Approaches to Quantum Gravity

17The group field theory approach to Quantum GravityD. ORITI17.1 Introduction and motivationGroup field theories (GFTs) [1; 2] were developed at first as a generalization ofmatrix models for 2d Quantum Gravity to 3 and 4 spacetime dimensions to producea lattice formulation of topological theories. More recently, they have been developedfurther in the context of spin foam models for Quantum Gravity, as a toolto overcome the limitations of working with a fixed lattice in the non-topologicalcase. In our opinion, however, GFTs should be seen as a fundamental formulationof Quantum Gravity and not just as an auxiliary tool. The bottom line of thisperspective, here only tentatively outlined and still to be fully realized, hopefully,after much more work, can be summarized as follows: GFTs are quantum fieldtheories of spacetime (as opposed to QFTs on spacetime), that describe the dynamicsof both its topology and geometry in local, simplicial, covariant, algebraicterms, and that encompass ideas and insights from most of the other approachesto non-perturbative Quantum Gravity. We have just began to explore the structureof these models, but there is already some evidence, in our opinion, that in the GFTframework lies the potential for important developments.The idea of defining a quantum field theory of geometry, i.e. a QFT on superspace(the space of 3-geometries) for given spatial topology, say S 3 , was alreadyexplored in the past [3; 4; 5]. The context was then a global or “quantum cosmology”one. Such a theory would produce, in its perturbative expansion, a sumover different topologies each corresponding to a possible Feynman graph, i.e. toa possible interaction process for “universes” represented by the basic 3-sphere.The spatial topology change would be limited therefore to a changing numberof disjoint copies of S 3 . The field would represent a second quantization of thecanonical wave function on superspace, here describing the “one-particle sector”of the theory. 1 The quantum amplitude for each Feynman graph, corresponding to1 The 3-metric being itself a field, this second quantization of what is already a field theory was dubbed “thirdquantization”.Approaches to Quantum Gravity: Toward a New Understanding of Space, Time and Matter, ed. Daniele Oriti.Published by Cambridge University Press. c○ Cambridge University Press 2009.