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I am following Sidney Coleman's lectures of Quantum Field Theory.

At the end of ch.32, he derived the Ward identity for the 1PI generating functional $\Gamma[\psi,\bar{\psi},A_{\mu}]$ for QED:

\begin{equation} ie \bar{\psi} \frac{\delta \Gamma}{\delta \bar{\psi}(x)} - ie \frac{\delta \Gamma}{\delta \psi(x)} \psi(x)- \partial^{\mu} \frac{\delta \Gamma}{\delta A^{\mu}(x)} = \frac{-1}{\xi} (\partial_{\nu}\partial^{\nu})(\partial_{\mu} A^{\mu}). \end{equation}

The term on the RHS is the gauge fixing term in the original QED Lagrangian. I am now wondering that whether all of the fields involved in the Ward identity are $c$-number fields. As $\psi$ and $\bar{\psi}$ represent Fermi field, it seems like we should interpret their classical correspondence as Grassmann fields. However, it is clear on the RHS, we have a $c$-number function. Then this equation seems to have both $c$-number and Grassmann number involved, which I think may not make sense?

I am wondering whether we should interpret both $\psi$ and $\bar{\psi}$ also as $c$-number fields? But if that is the case, how does the Fermi minus sign issue be properly handle under the above Ward identity?

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No, $\psi$ and $\bar{\psi}$ are $a$-number fields, while $A_{\mu}$ is a $c$-number field. The above Ward-Takahashi identity (WTI) is a supernumber-valued identity.

The WTI is often used by differentiating it a number of times wrt. the fields, and then set the remaining fields to zero.

See also this related Phys.SE post.

References:

  1. Bryce DeWitt, Supermanifolds, Cambridge Univ. Press, 1992.
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