Phase-Space Noncommutativity and the Dirac Equation
  We consider full phase-space noncommutativity in the Dirac equation, and find that in order to preserve gauge invariance, configuration space noncommutativity must be dropped. The resulting space structure gives rise to a constant magnetic field background and this effect is explicitly seen on the spectrum of the hydrogen atom. Computing this spectrum we find a bound on the momentum noncommutative parameter $\eta$, $\sqrt{\eta}\lsim2.26\mu eV/c$. 
Orfeu Bertolami, Raquel Queiroz
hep-th gr-qc hep-ph