Approaches to Quantum Gravity

322 D. Oritior SO(3, 1)) [8; 9; 10]. This restriction can be imposed at the GFT level, startingfrom the GFT describing 4D BF theory, by projecting down the arguments of thefield from G = SO(4) (SO(3, 1)) to the homogeneous space SO(4)/SO(3) ≃ S 3(SO(3, 1)/SO(3) or SO(3, 1)/SO(2, 1) in the Lorentzian case), exploiting thefact that only simple representations of G appear in the harmonic decompositionof functions on these spaces. The GFT action is then defined [7] as:(S[φ] = 1 ∏∫ )dg i P g P h φ(g 1 , g 2 , g 3 , g 4 )P g P h φ(g 1 , g 2 , g 3 , g 4 )2iSO(4)( 10∏+ λ ∫ )[Pgdg i P h φ(g 1 , g 2 , g 3 , g 4 )P g P h φ(g 4 , g 5 , g 6 , g 7 )5!i=1SO(4)P g P h φ(g 7 , g 8 , g 3 , g 9 )P g P h φ(g 9 , g 5 , g 2 , g 10 )P g P h φ(g 10 , g 8 , g 6 , g 1 ) ](17.6)where the projection P h φ(g i ) = ∏ ∫i SO(3) dh i φ(g i h i ) from the group to thehomogeneous space imposes the wanted constraints on the representations, andthe projection P g φ(g i ) = ∫ SO(4) dg φ(g i g) ensures that gauge invariance is maintained.Different variations of this model, resulting in different edge amplitudes A e ,can be constructed [18; 19; 9; 10] by inserting the two projectors P h and P g in theaction in different combinations. The corresponding Feynman amplitudes are:Z(Ɣ) = ∑ ∏dim(J f ) ∏ A e (J f |e ) ∏ V BC (J f |v ), (17.7)J f fevwhere dim(J f ) is the measure for the representation J f , labeling the faces ofthe 2-complex/Feynman graph, entering the harmonic decomposition of the deltafunction on the group, and the function V BC (J f |v ), depending on the ten representationslabeling the ten faces of Ɣ incident to the same vertex v is the so-calledBarrett–Crane vertex [8; 9; 10].The above Feynman amplitudes can be justified in various ways, e.g. startingfrom a discretization of classical BF theory and a subsequent imposition of theconstraints [9; 10], and there is a good consensus on the fact that the Barrett–Cranevertex amplitude captures at least some of the properties needed by a spin foamdescription of 4D Quantum Gravity. Also [27], for configurations correspondingto non-degenerate simplicial geometries the asymptotic limit of the Barrett–Craneamplitude V BC (J) is proportional to the cosine of the Regge action, i.e. a correctdiscretization of General Relativity.All the above models share the following properties: (1) their Feynman amplitudesare real; (2) no unique orientation for the (various elements of the) triangulationdual to any Feynman graph can be reconstructed from the amplitudeassociated with it; (3) in Quantum Gravity models, the asymptotic limit of the