If one modifies the Dirac equation in momentum space to [Formula: see text], the symmetry of positive and negative energy eigenvalues is lifted by [Formula: see text] for a small [Formula: see text]. The mass degeneracy of the particle and antiparticle is thus lifted in a Lorentz invariant manner since the combinations [Formula: see text] with step functions are manifestly Lorentz invariant. We explain an explicit construction of this CPT breaking term in coordinate space, which is Lorentz invariant but nonlocal at the distance scale of the Planck length. The application of this Lorentz invariant CPT breaking mechanism to the possible mass splitting of the neutrino and antineutrino in the Standard Model is briefly discussed.