When can the Ward Identity be used in quantum field theory?

[MathematicalPhysicist]

I don't understand from Peskin when can I use Ward Identity?
I mean I can see that this identity isn't always valid to use, but when it is?

Take for example equation (16.10) page 508 of Peskin's and Schroeder's.

 

[jedishrfu]

I can't answer your question directly but perhaps you'll find something here that will be useful until someone more knowledgeable chimes in.

https://en.wikipedia.org/wiki/Ward–Takahashi_identity

 

[topochu]

Maybe I'm not the one who is knowledgeable, I will try giving you answer. I hope someone else can supplement or even correct me.

Some background: The Ward identity is a quantum manifestation of the Noether's first theorem, and it is more general. In classical field theory, the Noether's theorem tells us that each continuous symmetry correspond to a conservation law, and the conserved charge is the generator of that symmetry transformation. In quantum field theory, the Ward identity does more or less that same thing. The former works only for the on-shell case whereas the latter one works well for both on-shell and off-shell cases, because it does not rely on the equation of motion.

Examples: You can read p.186 of Peskin and Schroder to see how Ward identity is used to constraint the form of $\Gamma$ , and an example as you suggested on p.508. I think in a narrow sense, in QED if you have an external photon leg, then you will have $\epsilon * M$ , and then you can already use the condition of $k * M = 0$. I think it is not the only situation that you can use the Ward identity.

FYI: In conformal field theory (at least in 2d), the Ward identity plays a much more important role as it highly constrain the form of correlation functions, and also tell much information among these correlators.