Spin foam quantization of SO(4) Plebanski's action

TL;DR

This paper develops a spin foam model for Riemannian general relativity based on Plebanski's action, connecting it to the Barrett-Crane model and introducing a simpler alternative model for degenerate sectors.

Contribution

It systematically implements constraints in the BF theory to derive the Barrett-Crane model and proposes a new, simpler spin foam model for degenerate sectors of Plebanski's theory.

Findings

The Barrett-Crane model is derived from constrained BF theory.

A new simple spin foam model for degenerate sectors is introduced.

The new model aligns with previously proposed alternative models.

Abstract

The goal of this work is two-fold. In the first part of this paper we regard classical Plebanski's action as a BF action supplemented by constraints. We introduce a spin foam model for Riemannian general relativity by systematically implementing these constraints as restrictions on paths in the state-sum of the BF theory. The spin foam model obtained is precisely the Barrett-Crane model. This provides a clear-cut connection of the model with a simplicial action. In the second part of the paper we study the quantization of the effective action corresponding to the degenerate sectors of Plebanski's theory and obtain a very simple spin foam model. This model turns out to be precisely the one introduced by De Pietri et al. as an alternative to the one proposed by Barrett and Crane.