Construction of the General Spin Foam Model of Lorentzian BF theory and Gravity

In this article we report our progress in the construction of a general Lorentzian spin foam model based on the Gelfand-Naĭmark theory of the representations of SL(2,C). First we construct the simplex amplitude for the BF SL(2,C) model. Then we discuss the asymptotic limit of this model. Next we discuss the implemention of the Barrett-Crane constraints on this model. We derive an equation that the general Lorentzian spin foam model has to satisfy. In the appendix we give a simple derivation of the Clebsch-Gordan coefficients for SL(2,C). We have tried to make this article as much self contained as possible