2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/101821A theory based on the Aharonov -Bohm effect in the momentum space for the Spin-Hall conductivity without a magnetic field is presented. The two dimensional Rashba Hamiltonian is diagonalized in the momentum spinor basis. This spinor is singular at K=0. The representation of the cartesian coordinates in the spinor momentum basis obey non-commutative rules. The non-commuting relations are a result of an effective Aharonov-Bohm vortex at K=0. We find the exact value of $\frac{e}{4π}$ for the Spin-Hall conductivity. The effect of a time reversal scattering potential on the Spin-Hall current causes the current to vanishes for an infinite system.submitted to P.R.LMesoscale and Nanoscale Physicstext