Fermions and noncommutative theories

Abstract

By using a framework where the object of noncommutativity $\theta^{\mu\nu}$ represents independent degrees of freedom, we study the symmetry properties of an extended $x+\theta$ space-time, given by the group $P$', which has the Poincar\'{e} group $P$ as a subgroup. In this process we use the minimal canonical extension of the Doplicher, Fredenhagen and Roberts algebra. It is also proposed a generalized Dirac equation, where the fermionic field depends not only on the ordinary coordinates but on $\theta^{\mu\nu}$ as well. The dynamical symmetry content of such fermionic theory is discussed, and we show that its action is invariant under $\cal P$'.