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New basis for atoms of spatial geometry (intertwiners)

  1. May 29, 2013 #1


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    Here's the introduction of the paper by Freidel and Hnybida. Quantum geometry is built up of chunks of geometry that contain information relating to volume, areas, angles made with neighbor chunks, etc. The Hilbert space that these chunks (called intertwiners) live in needs a set of basis vectors. The two sets of basis vectors proposed so far have drawbacks. Freidel and Hnybida have discovered a third set of basis vectors that will be better to work with and reveals a Regge limit.
    ==quote http://arxiv.org/abs/1305.3326 page 1==
    It has been observed long ago that the compositions of quantum states of angular momentum are related to geometrical objects [1–3]. The simplest example is the Clebsch-Gordan coefficients which vanish unless the spins satisfy the so called triangle relations. A less trivial example is the Wigner 6j symbol which vanishes unless the spins can represent the edge lengths of a tetrahedron. This insight was one of the motivations which led Ponzano and Regge to use the 6j symbol as the building block for a theory of quantum gravity in three dimensions [4, 5], together with the fact that the asymptotic limit of the 6j symbol is related to the discretized version of the Einstein-Hilbert action. In higher dimensions this line of thought led to the idea of spin foam models which are a generalization of Ponzano and Regge’s idea to a four dimensional model of quantum General Relativity. For a review see [6].
    Following a canonical approach, Loop Quantum Gravity came to the same conclusion: Geometrical quantities such area and volume are quantized [7]. In fact there are many remarkable similarities between LQG and spin foam models. For instance, the building blocks of both models are SU(2) intertwiners which represent quanta of space [8–10]. An intertwiner is simply an invariant tensor on the group. In other words if we denote by Vj the 2j + 1 dimensional vector space representing spin j then an intertwiner is an element of the SU(2) invariant subspace of this tensor product which we will denote
    Hj1,...,jn ≡InvSU(2)[Vj1 ⊗···⊗Vjn].
    The vectors in this Hilbert space will be referred to as n-valent intertwiners since they are represented graphically by an n-valent node. The legs of this node carry the spins ji which can be interpreted as the areas of the faces of a polyhedron [11–14] which is a consequence of the celebrated Guillemin-Sternberg theorem [15]. In this paper we will be focused on 4-valent intertwiners but many of the methods developed here can be extended to the n-valent case.
    Last edited: May 29, 2013
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  3. May 29, 2013 #2


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    It's sort of boring that there aren't any spare structures "chunk of geometry" free to braid into specific structures that can yield specific fields, like string theory does.
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