Barrett-Crane spin foam model from generalized BF-type action for gravity

TL;DR

This work shows that gravity expressed as a generalized BF-type action (CMPR) discretizes and quantizes to the Barrett-Crane spin foam model in both Euclidean and Lorentzian settings. By mapping the B field to a tetrad-based E field, the authors demonstrate that the resulting constraints reduce to the Plebanski simplicity constraints, yielding Barrett-Crane simple representations at the quantum level and precluding an Immirzi parameter in the spin foam. They discuss an alternative Reisenberger discretization and contrast the covariant spin foam results with canonical loop quantum gravity, particularly regarding the area spectrum and the appearance of the Immirzi parameter. The results suggest a universality of the Barrett-Crane framework across generalized BF actions and raise important questions about connections between models and the physical interpretation of geometric observables.

Abstract

We study a generalized action for gravity as a constrained BF theory, and its relationship with the Plebanski action. We analyse the discretization of the constraints and the spin foam quantization of the theory, showing that it leads naturally to the Barrett-Crane spin foam model for quantum gravity. Our analysis holds true in both the Euclidean and Lorentzian formulation.