# Complex conjugate of the Dirac equation

(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?

## 1 Answer

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.).

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