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I have some very basic questions about Quantum Field Theory. So let's assume we have massless fermions. In 4 spacetime dimensions, due to the Group Structure of $SO(3,1)$ there exists the famous $\gamma_5$ matrix which can be used to reduce any representation of the group to the so--called left-handed and right-handed parts. As far as I understand the eigenvalues of $\gamma_5$ are good quantum numbers for massless fermions; much in the same way that charge is - for a theory satisfying global gauge invariance.

Now to the point: Weinberg, in his extremely lucid presentation of Quantum Field Theory, argues that the fields in order to conserve quantum numbers, should be consisted of creation operators of particle and annihilation operators for antiparticle. In particular

The Dirac field $\psi (x)$ for massless fermions, should create particles of charge $Q=+1$ and destroy antiparticles of charge $Q = -1$, so that global charge conservation is maintained.

As far as I understand; the field

$\psi_L (x) = \dfrac{1}{2}\left( 1 - \gamma_5 \right)\psi (x)$ (assuming it is charged) should: create left-handed particles of charge $Q=+1$ and annihilate right-handed antiparticles of charge $Q=-1$, so that both chirality (eigenvalue of $\gamma_5$) and charge are globally conserved.

Similarly:

$\psi_R (x) = \dfrac{1}{2}\left( 1 + \gamma_5 \right)\psi (x)$ (assuming it is charged) should: create right-handed particles of charge $Q=+1$ and annihilate left-handed antiparticles of charge $Q=-1$, so that both chirality (eigenvalue of $\gamma_5$) and charge are globally conserved.

Now $SO(3,1)$ has an invariant tensor which acts as $CP$. I want to understand first of all if the above description of the field is correct, and second of all what would the effect of $CP$ be on say $\psi$, $\psi_{L,R}$.

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