Dynamics of a Dirac fermion in the presence of spin noncommutativity

In this work, we gain further insight in the physical aspects of the spin noncommutativity, which mix spacetime and spin degrees of freedom in a noncommutative scenario, in a Lorentz invariant way. Due to the combination of noncommutativity and Lorentz invariance, time also becomes nonlocal so the modified Dirac equation in general contains an infinite tower of time derivatives. Nevertheless, we prove the existence of a conserved probability current in any order in the noncommutativity parameter and for any background field Aμ. Finally, we study the Landau problem in the presence of spin noncommutativity. In this case, we are able to derive a simple Hermitean noncommutative correction to the Hamiltonian operator. We show that the degeneracy of the excited states is lifted by the noncommutativity at the second order of perturbation theory. This is to be contrasted to the case of canonical noncommutativity, where these corrections are of the first order, imposing much stronger restrictions to the noncommutativity.