Thomas Thiemann, Antonia Zipfel
In a seminal paper, Kaminski, Kisielowski an Lewandowski for the first time extended the definition of spin foam models to arbitrary boundary graphs. This is a prerequisite in order to make contact to the canonical formulation of Loop Quantum Gravity (LQG) and allows to investigate the question whether any of the presently considered spin foam models yield a rigging map for any of the presently defined Hamiltonian constraint operators. The KKL extension cannot be described in terms of Group Field Theory (GFT) since arbitrary foams are involved while GFT is tied to simplicial complexes. Therefore one has to define the sum over spin foams with given boundary spin networks in an independent fashion using natural axioms, most importantly a gluing property for 2-complexes. These axioms are motivated by the requirement that spin foam amplitudes should define a rigging map (physical inner product) induced by the Hamiltonian constraint. This is achieved by constructing a spin foam operator based on abstract 2-complexes that acts on the kinematical Hilbert space of Loop Quantum Gravity. In the analysis of the resulting object we are able to identify an elementary spin foam transfer matrix that allows to generate any finite foam as a finite power of the transfer matrix. It transpires that the sum over spin foams, as written, does not define a projector on the physical Hilbert space. This statement is independent of the concrete spin foam model and Hamiltonian constraint. However, the transfer matrix potentially contains the necessary ingredient in order to construct a proper rigging map in terms of a modified transfer matrix.
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http://arxiv.org/abs/1307.5885
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