Unclear formulation of Ward identity

[IRobot]

Hello,

I am really familiar with the Ward-Takashi identity formulated in the form k_{\mu}M^{\mu\nu}=0 applying the fact that the longitudinal polarization of the 4 vector A is nonphysical (redundant) and should not contribute to the physical amplitudes. But, by opening a test subject on QED, I ran into this formula: -\frac{1}{\alpha}\Box\partial_{\mu}A^{\mu} + \partial^{\mu}\frac{\delta\Gamma}{\delta A^{\mu}} + ie\psi\frac{\delta\Gamma}{\delta\psi} -ie\bar{\psi}\frac{\delta\Gamma}{\delta\bar{\psi}}=0 which is quite unclear for me. \Gamma [\psi,\bar{\psi},A] is the generator of 1PI graphs. Does someone have a reference on the derivation of that, or could show me how to get this?

 

[torquil]

Section 12.1, eq. 12.13 in Kaku "Quantum Field Theory - A modern introduction" (1993)

 

[IRobot]

Thanks, I checked in Kaku's book, and indeed it's not hard to derive, by playing with Z[\eta,\bar{\eta},J^{\mu}] and \Gamma[\psi,\bar{\psi},A^{\mu}].