2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/24256We present an exact derivation for non-commuting coordinates induced by the SU(2) transformation used to diagonalize the spin-orbit hamiltonian in two dimension.As a result an exact non-dissipative Hall current less sensitive to disorder and complementary to the dissipative conductivity is found. We compute the non-dissipative charge and spin-Hall conductance for the spin-orbit problem.We find that the spin-Hall conductance is quantized in units of $\frac{egμ_{B}}{h}$ .In the presence of a Zeeman interaction the charge-Hall conductivity is proportional to the magnetic field. We propose an experiment to test our theory.Mesoscale and Nanoscale Physicstext