Quantum simplicial geometry in the group field theory formalism: reconsidering the Barrett-Crane model
Aristide Baratin, Daniele Oriti
(Submitted on 4 Aug 2011)
Using the non-commutative metric formulation of group field theories (GFT), we define a model of 4-dimensional quantum gravity as a constrained BF theory, without Immirzi parameter, encoding the quantum simplicial geometry of any triangulation used to define its quantum amplitudes. This involves a generalization of the usual GFT framework, where the usual field variables, associated to the four triangles of a tetrahedron, are supplemented by an S^3 vector playing the role of the normal to the tetrahedron. This leads naturally to projected spin network states. We give both a simplicial path integral and a spin foam formulation of the Feynman amplitudes, which correspond to a variant of the Barrett-Crane amplitudes. We then re-examin the arguments against the Barrett-Crane model(s), in light of our construction. We argue that it can still be considered a plausible quantization of 4d gravity, and that further work is needed to either confirm or refute its validity.