  An extension of QED is considered in which the Dirac fermion has both
Hermitian and anti-Hermitian mass terms, as well as both vector and
axial-vector couplings to the gauge field. Gauge invariance is restored when
the Hermitian and anti-Hermitian masses are of equal magnitude, and the theory
reduces to that of a single massless Weyl fermion. An analogous non-Hermitian
Yukawa theory is considered, and it is shown that this model can explain the
smallness of the light-neutrino masses and provide an additional source of
leptonic CP violation.
