# Ward identity of QED - whether the fields are all $c$-number fields

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:

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

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?

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.