Spin foam quantization of SO(4) Plebański's action
In the present paper Plebanski's action is considered as a BF action supplemented by constraints. The author introduces a spin foam model for Riemannian general relativity by implementing these constraints as restrictions on paths in the state-sum of the BF theory. It is shown that the model obtained is related to the Barrett-Crane model. The author proves that all tetrahedra in the allowed configurations have zero volume. In the second part of the paper the author studies the quantization of the effective action corresponding to the degenerate sectors of Plebanski's theory and ends up with a spin foam model that is precisely the same as introduced by De Pietri et al. It is shown that the four-simplex configurations of the model corresponding to the full Plebanski's action turn out to be entirely contained in the set of configurations of the model of the degenerate sector.
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