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(Following the calculations done in 'Quantum Field Theory in a Nutshell' [Second Edition] by Zee, Page 101)

The Dirac equation in the presence of an electromagnetic field is given by:

$$ [i \gamma^{\mu} (\partial_{\mu} - i e A_{\mu}) - m]\psi = 0 $$

where $\gamma^{\mu}$ are the gamma matrices, $A_{\mu}$ is the gauge field, $e$ is charge, $m$ is mass and $\psi$ is a spinor.

The complex conjugate of this equation is given by:

$$ [-i \gamma^{\mu *} (\partial_{\mu} + i e A_{\mu}) - m]\psi^{*} = 0 $$

Zee defines:

$$ -\gamma^{\mu *} = (C \gamma^{0})^{-1} \gamma^{\mu} (C \gamma^{0}) $$

where $C$ is the charge conjugation operator. Zee states that you can plug this into the complex conjugated Dirac equation to get:

$$ [i \gamma^{\mu} (\partial_{\mu} + i e A_{\mu}) - m]\psi_{c} = 0 $$

where $\psi_{c} = C \gamma^{0} \psi^{*}$. When I try to do this I get the following:

$$ [i (C \gamma^{0})^{-1} \gamma^{\mu} (C \gamma^{0}) (\partial_{\mu} + i e A_{\mu}) - m]\psi^{*} = 0 $$

$$ (C \gamma^{0}) \cdot [i (C \gamma^{0})^{-1} \gamma^{\mu} (C \gamma^{0}) (\partial_{\mu} + i e A_{\mu}) - m]\psi^{*} = (C \gamma^{0}) \cdot 0 $$

$$ [-i (C \gamma^{0})(C \gamma^{0})^{-1} \gamma^{\mu} (C \gamma^{0}) (\partial_{\mu} + i e A_{\mu}) - m(C \gamma^{0})]\psi^{*} = 0 $$

$$ [-i (C \gamma^{0})(C \gamma^{0})^{-1} \gamma^{\mu} (\partial_{\mu} - i e A_{\mu})(C \gamma^{0}) - m(C \gamma^{0})]\psi^{*} = 0 $$

$$ [-i (C \gamma^{0})(C \gamma^{0})^{-1} \gamma^{\mu} (\partial_{\mu} - i e A_{\mu}) - m]C \gamma^{0}\psi^{*} = 0 $$

$$ [i (C \gamma^{0})(C \gamma^{0})^{-1} \gamma^{\mu} (-\partial_{\mu} + i e A_{\mu}) - m]\psi_{c} = 0 $$

where I used the antilinear property $Ci = -iC$.

Now I have run into two issues. I am unsure of how to treat $(C \gamma^{0})(C \gamma^{0})^{-1}$ and also the sign of the $\partial_{\mu}$ term does not match the sign in Zee's expression. Can anyone point me in the right direction?

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The charge conjugation matrix $C$ is not an antilinear map. It is ordinary matrix with the property that $$ C\gamma^\mu C^{-1} =-(\gamma^\mu)^T. $$
(or maybe $C^{-1} \gamma^\mu C = -(\gamma^\mu)^T$. Conventions differ.).

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  • $\begingroup$ Do you have a reference for the derivation of this property? $\endgroup$ – NahPlsMan Dec 31 '20 at 1:18
  • $\begingroup$ Also I don't think it solves the issue of the negative sign on $\partial_{\mu}$ $\endgroup$ – NahPlsMan Dec 31 '20 at 1:19
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    $\begingroup$ Try hitoshi.berkeley.edu/230A/clifford.pdf or my arXiv:2009.00518 article which is based on that. $\endgroup$ – mike stone Dec 31 '20 at 13:32

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