  An idea of reality conditions in the context of spin foams (Barrett-Crane
models) is developed. The square of areas are the most elementary observables
in the case of spin foams. This observation implies that simplest reality
conditions in the context of the Barrett-Crane models is that the all possible
scalar products of the bivectors associated to the triangles of a four simplex
be real. The continuum generalization of this is the area metric reality
constraint: the area metric is real iff a non-degenerate metric is real or
imaginary. Classical real general relativity (all signatures) can be extracted
from complex general relativity by imposing the area metric reality constraint.
The Plebanski theory can be modified by adding a Lagrange multiplier to impose
the area metric reality condition to derive classical real general relativity.
I discuss the SO(4,C) BF model and SO(4,C) Barrett-Crane model. It appears that
the spin foam models in 4D for all the signatures are the projections of the
SO(4,C) spin foam model using the reality constraints on the bivectors.

General Relativity and Quantum Cosmology