Approaches to Quantum Gravity

The group field theory approach to Quantum Gravity 323vertex amplitude gives (in the non-degenerate sector) the cosine of the Reggeaction instead of the exponential of it. These properties suggest the interpretation[24] of the corresponding models as defining the Quantum Gravity analog of theHadamard function for a relativistic particle, and, as said, are the wanted propertiesif we seek a GFT definition of the canonical/Hamiltonian inner product. 4 However,there are several reasons why one may want to go beyond this type of structure.(1) From the point of view of a field theory on the simplicial superspace we areadvocating here, the most natural object one would expect a GFT to define with its2-point functions is not a canonical inner product, solution of the Hamiltonian constraint,but a Green function for it. This is what happens in ordinary QFT, for thefree theory, and in the formal context of continuum third quantization for QuantumGravity, where the (free theory) Feynman amplitudes correspond to the usual pathintegral for Quantum Gravity, with amplitude given by the exponential of the GRaction [3; 4; 5], which is a Green function for the Hamiltonian constraint, and nota solution of the same, in each of its arguments. (2) The orientation of the GFT2-complexes can be given, for Lorentzian models, a causal interpretation [28; 29],and thus the orientation independence of the usual models suggests that one shouldbe able to construct other types of models defining causal Quantum Gravity transitionamplitudes [28; 29] and corresponding GFTs. (3) No clear meaning can begiven from the Hamiltonian/canonical perspective to the GFT amplitudes for Feynmangraphs beyond the tree level, when spatial topology change is present. For allthese reasons one would like to have a more general class of GFT models thatdo depend on the orientation of the GFT Feynman graphs, that can be interpretedconsistently as analogs of causal transition amplitudes of QFT, that are in moredirect contact with usual path integral formulations of (simplicial) gravity, and thatreduce to the above type of models when suitably restricted. A class of modelsthat achieves this was constructed in [24]. Here a generalized version of the GFTformalism was defined, for a field φ(g i , s i ) : (G × R) ⊗4 → C:S gen = ∑ μ,α+ ∑ μ144∏∫i=1∑α idg i∫λ {αi ,μ}5!R[ ∏ ( )ds i{φ ] }−μα (g i , s i ) −iμα∂si +∇ i φ μα (g i , s i )i5∏∫ ∫{dg ij ds ij Ph φ μα 1(g 1 j , s 1 j )P h φ μα 2(g 2 j , s 2 j )i̸= j=1GR...P h φ μα 5(g 5 j , s 5 j ) ∏ θ(α i s ij + α j s ji )K ( g ij , g ji ; μ(α i s ij + α j s ji ) )} ,4 In other words, the Feynman amplitudes of these GFT models would correspond not to a simplicial versionof the path integral formalism for Quantum Gravity, but to the symmetrized version of the same over oppositespacetime orientations, that indeed gives a path integral definition of solutions of the Hamiltonian constraintoperator of canonical Quantum Gravity [26].