Approaches to Quantum Gravity

The group field theory approach to Quantum Gravity 313along the face labeled by the representation J 2 , and effected by the contractionof the corresponding vector indices (of course, states corresponding to disjoint(D − 1)-simplices are also allowed). The corresponding state in configuration variablesis: ∫ dg 2 φ(g 1 , g 2 , ..., g D )φ( ˜g 1 , g 2 , ..., ˜g D ). We see that states of the theoryare then labeled, in momentum space, by spin networks of the group G (see chapter13 by Thiemann and chapter 15 by Perez). The second quantization of the theorypromotes these wave functions to operators, and the field theory is specified by achoice of action and by the definition of the quantum partition function. The partitionfunction is then expressed perturbatively in terms of Feynman diagrams, aswe are going to discuss. This implicitly assumes a description of the dynamics interms of creation and annihilation of (D − 1)-simplices, whose interaction generatesa (discrete) spacetime as a particular interaction process (Feynman diagram)[7]. This picture has not been worked out in detail yet, and no clear Fock structureon the space of states has been constructed. Work on this is in progress [11].Spacetime, represented by a D-dimensional simplicial complex, emerges in perturbativeexpansion as a particular interaction process among (D − 1)-simplices,described as an ordinary QFT Feynman diagram. It is then easy to understand thechoice of classical field action in group field theories. This action, in configurationspace, has the general structure:S D (φ, λ) = 1 2+(∏ D ∫i=1λ(D + 1)!)dg i d ˜g i φ(g i )K(g i ˜g −1i)φ( ˜g i )⎛D+1∏∫⎝i̸= j=1dg ij⎞⎠ φ(g 1 j )...φ(g D+1 j ) V(g ij g −1ji), (17.1)where the choice of kinetic and interaction functions K and V define the specificmodel. The interaction term describes the interaction of D + 1 (D−1)-simplices toform a D-simplex by gluing along their (D−2)-faces (arguments of the fields). Thenature of this interaction is specified by the choice of function V. The (quadratic)kinetic term involves two fields each representing a given (D − 1)-simplex seenfrom one of the two D-simplices (interaction vertices) sharing it, so that the choiceof kinetic functions K specifies how the information and therefore the geometricdegrees of freedom corresponding to their D (D−2)-faces are propagated from onevertex of interaction (fundamental spacetime event) to another. What we have thenis an almost ordinary field theory, in that we can rely on a fixed background metricstructure, given by the invariant Killing–Cartan metric, and the usual splittingbetween kinetic (quadratic) and interaction (higher order) terms in the action, thatwill later allow for a straightforward perturbative expansion. However, the actionis also non-local in that the arguments of the D + 1 fields in the interaction term