Wolfgang Bietenholz (Zeuthen):

Simulating U(1) Gauge Theory on Non-Commutative Spaces

Abstract:

We first present a non-perturbative study of the φ⁴ model on a three dimensional non-commutative space, based on numerical simulations. If the extent of non-commutativity exceeds a lower limit, 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 infrared regime, in agreement with the property of UV/IR mixing. Next we address non-commutative U(1) gauge theory. Here the Wilson loop is complex on the non-perturbative level. We first consider the 2d case: small Wilson loops are almost 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 the behaviour is qualitatively similar for loops in the non-commutative plane, whereas the loops in other planes follow closely the commutative pattern. In both cases we also discuss the extrapolation of our results to the continuum and infinite volume by means of a double scaling limit. The 4d phase diagram reveals that the photon may survive in a non-commutative world, despite the perturbatively negative IR singularity, and its dispersion relation could be confronted with experimental data in the near future.

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