  We provide the Barrett-Crane spin foam model for quantum gravity with a
discrete action principle, consisting in the usual BF term with discretized
simplicity constraints which in the continuum turn topological BF theory into
gravity. The setting is the same as usually considered in the literature:
space-time is cut into 4-simplices, the connection describes how to glue these
4-simplices together and the action is a sum of terms depending on the
holonomies around each triangle. We impose the discretized simplicity
constraints on disjoint tetrahedra and we show how the Lagrange multipliers for
the simplicity constraints distort the parallel transport and the correlations
between neighbouring 4-simplices. We then construct the discretized BF action
using a non-commutative product between $\SU(2)$ plane waves. We show how this
naturally leads to the Barrett-Crane model. This clears up the geometrical
meaning of the model. We discuss the natural generalization of this action
principle and the spin foam models it leads to. We show how the recently
introduced spinfoam fusion coefficients emerge with a non-trivial measure. In
particular, we recover the Engle-Pereira-Rovelli spinfoam model by weakening
the discretized simplicity constraints. Finally, we identify the two sectors of
Plebanski's theory and we give the analog of the Barrett-Crane model in the
non-geometric sector.
