  The Barrett-Crane model for the SO(4,C) general relativity is systematically
derived. This procedure makes rigorous the calculation of the Barrett-Crane
intertwiners from the Barrett-Crane constraints of both real and complex
Riemannian general relativity. The reality of the scalar products of the
complex bivectors associated with the triangles of a flat four simplex is
equivalent to the reality of the associated flat geometry. Spin foam models in
4D for the real and complex orthogonal gauge groups are discussed in a unified
manner from the point of view of the bivector scalar product reality
constraints. Many relevant issues are discussed and generalizations of the
ideas are introduced. The asymptotic limit of the SO(4,C) general relativity is
discussed. The asymptotic limit is controlled by the SO(4,C) Regge calculus
which unifies the Regge calculus theories for all the real general relativity
cases. The spin network functionals for the 3+1 formulation of the spin foams
are discussed. The field theory over group formulation for the Barrett-Crane
models is discussed briefly. I introduce the idea of a mixed Lorentzian
Barrett-Crane model which mixes the intertwiners for the Lorentzian
Barrett-Crane models. A mixed propagator is calculated. I also introduce a
multi-signature spin foam model for real general relativity which is made by
splicing together the four simplex amplitudes for the various signatures.
Further research that is to be done is listed and discussed.
