Editing C-symmetry (section)

====Charge conjugation in the chiral basis====
Taking the [[Gamma matrices#Weyl representation|Weyl representation]] of the gamma matrices, one may write a (now taken to be massive) Dirac spinor as
:<math>\psi = \begin{pmatrix} \psi_\text{L}\\ \psi_\text{R} \end{pmatrix}</math>

The corresponding dual (anti-particle) field is
:<math>\overline{\psi}^\textsf{T}
    = \left( \psi^\dagger \gamma^0 \right)^\textsf{T}
    = \begin{pmatrix} 0 & I \\ I & 0\end{pmatrix} \begin{pmatrix} \psi_\text{L}^* \\ \psi_\text{R}^* \end{pmatrix}
    = \begin{pmatrix} \psi_\text{R}^* \\ \psi_\text{L}^* \end{pmatrix}
</math>

The charge-conjugate spinors are
:<math>\psi^c
    = \begin{pmatrix} \psi_\text{L}^c\\ \psi_\text{R}^c \end{pmatrix} 
    = \eta_c C \overline\psi^\textsf{T}
    = \eta_c \begin{pmatrix} -i\sigma^2 & 0 \\ 0 & i\sigma^2\end{pmatrix} \begin{pmatrix} \psi_\text{R}^* \\ \psi_\text{L}^* \end{pmatrix}
    = \eta_c \begin{pmatrix} -i\sigma^2\psi_\text{R}^* \\ i\sigma^2\psi_\text{L}^* \end{pmatrix}
</math>

where, as before, <math>\eta_c</math> is a phase factor that can be taken to be <math>\eta_c=1.</math>  Note that the left and right states are inter-changed. This can be restored with a parity transformation. Under [[P-symmetry|parity]], the Dirac spinor transforms as
:<math>\psi\left(t, \vec x\right) \mapsto \psi^p\left(t, \vec x\right) = \gamma^0 \psi\left(t, -\vec x\right)</math>

Under combined charge and parity, one then has
:<math>\psi\left(t, \vec x\right) \mapsto \psi^{cp}\left(t, \vec x\right) 
     =  \begin{pmatrix} \psi_\text{L}^{cp} \left(t, \vec x\right)\\ \psi_\text{R}^{cp}\left(t,\vec x\right) \end{pmatrix} 
     = \eta_c \begin{pmatrix} -i\sigma^2\psi_\text{L}^*\left(t, -\vec x\right) \\ i\sigma^2\psi_\text{R}^*\left(t, -\vec x\right) \end{pmatrix}</math>

Conventionally, one takes <math>\eta_c = 1</math> globally. See however, the note below.