We have used the Pauli-Schr\"{o}dinger equation in its covariant form, that is, written in the light-cone of a five-dimensional De Sitter space-time. Following standard procedures, the analogue of the Dirac equation is derived, standing for a galilean spin 1/2 particle in the presence of a external field. Some results are important to be mention, such as the expected g-factor, but in a galilean (not Lorentz) context. In addition, considering interaction, the Pauli-Hartree-Fock equation is obtained following in parallel to the ideas used to construct the Dirac-Hartree-Fock equation.