Field theory in noncommutative Minkowski superspace

There is much discussion of scenarios where the space-time coordinates $x^{\mu }$ are noncommutative. The discussion has been extended to include nontrivial anticommutation relations among spinor coordinates in superspace. A number of authors have studied field theoretical consequences of the deformation of $\mathcal{N}=1$ superspace arising from nonanticommutativity of coordinates $\theta$, while leaving $\overline{\theta }$’s anticommuting. This is possible in Euclidean superspace only. In this note we present a way to extend the discussion by making both $\theta$ and $\overline{\theta }$ coordinates nonanticommuting in Minkowski superspace. We present a consistent algebra for the supercoordinates, find a star-product, and give the Wess-Zumino Lagrangian ${\mathcal{L}}_{WZ}$ within our model. It has two extra terms due to non(anti)commutativity. The Lagrangian in Minkowski superspace is always manifestly Hermitian and for ${\mathcal{L}}_{WZ}$ it preserves Lorentz invariance.