Non-Perturbative Results in Non-Commutative Field Theory

We present a non-perturbative study of the phi^4 model
on a three dimensional non-commutative space, based on
numerical simulations. A new phase occurs where translation
symmetry is spontaneously broken, so that stripe patterns
dominate. In this phase the dispersion relation is deformed
in the IR sector, in agreement with the property of UV/IR mixing.

Next we address non-commutative U(1) gauge theory, where
the Wilson loop are complex. In d=2 small Wilson loops are real
and follow an area law, whereas for large Wilson loops the complex
phase rises linearly with the area, analogous to the Aharonov-Bohm
effect. In d=4 we investigate the phase diagram and its
extrapolation to the continuum and infinite volume by means of
a double scaling limit. Due to another phase of broken translation
symmetry the photon appears to be IR stable in non-commutative spaces,
despite the perturbatively negative IR singularity.
We evaluate the corresponding dispersion relation explicitly.