Group field theory with non-commutative metric variables

We introduce a dual formulation of group field theories, making them a type of non-commutative field theories. In this formulation, the variables of the field are Lie algebra variables with a clear interpretation in terms of simplicial geometry. For Ooguri-type models, the Feynman amplitudes are simplicial path integrals for BF theories. This formulation suggests ways to impose the simplicity constraints involved in BF formulations of 4d gravity directly at the level of the group field theory action. We illustrate this by giving a new GFT definition of the Barrett-Crane model.Comment: 4 pages; v3 published versio

Infinite-Dimensional Representations of 2-Groups

A "2-group" is a category equipped with a multiplication satisfying laws like those of a group. Just as groups have representations on vector spaces, 2-groups have representations on "2-vector spaces", which are categories analogous to vector spaces. Unfortunately, Lie 2-groups typically have few representations on the finite-dimensional 2-vector spaces introduced by Kapranov and Voevodsky. For this reason, Crane, Sheppeard and Yetter introduced certain infinite-dimensional 2-vector spaces called "measurable categories" (since they are closely related to measurable fields of Hilbert spaces), and used these to study infinite-dimensional representations of certain Lie 2-groups. Here we continue this work. We begin with a detailed study of measurable categories. Then we give a geometrical description of the measurable representations, intertwiners and 2-intertwiners for any skeletal measurable 2-group. We study tensor products and direct sums for representations, and various concepts of subrepresentation. We describe direct sums of intertwiners, and sub-intertwiners--features not seen in ordinary group representation theory. We classify irreducible and indecomposable representations and intertwiners. We also classify "irretractable" representations--another feature not seen in ordinary group representation theory. Finally, we argue that measurable categories equipped with some extra structure deserve to be considered "separable 2-Hilbert spaces", and compare this idea to a tentative definition of 2-Hilbert spaces as representation categories of commutative von Neumann algebras

Non-commutative flux representation for loop quantum gravity

The Hilbert space of loop quantum gravity is usually described in terms of cylindrical functionals of the gauge connection, the electric fluxes acting as non-commuting derivation operators. It has long been believed that this non-commutativity prevents a dual flux (or triad) representation of loop quantum gravity to exist. We show here, instead, that such a representation can be explicitly defined, by means of a non-commutative Fourier transform defined on the loop gravity state space. In this dual representation, flux operators act by *-multiplication and holonomy operators act by translation. We describe the gauge invariant dual states and discuss their geometrical meaning. Finally, we apply the construction to the simpler case of a U(1) gauge group and compare the resulting flux representation with the triad representation used in loop quantum cosmology.Comment: 12 pages, matches published versio

Group field theory and simplicial gravity path integrals: A model for Holst-Plebanski gravity

In a recent work, a dual formulation of group field theories as non-commutative quantum field theories has been proposed, providing an exact duality between spin foam models and non-commutative simplicial path integrals for constrained BF theories. In light of this new framework, we define a model for 4d gravity which includes the Immirzi parameter gamma. It reproduces the Barrett-Crane amplitudes when gamma goes to infinity, but differs from existing models otherwise; in particular it does not require any rationality condition for gamma. We formulate the amplitudes both as BF simplicial path integrals with explicit non-commutative B variables, and in spin foam form in terms of Wigner 15j-symbols. Finally, we briefly discuss the correlation between neighboring simplices, often argued to be a problematic feature, for example, in the Barrett-Crane model.Comment: 26 pages, 1 figur

Quantum gravity as a group field theory: a sketch

We give a very brief introduction to the group field theory approach to quantum gravity, a generalisation of matrix models for 2-dimensional quantum gravity to higher dimension, that has emerged recently from research in spin foam models.Comment: jpconf; 8 pages, 9 figures; to appear in the Proceedings of the Fourth Meeting on Constrained Dynamics and Quantum Gravity, Cala Gonone, Italy, September 12-16, 200

Bubbles and jackets: new scaling bounds in topological group field theories

We use a reformulation of topological group field theories in 3 and 4 dimensions in terms of variables associated to vertices, in 3d, and edges, in 4d, to obtain new scaling bounds for their Feynman amplitudes. In both 3 and 4 dimensions, we obtain a bubble bound proving the suppression of singular topologies with respect to the first terms in the perturbative expansion (in the cut-off). We also prove a new, stronger jacket bound than the one currently available in the literature. We expect these results to be relevant for other tensorial field theories of this type, as well as for group field theory models for 4d quantum gravity.Comment: v2: Minor modifications to match published versio

Group field theory formulation of 3d quantum gravity coupled to matter fields

We present a new group field theory describing 3d Riemannian quantum gravity coupled to matter fields for any choice of spin and mass. The perturbative expansion of the partition function produces fat graphs colored with SU(2) algebraic data, from which one can reconstruct at once a 3-dimensional simplicial complex representing spacetime and its geometry, like in the Ponzano-Regge formulation of pure 3d quantum gravity, and the Feynman graphs for the matter fields. The model then assigns quantum amplitudes to these fat graphs given by spin foam models for gravity coupled to interacting massive spinning point particles, whose properties we discuss.Comment: RevTeX; 28 pages, 21 figure

Hidden Quantum Gravity in 4d Feynman diagrams: Emergence of spin foams

We show how Feynman amplitudes of standard QFT on flat and homogeneous space can naturally be recast as the evaluation of observables for a specific spin foam model, which provides dynamics for the background geometry. We identify the symmetries of this Feynman graph spin foam model and give the gauge-fixing prescriptions. We also show that the gauge-fixed partition function is invariant under Pachner moves of the triangulation, and thus defines an invariant of four-dimensional manifolds. Finally, we investigate the algebraic structure of the model, and discuss its relation with a quantization of 4d gravity in the limit where the Newton constant goes to zero.Comment: 28 pages (RevTeX4), 7 figures, references adde

Regge calculus from a new angle

In Regge calculus space time is usually approximated by a triangulation with flat simplices. We present a formulation using simplices with constant sectional curvature adjusted to the presence of a cosmological constant. As we will show such a formulation allows to replace the length variables by 3d or 4d dihedral angles as basic variables. Moreover we will introduce a first order formulation, which in contrast to using flat simplices, does not require any constraints. These considerations could be useful for the construction of quantum gravity models with a cosmological constant.Comment: 8 page

The 1/N expansion of colored tensor models in arbitrary dimension

In this paper we extend the 1/N expansion introduced in [1] to group field theories in arbitrary dimension and prove that only graphs corresponding to spheres S^D contribute to the leading order in the large N limit.Comment: 4 pages, 3 figure