Approaches to Quantum Gravity

328 D. Oritiresearch. On the other hand, the overall picture of spacetime and of gravity that thisapproach is based on is definitely radical and suggests the following. There existfundamental building blocks or atoms of space, which can be combined to giverise to all sorts of geometry and topology of space. At the perturbative level spacetimeis the discrete (virtual) history of creation/annihilation of these fundamentalatoms; it has no real existence, at least no more real existence in itself than eachof the infinite possible interaction processes corresponding to individual Feynmandiagrams in any field theory. The description of this evolution is necessarily backgroundindependent (from the point of view of spacetime) because spacetime itselfis built from the bottom up and all of spacetime information has to be reconstructedfrom the information carried by the “atoms” and thus by the Feynman diagrams.At the non-perturbative level, for what we can see given the present status of thesubject, spacetime is simply not there, given that the non-perturbative properties ofQuantum Gravity would be encoded necessarily either in the GFT action, and inthe resulting equations of motion, or in the GFT partition function, and the relatedcorrelation functions, to be studied non-perturbatively, neither of which need anynotion of spacetime to be defined or analyzed. Spacetime information is thus necessarilyencoded in structures that do not use per se a notion of spacetime. Finally,therewouldbeafundamental discreteness of spacetime and a key role for causality,in the pre-geometric sense of ordering (so that it would probably be better totalk about “pre-causality”). Many of these ideas had been proposed several timesin the past, and occur in more than one other approach to Quantum Gravity, butthe GFT formalism brings all of them together within a unique framework and, assaid, expresses them in a rather conventional and powerful language.Let us sketch some examples of how traditional field theoretic methods canbe used to tackle within a new perspective some crucial open issue in QuantumGravity research. We have already mentioned some of these examples. The longstandingproblem of solving the Hamiltonian constraint equation of canonicalQuantum Gravity can be identified with the task of solving the classical GFT equationsof motion. The other long-standing issue of defining a canonical inner productfor Quantum Gravity states is turned into the task of analyzing the tree level truncationof the (perturbative expansion of the) appropriate GFT. Also, the perturbationtheory around such Quantum Gravity states would be governed, according to theabove results, by the approximation of the GFT partition function around its classicalsolutions, and this suggests a new strategy for investigating the existence ofgravitons (propagating degrees of freedom) in specific GFT/spin foam models. Themost outstanding open issue that most of the discrete non-perturbative approachesto Quantum Gravity still face is, however, that of the continuum approximation.This problem has been formulated and tackled in a variety of ways. Obviously,given the role that formalisms like dynamical triangulations, quantum Regge