  In a recent work, a dual formulation of group field theories as
non-commutative quantum field theories has been proposed, providing an exact
duality between spin foam models and non-commutative simplicial path integrals
for constrained BF theories. In light of this new framework, we define a model
for 4d gravity which includes the Immirzi parameter gamma. It reproduces the
Barrett-Crane amplitudes when gamma goes to infinity, but differs from existing
models otherwise; in particular it does not require any rationality condition
for gamma. We formulate the amplitudes both as BF simplicial path integrals
with explicit non-commutative B variables, and in spin foam form in terms of
Wigner 15j-symbols. Finally, we briefly discuss the correlation between
neighboring simplices, often argued to be a problematic feature, for example,
in the Barrett-Crane model.
