  Non commutative quantum field theory is a possible candidate for the
quantization of gravity. In our thesis we study in detail the $\phi 4$ model on
the Moyal plane with an harmonic potential. Introduced by Grosse and
Wulkenhaar, this model exhibits the Langmann-Szabo duality not only for the
vertex but also for the propagator. We have obtained several results concerning
this model. We have proved the renormalisability of this theory at all orders
in the position space. We have introduced the parametric and Complete Mellin
representation for the model. Furthermore we have proved that the coupling
constant has a bounded flow at all orders in perturbation theory. Finally we
have achieved the dimensional regularization and renormalization of the model.
Further possible studies include the study of gauge theory on the Moyal plane
and there possible applications for the quantization of gravity. The
connections with string theory and loop quantum gravity should also be
investigated.
