The most popular noncommutative field theories are characterized by a matrix parameter θ µν that violates Lorentz invariance. We consider the simplest algebra in which the θ-parameter is promoted to an operator and Lorentz invariance is preserved. This algebra arises through the contraction of a larger one for which explicit representations are already known. We formulate a star product and construct the gauge-invariant Lagrangian for Lorentz-conserving noncommutative QED. Three-photon vertices are absent in the theory, while a four-photon coupling exists and leads to a distinctive phenomenology.

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For each m ≥ 1 and p > 2 we characterize bounded simply connected Sobolev L m p -extension domains Ω ⊂ R 2 . Our criterion is expressed in terms of certain intrinsic subhyperbolic metrics in Ω. Its proof is based on a series of results related to the existence of special chains of squares joining given points x and y in Ω.An important geometrical ingredient for obtaining these results is a new "Square Separation Theorem". It states that under certain natural assumptions on the relative positions of a point x and a square S ⊂ Ω there exists a similar square Q ⊂ Ω which touches S and has the property that x and S belong to distinct connected components of Ω \ Q.