Approaches to Quantum Gravity

316 D. Oritidiagrams correspond to manifolds is identified and discussed at length in [17]. Allthe relevant conditions can be checked algorithmically on any given Feynman diagram.It is not clear, at present, whether one can construct suitably constrained GFTmodels satisfying these conditions, thus generating only manifold-like complexesin their Feynman expansion.Each strand carries a field variable, i.e. a group element in configuration spaceor a representation label in momentum space. Therefore in momentum space eachFeynman diagram is given by a spin foam (a 2-complex with faces labeled byrepresentation variables), and each Feynman amplitude (a complex function of therepresentation labels, obtained by contracting vertex amplitudes with propagatorfunctions) by a spin foam model (see chapter 15 by Perez):Z(Ɣ) = ∑ ∏A(J f ) ∏ A e (J f |e ) ∏ A v (J f |v ).J f fevAs in all spin foam models, the representation variables have a geometricinterpretation (edge lengths, areas, etc.) (see [9; 10]) and so each of theseFeynman amplitudes corresponds to a definition of a sum-over-histories for discreteQuantum Gravity on the specific triangulation dual to the Feynman diagram,although the quantum amplitudes for each geometric configuration are not necessarilygiven by the exponential of a discrete gravity action. For more on thequantum geometry behind spin foam models we refer to the literature [9; 10; 28].One can show that the inverse is also true: any local spin foam model can beobtained from a GFT perturbative expansion [13; 2]. This implies that the GFTapproach subsumes the spin foam approach at the perturbative level, while at thesame time going beyond it, since there is of course much more in a QFT thanits perturbative expansion. The sum over Feynman diagrams gives then a sumover spin foams (histories of the spin networks on the boundary in any scatteringprocess), and equivalently a sum over triangulations, augmented by a sum overalgebraic data (group elements or representations) with a geometric interpretation,assigned to each triangulation. Expectation values of GFT observables can also beevaluated perturbatively. These are given [2] by gauge invariant combinations ofthe basic field operators that can be constructed in momentum space using spinnetworks according to the formula⎛⎞O =(γ, je ,i v )(φ) = ⎝ ∏ ∫dg ij dg ji⎠ (γ, je ,i v )(g ij g −1ji) ∏ φ(g ij ),(ij)iwhere (γ, je ,i v )(g) identifies a spin network functional for the spin network labeledby a graph γ with representations j e associated to its edges and intertwiners i v associatedto its vertices, and g ij are group elements associated to the edges (ij) of γ