Spin foam model from canonical quantization

We suggest a modification of the Barrett-Crane spin foam model of four-dimensional Lorentzian general relativity motivated by the canonical quantization. The starting point is Lorentz covariant loop quantum gravity. Its kinematical Hilbert space is found as a space of the so-called projected spin networks. These spin networks are identified with the boundary states of a spin foam model and provide a generalization of the unique Barrett-Crane intertwiner. We propose a way to modify the Barrett-Crane quantization procedure to arrive at this generalization: the B field (bivectors) should be promoted not to generators of the gauge algebra, but to their certain projection. The modification is also justified by the canonical analysis of the Plebanski formulation. Finally, we compare our construction with other proposals to modify the Barrett-Crane model.