Noncommutative gauge theory without Lorentz violation

The most popular noncommutative field theories are characterized by a matrix parameter theta(munu) that violates Lorentz invariance. We consider the simplest algebra in which the theta 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.