Group Field Theory and Loop Quantum Gravity

Abstract

In this contribution, we introduce the group field theory (GFT) formalism for quantum gravity [1], mainly from the point of view of loop quantum gravity, arguing why it represents, in our opinion, a most promising setting for future developments. We describe the kinematical Hilbert space and its relation to the LQG one, and how the GFT quantum dynamics connects to the canonical one as well as completes spin foam models. We also discuss the problem of defining the continuum limit of such theories and of extracting effective continuum physics, highlighting the important role that GFTs can play in this respect. This is not an in-depth introduction, nor a complete review of the literature. We only outline the foundations of the formalism, survey recent results and offer a perspective on future developments. An historical prelude - Group field theories can be approached from different angles, coming from different lines of research in quantum gravity. Historically, their first appearance [2, 3] came as a development of tensor models [4] (themselves a generalisation of matrix models [5], which provided a successful quantisation of (pure) 2d gravity), allowing to make contact with state sum formulations of 3d quantum gravity (Ponzano-Regge and Turaev-Viro model), whose relation with simplicial quantum gravity, e.g. quantum Regge calculus [6], was already known, and more generally topological BF theory in any dimension. These first models were obtained by taking the simplest tensor model for 3d simplicial gravity and: 1) replacing the domain set for the tensor indices with a group manifold (SU(2)); 2) adding a gauge invariance property to the field (tensor), with the effect of introducing a gauge connection on the lattices generated by the perturbative expansion of the model. The triviality of the kinetic and interaction kernels (simple delta functions on the group) in the GFT action resulted in the amplitudes being exactly those of BF theory discretised on the same lattices (imposing flatness of the connection). Written in terms of group representations, the same amplitudes took the form of the mentioned state sums. This is the first way to understand group field theories: GFTs can be seen as tensor models enriched by algebraic data with a quantum geometric interpretation (allowing a nice encoding of discrete gravity degrees of freedom), or, equivalently, as more general class of combinatorially non-local field theories of tensorial type. The relation between state sum models of topological field theory, and their GFT formulation, and loop quantum gravity was soon pointed out in [7] (where the link to the dynamical triangulations approach [8] was also mentioned): the boundary states of such models matched the newly developed loop representation for quantum gravity [9]. Indeed, spin networks (introduced in LQG immediately afterwards) describe also the Hilbert space of GFT models. The latter acquire then a nice interpretation from the LQG perspective: GFTs are quantum field theories for spin networks, providing them with a covariant dynamics. This covariant definition started being developed a few years later [10–12]. Indeed, more interest in GFTs as quantum gravity models came with the realisation that they provide a complete definition of spin foam models for 4d gravity [13]: they capture the same quantum amplitudes as Feynman amplitudes,