Two classes of Majorana neutrinos in the seesaw model

The commonly used pseudo-C symmetry (νL)ᶜ=CνLT is not defined in Lagrangian field theory. In general, there exist two classes of Majorana fermions; the first is associated with the chiral fermion with the conventional C and P symmetries, and the second is associated with the Weyl fermion defined by CP symmetry only and formally characterised by the pseudo-C symmetry. Taking the seesaw model as an example, it is shown that a generalized Pauli--G\"{u}rsey (or Bogoliubov-type) canonical transformation converts the neutrino defined by the Weyl fermion to the neutrino defined by the chiral fermion and thus to the conventional Majorana fermion, while preserving the anti-commutation relations. The mixing angles in the weak lepton sector are not modified by this generalized Pauli--G\"{u}rsey transformation.