Approaches to Quantum Gravity

The group field theory approach to Quantum Gravity 319of such a canonical inner product, if the resulting 2-point functions result in beingreal and positive, as for example those of the BF or Barrett–Crane models. Thedefinition is well posed, because at tree level every single amplitude Z(Ɣ) is finitewhatever the model considered due to the absence of infinite summation. Moreover,it possesses all the properties one expects from a canonical inner product: (1)it involves a sum over Feynman diagrams, and therefore triangulations, with thecylindrical topology S D−1 ×[0, 1], for closed spin networks i associated withthe two boundaries, as is easy to verify; (2) it is real and positive, but not strictlypositive; it has a non-trivial kernel that can be shown [2] to include all solutions ofthe classical GFT equations of motion, as expected. This means that the physicalHilbert space for canonical spin network states can be constructed, using the GNSconstruction, from the kinematical Hilbert space of all spin network states by quotientingout those states belonging to this kernel. This represent a concrete testableproposal for completing the definition of a loop formulation of Quantum Gravity,and a proof of the usefulness of GFT ideas and techniques. At the same time, itshows that the GFT formalism contains much more than any canonical quantumtheory of gravity, given that the last is fully contained at the “classical” level onlyof the former.17.3 Some group field theory modelsLet us now discuss some specific GFT models. The easiest example is the straightforwardgeneralization of matrix models for 2d Quantum Gravity to a GFT [20],given by the action:∫1S[φ] = dg 1 dg 2G 2 φ(g 1, g 2 )φ(g 1 , g 2 )+ λ ∫dg 1 dg 2 dg 3 φ(g 1 , g 2 )φ(g 1 , g 3 )φ(g 2 , g 3 ) (17.2)3!where G is a generic compact group, say SU(2), and the symmetries mentionedabove are imposed on the field φ implying, in this case: φ(g 1 , g 2 ) = ˜φ(g 1 g −12 ).Therelation with matrix models is apparent in momentum space, expanding the field inrepresentations j of G to give:S[φ] = ∑ j( 1dim( j)2 tr( ˜φ 2 j ) + λ )3! tr( ˜φ 3 j )(17.3)where the field modes ˜φ j are indeed matrices with dimension dim( j), sothattheaction is given by a sum of matrix models actions for increasing dimensions, or,