  The Majorana neutrino $\psi_{M}(x)$ when constructed as a superposition of
chiral fermions such as $\nu_{L} + C\overline{\nu_{L}}^{T}$ is characterized by
$ ({\cal C}{\cal P}) \psi_{M}(x)({\cal C}{\cal P})^{\dagger}
=i\gamma^{0}\psi_{M}(t,-\vec{x})$, and the CP symmetry describes the entire
physics contents of Majorana neutrinos. Further specifications of C and P
separately could lead to difficulties depending on the choice of C and P. The
conventional $ {\cal C} \psi_{M}(x) {\cal C}^{\dagger} = \psi_{M}(x)$ with
well-defined P is naturally defined when one constructs the Majorana neutrino
from the Dirac-type fermion. In the seesaw model of Type I or Type I+II where
the same number of left- and right-handed chiral fermions appear, it is
possible to use the generalized Pauli-Gursey transformation to rewrite the
seesaw Lagrangian in terms of Dirac-type fermions only; the conventional C
symmetry then works to define Majorana neutrinos. In contrast, the "pseudo
C-symmetry" $\nu_{L,R}(x)\rightarrow C\overline{\nu_{L,R}(x)}^{T}$ (and
associated "pseudo P-symmetry"), that has been often used in both the seesaw
model and Weinberg's model to describe Majorana neutrinos, attempts to assign a
nontrivial charge conjugation transformation rule to each chiral fermion
separately. But this common construction is known to be operatorially
ill-defined and, for example, the amplitude of the neutrinoless double beta
decay vanishes if the vacuum is assumed to be invariant under the pseudo
C-symmetry.
