  Recently, it has been proposed a spacetime noncommutativity that involves
spin degrees of freedom, here called "spin noncommutativity". One of the
motivations for such a construction is that it preserves Lorentz invariance,
which is deformed or simply broken in other approaches to spacetime
noncommutativity. In this work, we gain further insight in the physical aspects
of the spin noncommutativity. The noncommutative Dirac equation is derived from
an action principle, and it is found to lead to the conservation of a modified
current, which involves the background electromagnetic field. Finally, we study
the Landau problem in the presence of spin noncommutativity. For this scenario
of a constant magnetic field, we are able to derive a simple Hermitean
non-commutative correction to the Hamiltonian operator, and show that the
degeneracy of the excited states is lifted by the noncommutativity at the
second order or perturbation theory.
