Spin Hall effect and Berry phase of spinning particles

We consider the adiabatic evolution of the Dirac equation in order to compute its Berry curvature in momentum space. It is found that the position operator acquires an anomalous contribution due to the non Abelian Berry gauge connection making the quantum mechanic algebra noncommutative. A generalization to any known spinning particles is possible by using the Bargmann-Wigner equation of motions. The noncommutativity of the coordinates is responsible of the topological spin transport of spinning particle similarly to the spin Hall effect in spintronic physics or the Magnus effect in optics. As an application we predict new dynamics for nonrelativistic particles in an electric field and for photons in a gravitational field.