  We present numerical results for U(1) gauge theory in 2d and 4d spaces
involving a non-commutative plane. Simulations are feasible thanks to a mapping
of the non-commutative plane onto a twisted matrix model. In d=2 it was a
long-standing issue if Wilson loops are (partially) invariant under
area-preserving diffeomorphisms. We show that non-perturbatively this
invariance breaks, including the subgroup SL(2,R). In both cases, d=2 and d=4,
we extrapolate our results to the continuum and infinite volume by means of a
Double Scaling Limit. In d=4 this limit leads to a phase with broken
translation symmetry, which is not affected by the perturbatively known IR
instability. Therefore the photon may survive in a non-commutative world.
