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I'm reading Maggiore's book and I got to the part of charge conjugation symmetry for Dirac spinor. I get that the definition of charge conjugation is representation-dependent, however I couldn't find anywhere nor carry out by myself the explicit calculations necessary to define in the proper way the C operator. In Maggiore's errata corrige C is defined on the annihilation and creation operators as: $$Ca_{{p}, 1}C = \eta_{C}^{a} b_{{p}, 2}\,\,\, \, \, \, Ca_{{p}, 2}C = -\eta_{C}^{a} b_{{p}, 1}$$ and similar expressions for the b operators: $$Cb_{{p}, 1}C = -\eta_{C}^{b} b_{{p}, 2}\,\,\, \, \, \, Cb_{{p}, 2}C = \eta_{C}^{} b_{{p}, 1}$$ Its then stated that from the expressions: $$[-i \gamma^2 u^{1}(p)]^{*} = v^{2}(p) \,\,\,\,\,\, [-i \gamma^2 u^{2}(p)]^{*} = -v^{1}(p)$$ $$[-i \gamma^2 v^{1}(p)]^{*} = -v^{2}(p) \,\,\,\,\,\, [-i \gamma^2 v^{2}(p)]^{*} = +u^{1}(p)$$ one could derive $\psi^{c} = -i \gamma^2 \psi^{*}$.

We have defined $\zeta^{1}, \zeta^{2}$ as the base of the 2d complex vector space and $u^{s} (p) = (\sqrt{p \cdot \sigma} \zeta^{s}, \sqrt{p \cdot \bar{\sigma}} \zeta^{s})$. So $v^{1}(p)$ and $u^{2}(p)$ carry the same spin and the charge cojugation doesn't change it. Its absolutely not obvious to me how to derive the spinor relations in the chiral representation. Moreover I see similar arguments are used for the time reversal symmetry and I think they rely on this expressions. Can someone please give me a clear derivation of them?

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  • $\begingroup$ Can anyone help me? $\endgroup$ – MrRobot Oct 20 '17 at 10:59

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