Noncommutative gauge theory without Lorentz violation

The most popular noncommutative field theories are characterized by a matrix parameter ${\theta }^{\mu \nu }$ 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.