While I was reading about charge conjugation I found some (apparently) contradictory facts. For example Itzykson & Zuber says (page 153) "Up to a phase, $\cal C$ interchanges particles and antiparticles with the same momentum, energy and helicity" while Zee (pag. 101) "You can easily convince yourself that the charged conjugated of a left handed field is a right handed field and vice versa" How can this be possible (in the case with $m=0$ when chirality coincides with helicity)?
In conclusion what is the handedness of the charge conjugated of a left handed field?
In order to convince myself I worked out two contradictory proofs: Let $U$ be the charge conjugation operator in Fock space, and $C$ the matrix that realizes charge conjugation on spinors: $\Psi^c = U^\dagger\Psi U = C\bar{\Psi}^t$, then:
$U^\dagger P_L\Psi U = P_L U^\dagger\Psi U = P_L C\bar{\Psi}^t$, (because $P_L$ is acting only upon creation and annihilation operators) and so here we proved that the charge conjugated of a left handed field is still left handed;
$U^\dagger P_L\Psi U = C\overline{(P_L\Psi)}^t = C\gamma^0 P_L\Psi^* = P_RC\bar{\Psi}^t$, (using Pauli-Dirac as well as Weyl representation of gamma matrices) and so here we proved that the charge conjugated of a left handed field is instead right handed.
Can you help me?
Note added: does a Majorana massless fermion exist?
