  If one modifies the Dirac equation in momentum space to
$[\gamma^{\mu}p_{\mu}-m-\Delta m(\theta(p_{0})-\theta(-p_{0}))
\theta(p_{\mu}^{2})]\psi(p)=0$, the symmetry of positive and negative energy
eigenvalues is lifted by $m\pm \Delta m$ for a small $\Delta m$. The mass
degeneracy of the particle and antiparticle is thus lifted in a Lorentz
invariant manner since the combinations $\theta(\pm p_{0})\theta(p_{\mu}^{2})$
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 non-local at a 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.
