Dynamics of a Dirac Fermion in the presence of spin noncommutativity

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.