Approaches to Quantum Gravity

The group field theory approach to Quantum Gravity 329calculus, causal sets or loop quantum gravity can play within the group field theoryframework, the various techniques developed for them can be adapted to the GFTs.However, the field theory language that is at the forefront of the GFT approachsuggests once more new perspectives. Let us sketch them briefly.The continuum approximation issue can be seen as the search for the answer totwo different types of questions. (a) What is the best procedure to approximate adiscrete spacetime, e.g. a simplicial complex, with a continuum manifold, and toobtain some effective quantum amplitude for each geometric configuration fromthe underlying fundamental discrete model? In the context of spin foam models,this amounts to devising a background independent procedure for “coarse graining”the spin foam 2-complexes and the corresponding amplitudes [40; 41] to obtaina smooth approximation of the same. (b) If a continuum spacetime or space arenothing else than some sort of “condensate” of fundamentally discrete objects, asin some “emergent gravity” approaches (see chapter 7 by Dreyer and chapter 9by Markopoulou) and, as suggested by condensed matter analog models of gravity[42; 43], what are these fundamental constituents? What are their properties? Whatkind of (necessarily background independent) model can describe them and thewhole process of “condensation”? What are the effective hydrodynamic variablesand what is their dynamics in this “condensed or fluid phase”? How does it compareto GR?For what concerns the first (set of) question(s), the GFT approach offers a potentiallydecisive reinterpretation: since spin foam are nothing else than Feynmandiagrams of a GFT, and that spin foam models are nothing else than their correspondingFeynman amplitudes, the coarse graining of a spin foam model [40; 41],is exactly the perturbative renormalization of the corresponding GFT. On the onehand this suggests that one deal with the problem of continuum approximation ofspin foams using all the perturbative and non-perturbative renormalization grouptechniques from ordinary field theory adapted to the GFT case. On the other handgives a further justification for the idea, proposed in [41], that the Connes–KreimerHopf algebra of renormalization developed for QFT could be the right type offormalism to use in such a Quantum Gravity context.As for the second (set of) question(s), the GFT approach identifies uniquelythe basic building blocks of a quantum space, those that could be responsible forthe kind of “condensation” process or the transition to a fluid phase at the root of theemergence of a smooth spacetime in some approximation and physical regime, andgives a precise prescription for their classical and quantum dynamics, that can nowbe investigated. From this perspective, it is best interpreted as a theory of “pregeometry”in the sense discussed in the chapters by Markopoulou and Dreyer. Inparticular, one could develop a statistical mechanics picture for the dynamics of theGFT “atoms” of space, and then the above idea of a “condensation” or in general