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When we derive the Dirac equation from the Lagrangian, $$ \mathcal{L}=\overline{\psi}i\gamma^{\mu}\partial_{\mu}\psi-m\overline{\psi}\psi, $$ we assume $\psi$ and $\overline{\psi}=\psi^{*^{T}}\gamma^{0}$ are independent. So when we take the derivative of the Lagrangian with respect to $\overline{\psi}$, we get the Dirac equation $$ 0=\partial_{\mu}\frac{\partial\mathcal{L}}{\partial\left(\partial_{\mu}\overline{\psi}\right)}=\frac{\partial\mathcal{L}}{\partial\overline{\psi}}=\left(i\gamma^{\mu}\partial_{\mu}-m\right)\psi. $$

Now if we include a term with charge conjugation, $\psi^{c}=-i\gamma^{2}\psi^{*}$, into the Lagrangian (like $\Delta\mathcal{L}=\overline{\psi}\psi^{c}$), does this $\psi^c$ depend on $\overline{\psi}$ or $\psi$? Why or why not?

If $\psi^{c}$ depends on $\psi$, why wouldn't the reason that $\overline{\psi}$ and $\psi$ are independent apply for $\psi^{c}$ and $\psi$?

If $\psi^{c}$ depends on $\overline{\psi}$, how should we take derivative of $\Delta\mathcal{L}$ with respect to $\overline{\psi}$?

Qmechanic
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Louis Yang
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2 Answers2

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Yes, when we want to obtain the equation of motion using Euler-Lagrange equation, we should treat $\psi$ and $\psi^c$ independent, but $\overline{\psi}$ and $\psi^c$ dependent. The reason for this is that we can simply expressed $\psi^c$ in terms of $\overline{\psi}$ by $$ \psi^{c}=C\overline{\psi}^{T}, $$ where $C=-i\gamma^{2}\gamma^{0}$ is the charge conjugation matrix. So $\overline{\psi}$ and $\psi^c$ are the same degree of freedom.

For the derivative of $\overline{\psi}\psi^{c}$ with respect to $\overline{\psi}$, one should be really careful because $\psi$ is anticommuting. Since the derivative in Euler-Lagrange equation actually comes from the variation of Lagrangian, We should start from the variation \begin{eqnarray} \delta\left(\overline{\psi}\psi^{c}\right) & = & \delta\left(\overline{\psi}C\overline{\psi}^{T}\right)=\delta\left(\overline{\psi_{i}}C_{ij}\overline{\psi_{j}}\right)=\delta\left(\overline{\psi_{i}}\right)C_{ij}\overline{\psi_{j}}+\overline{\psi_{i}}C_{ij}\delta\overline{\psi_{j}}\\ & = & \delta\left(\overline{\psi_{i}}\right)C_{ij}\overline{\psi_{j}}-\delta\left(\overline{\psi_{j}}\right)C_{ij}\overline{\psi_{i}}, \end{eqnarray} where I use the anticommutation of the fields to get the minus sign for the last step. Now notice that $C^{T}=C^{+}=-C$. So the last term is \begin{equation} -\delta\left(\overline{\psi_{j}}\right)C_{ij}\overline{\psi_{i}}=\delta\left(\overline{\psi_{j}}\right)C_{ji}\overline{\psi_{i}}=\delta\left(\overline{\psi_{i}}\right)C_{ij}\overline{\psi_{j}}. \end{equation} and we get $\delta\left(\overline{\psi}\psi^{c}\right)=2\delta\left(\overline{\psi}\right)C\overline{\psi}^{T}.$ Therefore, the equation of motion from this term is \begin{equation} \frac{\partial}{\partial\overline{\psi}}\left(\overline{\psi}\psi^{c}\right)=2C\overline{\psi}^{T}=2\psi^{c}. \end{equation}

Louis Yang
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2

I) The Dirac spinor $\psi$ and its complex conjugate $\psi^*$ are not independent variables, but in some calculations one can treat them as such.

For the similar question about a complex scalar field $\phi$ and its complex conjugate $\phi^*$, see e.g. this Phys.SE post.

II) The charge conjugated field $\psi^{c}=-i\gamma^{2}\psi^{*}$ is tied to the complex conjugate $\psi^*$ by a bijective transformation, so that they are not independent.

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