does it have something like building blocks? If so, how should we imagine them?