  For more than three decades, clear discrepancies have existed between spin
densities in momentum space revealed by Magnetic Compton scattering experiments
and theoretical calculations based on density functional theory (DFT). Here by
making a wide comparison between different theoretical methods, including DFT,
DFT combined with dynamical mean field theory, and Hedin's $GW$ approximation,
we discover how the magnetic Compton profiles of Ni can be predicted remarkably
well. We find that the essential ingredients missing in DFT are (i) local spin
fluctuations and (ii) a non-local treatment of electron correlations.
