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In Schwartz's QFT, in the Weyl basis, Majorana fermions are written on the same footing as Dirac fermions, as matrix $$ \psi_{Majorana}=(\psi_L \quad i\sigma_2\psi_L^*)^T $$ I don't understand the reason behind this combination. One reason is given, $\sigma_2\psi_L^*$ transforms like $\psi_R$, but how to prove this? This $i\gamma^2$ can be said as the charge conjugation operator. I understand it is real in the Weyl basis but is the charge conjugation operator necessarily real?

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Hint: if $$ U=\left(\matrix{a^*&-b\cr b^* &a}\right) \in {\rm SU}(2) $$ and $T= i\sigma_2$, then $T^{-1}UT= U^*$ where $U^*$ is the matrix with its entries complex-conjugated.

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