  On the basis of a previously proposed mechanism of neutrino-antineutrino mass
splitting in the Standard Model, which is Lorentz and $SU(2)\times U(1)$
invariant but non-local to evade $CPT$ theorem, we discuss the possible
implications of neutrino-antineutrino mass splitting on neutrino physics and
baryogenesis. It is shown that non-locality within a distance scale of the
Planck length, that may not be fatal to unitarity in generic effective theory,
can generate the neutrino-antineutrino mass splitting of the order of observed
neutrino mass differences, which is tested in oscillation experiments, and
non-negligible baryon asymmetry depending on the estimate of sphaleron
dynamics. The one-loop order induced electron-positron mass splitting in the
Standard Model is shown to be finite and estimated at $\sim 10^{-20}$ eV, well
below the experimental bound $< 10^{-2}$ eV. The induced $CPT$ violation in the
$K$-meson in the Standard Model is expected to be even smaller and well below
the experimental bound $|m_{K}-m_{\bar{K}}|<0.44\times 10^{-18}$ GeV.
