Approaches to Quantum Gravity

324 D. Oritiwhere: g i ∈ G, s i ∈ R, μ =±1andα i =±1 are orientation data that allowone to reconstruct the orientation of the Feynman graph from the complex amplitudeassociated to it, φ + (g i , s i ) = φ(g 1 , s 1 ; ..., g 4 , s 4 ) and φ − (g i , s i ) = φ † (g i , s i ),P h is the projector imposing invariance under the SO(3) subgroup, ∇ is theD’Alembertian operator on the group G, θ(s) is the step function and K (g, s)is the evolution kernel for a scalar particle on the group manifold G with evolutionparameter s. The field is assumed invariant under the diagonal action of G asdescribed above. The form of the kinetic and vertex operator impose a non-trivialdependence on the orientation data in fully covariant way. The resulting Feynmanamplitudes [24] have all the properties wanted, being complex and orientationdependent,and have the natural interpretation as analogs of Feynman transitionamplitudes for Quantum Gravity [24; 29]. Also, when expressed in terms of thevariables conjugate to the s i , the amplitude for each vertex is given by the exponentialof the Regge action in first order formalism, times an appropriate measurefactor [24]. It remains to be proven that this also holds for the amplitude associatedto the whole Feynman graph [30; 31]. Other models based on the same formalismand same type of field, but differing, for example, in the expression for the vertexterm can also be constructed, and share similar properties [31].Other types of GFTs have been constructed in the literature, ranging from aBoulatov-like model for 3d gravity based on the quantum group DSU(2) [32],with links to models of 3d Quantum Gravity coupled to matter mentioned below,to a modified version [33] of the GFTs for the Barrett–Crane models, with a tunableextra coupling among the 4-simplices and a possible use in the renormalization ofspin foam models. For all this we refer to the literature. We refer to the literaturealso for the recent construction of group field theory models for Quantum Gravitycoupled to matter fields of any mass and spin in 3d [34; 35; 36], for work inprogress concerning the 4d case (coupling of Quantum Gravity and gauge fields,of topological gravity and strings, etc.), and for the proposal of re-interpreting theconical singularities appearing in non-manifold-like Feynman graphs of GFTs asmatter fields [37].17.4 Connections with other approachesWe would like to recapitulate here some links to other approaches, and sketch a(rather speculative, at present) broader picture of GFTs as a generalized formalismfor Quantum Gravity, in which other discrete approaches can be subsumed.GFTs seek to realize a local simplicial third quantization of gravity, withdiscrete gravity path integrals as Feyman amplitudes and a sum over simplicialspacetimes of all topologies realized as a Feynman expansion. What is the exactrelationship with the more traditional path integral quantizations of simplicial