Coupling of spacetime atoms in 4D spin foam models from group field theory

Abstract

We study the issue of coupling among 4-simplices in the context of spin foam models obtained from a group field theory formalism. We construct a generalisation of the Barrett-Crane model in which an additional coupling between the normals to tetrahedra, as defined in different 4-simplices that share them, is present. This is realised through an extension of the usual field over the group manifold to a five argument one. We define a specific model in which this coupling is parametrised by an additional real parameter that allows to tune the degree of locality of the resulting model, interpolating between the usual Barrett-Crane model and a flat BF-type one. Moreover, we define a further extension of the group field theory formalism in which the coupling parameter enters as a new variable of the field, and the action presents derivative terms that lead to modified classical equations of motion. Finally, we discuss the issue of renormalisation of spin foam models, and how the new coupled model can be of help regarding this. Contents

Group field theory and simplicial quantum gravity

“…This requirement is important for the complete inversion of the connection variables as a function of the bivector variables; thus, for the definition of geometric dihedral angles from them, and it is not clear, at this stage, how much of this inversion one can still perform given our weaker form of the constraints. Note also that this weaker form of the constraints is the same imposed in the Barrett-Crane model (within a different formalism), and possibly the origin of the problems faced by it (see also [35]). How the stronger form of constraints can be imposed in a GFT context is the subject of work in progress [40].…”

Spin Foam Models with Finite Groups

“…We assume for simplicity 24 that the lattice is hypercubical. One lattice direction is designated as a time direction, in which there are periodic boundary conditions.…”

“…Spin foam diagrams naturally arise as the Feynman graphs of TGFTs [1][2][3][4][5], and indeed various quantumgravity-inspired TGFTs exist in the literature, such as the Boulatov-Ooguri theories [204,205], the EPRL and FK theories [22][23][24]206], and the Baratin-Oriti theories [25,26]. As a result, they have garnered increasing attention from the quantum gravity community, since, inherently, all current spin foam models for 4d quantum gravity involve a truncation of the degrees of freedom (due to the use of a single lattice structure) and a manifest loss of diffeomorphism symmetry.…”

“…A more in-depth analysis has yet to be completed, although these preliminary results are very promising. One may modify the propagator further to obtain the EPRL, FK [22][23][24], and BO [25,26] quantum gravity spin foam amplitudes.…”

“…This is one motivation for the dynamical triangulation approach [13][14][15], causal dynamical triangulations [16,17], and tensor model/tensor group field theory [18][19][20][21][22][23][24][25][26]. See also [27,28] for another possibility to introduce a dynamical lattice.…”