Group Field Theory and Loop Quantum Gravity

Abstract

We introduce the group field theory formalism for quantum gravity, mainly from the point of view of loop quantum gravity, stressing its promising aspects. We outline the foundations of the formalism, survey recent results and offer a perspective on future developments. GFT FROM LQG PERSPECTIVE: THE GENERAL IDEAIn 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...

Group field theories for all loop quantum gravity

“…Let us first recount the general definitions and structures of GFTs, as one finds them in the literature [14][15][16][17][18][19][20].…”

“…Group field theory (GFT) [14][15][16][17][18][19][20] also took a more central role in the quantum gravity landscape, in connection to loop quantum gravity and spin foam models. These are quantum field theories on group manifolds characterized by a peculiar non-local pairing of field variables in their interactions and motivated from both the canonical and the covariant perspective.…”

Semiclassical solutions of generalized Wheeler-DeWitt cosmology

“…In the latter, the dimensionless length of the smallest plaquette isμ. 5 In the large volume limit it will be formally self-adjoint w.r.t. the measure B(ν)dν.…”

The closure constraint for the hyperbolic tetrahedron as a Bianchi identity

“…Several proposals exist to date, such as Thiemann's original regularization [15] or the Master Constraint Program [16]. Other proposals from a more covariant perspective have been studied, as in spinfoams (for a review, see [17] and references therein), with the EPRL vertex [18], or in group field theory (see [19] for a review). One of the salient point of the dynamics, beside its definition, is its compatibility with the classical theory.…”