Spacetime as a Feynman diagram: the connection formulation

Abstract

Spin foam models are the path integral counterparts to loop quantized canonical theories. In the last few years several spin foam models of gravity have been proposed, most of which live on finite simplicial lattice spacetime. The lattice truncates the presumably infinite set of gravitational degrees of freedom down to a finite set. Models that can accomodate an infinite set of degrees of freedom and that are independent of any background simplicial structure, or indeed any a priori spacetime topology, can be obtained from the lattice models by summing them over all lattice spacetimes. Here we show that this sum can be realized as the sum over Feynman diagrams of a quantum 1 field theory living on a suitable group manifold, with each Feynman diagram defining a particular lattice spacetime. We give an explicit formula for the action of the field theory corresponding to any given spin foam model in a wide class which includes several gravity models. Such a field theory was recently found for a particular gravity model [1]. Our work generalizes this result as well as Boulatov's and Ooguri's models of three and four dimensional topological field theories, and ultimately the old matrix models of two dimensional systems with dynamical topology. A first version of our result has appeared in a companion paper [2]: here we present a new and more detailed derivation based on the connection formulation of the spin foam models.

Group Field Theory: An Overview

“…Conversely, given a spin foam model we can reconstruct a GFT via (12,13). This establishes the equivalence or duality between spin foam models and GFT, which was first proven in [29]. One can now check that the example (7) gives in dimension 3 the Ponzano-Regge model and in higher dimensions the discretization of topological BF .…”

Homogeneous cosmologies as group field theory condensates

“…As a consequence of this, 29) and so the states |ξ are squeezed states if the operatorsφ(g I ) −ξ g I and their Hermitian conjugates satisfy the (suitably gauge-invariant) algebra of creation and annihilation operators, 30) which will in general only be true for specific choices of ξ. Evaluating the commutator we find that…”

Group field theory and simplicial quantum gravity

“…The geometric content of this condition is revealed by taking the simplicity constraints into account as well. First of all, note that the presence of this delta function reduces the gauge invariance of the Feynman amplitude Z from the invariance under B ef → GB ef G −1 , H f →ḠH fḠ −1 for arbitrary G,Ḡ 8 , that could be deduced from the action term alone and the simplicity constraints (we will see that also the other terms μ and S c would allow for this large D Oriti symmetry) to the smaller B ef → GB ef G −1 , H f → GH f G −1 for the same group element G. This is indeed the symmetry of BF theory (see, for example, [24,25]). Consider now the simplicity constraints with a negative sign B ef = (b ef , −n e b ef n −1 e ) implying that B ef is an area bivector A f .…”