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 invari-ance breaks, including the subgroup SL 2 R . In both cases, d 2 and d 4, we extrapolate our results to the continuum and innite 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. The XXV International Symposium on Lattice Field Theor
