Approaches to Quantum Gravity

The group field theory approach to Quantum Gravity 317that meet at the vertex i. In particular, the transition amplitude (probability amplitudefor a certain scattering process) between certain boundary data represented bytwo spin networks, of arbitrary combinatorial complexity, can be expressed as theexpectation value of the field operators having the same combinatorial structure ofthe two spin networks [2]:∫〈 1 | 2 〉=Dφ O 1 O 2 e −S(φ) =∑Ɣ/∂Ɣ=γ 1 ∪γ 2λ Nsym[Ɣ] Z(Ɣ),where the sum involves only 2-complexes (spin foams) with boundary given by thetwo spin networks chosen.The above perturbative expansion involves therefore two very different types ofsums: one is the sum over geometric data (group elements or representations ofG) which is the GFT analog of the integral over momenta or positions of usualQFT; the other is the sum over Feynman diagrams. This includes a sum over alltriangulations for a given topology and a sum over all topologies (since all possiblegluings of D-simplices and face identifications are present by constructionin the GFT Feynman expansion). Both sums are potentially divergent. First of allthe naive definition of the Feynman amplitudes implies a certain degree of redundancy,resulting from the symmetries of the defining GFT. A proper gauge fixingof these symmetries, especially those whose group is non-compact, is needed toavoid divergences [16]. Even after gauge fixing, the sum over geometric data hasa potential divergence for every “bubble” of the GFT Feynman diagram, i.e. forevery closed collection of 2-cells. This is the GFT analog of loop divergences ofthe usual QFT. Of course, whether the GFT amplitudes are divergent or not dependson the specific model. 3 In general a regularization and perturbative renormalizationprocedure would be needed, but no systematic study of GFT renormalization hasbeen carried out to date, despite its obvious importance. The sum over Feynmandiagrams, on the other hand, is most certainly divergent. This is not surprising.The sum over Feynman diagrams gives a sum over all triangulations for all topologies,each weighted by a (discrete) Quantum Gravity sum-over-histories. That sucha sum can be defined constructively thanks to the simplicial and QFT setting isalready quite an achievement, and to ask for it to be finite would be really toomuch! Also, from the strictly QFT perspective, it is to be expected that the expansionin Feynman diagrams of a QFT would produce at most an asymptotic seriesand not a convergent one. This is the case for all the interesting QFTs we know of.What makes the usual QFT perturbative expansion useful in spite of its divergence3 For example, while the most natural definition of the group field theory for the Barrett–Crane spin foammodel [7], presents indeed bubble divergences, a simple modification of it [18; 19], possesses finite Feynmanamplitudes, i.e. it is perturbatively finite without the need for any regularization.