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Following the derivation of Ward Identity by Weinberg book, you get it in the form

$$ (l-k)_\mu S'(k)\Gamma^\mu(k,l)S'(l) = i S'(l) - iS'(k) $$

Can anyone explain the physical meaning of this identity? And how to understand it from this form?

$S'(k)$ is the complete propagator and $\Gamma^\mu(k,l)$ the vertex function.

EDIT (after comments and with more details and questions)

For instance, let's consider the following Lagrangian density

$\mathcal{L} = -\frac{1}{4}F^{\mu\nu}F_{\mu\nu} - \frac{1}{2}(\partial_\mu A^\mu)^2 - (D_\mu \phi)^*(D^\mu \phi) - m^2\phi^*\phi - \frac{\lambda}{4}(\phi^*\phi)^2$

where $D_\mu = \partial_\mu +ieA_\mu$, $\phi$ is a complex scalar field( of charge $-e$) and $A_\mu$ is the electromagnetic field.

This lagragian is invariant under the Gauge transformation $A_\mu \rightarrow A_\mu + \partial_\mu \alpha(x)$ if $\partial_\mu\partial^\mu \alpha(x) = 0$.

Suppose fields in the lagrangian are the "bare" fields. Then, I normalize them in a proper way and I obtain $$ \mathcal{L} = \mathcal{L}_0 + \mathcal{L}_c + \mathcal{L}_{int} $$ where $$ \mathcal{L}_0 =-\frac{1}{4}F^{\mu\nu}F_{\mu\nu} - \frac{1}{2}(\partial_\mu A^\mu)^2 - \partial^\mu\phi^*\partial_\mu\phi - m^2\phi^*\phi $$ and $$ \mathcal{L}_c = -i(Z_1-1)eJ_\mu A^\mu - (Z_4-1) e^2 \phi^* \phi A_\mu A^\mu - (Z_\lambda-1)\frac{\lambda}{4}(\phi^*\phi)^2 - (Z_2-1)\partial^\mu\phi^*\partial_\mu\phi -(Z_m - 1) m^2\phi^*\phi -\frac{1}{4}(Z_3-1)F^{\mu\nu}F_{\mu\nu} $$

where $J_\mu = \partial_\mu\phi^*\cdot \phi-\phi^*\partial_\mu\phi$

and $$ \mathcal{L}_{int} = -ieJ_\mu A^\mu -e^2 \phi^* \phi A_\mu A^\mu - \frac{\lambda}{4}(\phi^*\phi)^2 $$ Then, I'm working with these vertices

vertices

Which is the vertex function in this case? Is it only a 3-point irreducible diagram? I should draw also the following diagrams (and more diagrams that I don't upload)

vertex fnction

The third is the vertex function as I found on Weinberg's book. The first and the second seems the 4-point Green Functions.

Question 1) Can you explain me the differences among the diagrams and can you show me the mathematical expression of the vertex function?

Question 2) Can I use the Ward identity and the gauge invariance to say something about the $Z$ factors on $\mathcal{L}_c$?

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    $\begingroup$ In general, Ward-Takahashi identities represent the symmetries of the system at the quantum level; i.e., the symmetries put some restrictions on the Green and vertex functions. To understand the physical meaning for the identity above, one should see which symmetry it represents. $\endgroup$ – AlQuemist Dec 7 '15 at 11:08
  • $\begingroup$ Could you give an example how to use the identity in this form? $\endgroup$ – apt45 Dec 7 '15 at 11:15
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    $\begingroup$ Well, that's a very broad question, user13653! You can use such identities to reduce other equations (for observables) or check that an approximate vertex satisfies the symmetries (or conservations laws) of the system. $\endgroup$ – AlQuemist Dec 7 '15 at 11:22
  • $\begingroup$ Editing tip: meta.physics.stackexchange.com/q/5886/2451 $\endgroup$ – Qmechanic Dec 30 '15 at 14:11
  • $\begingroup$ @Qmechanic I just posted an example to clarify what I don't understand. Do you suggest me to open a new question and close this one? $\endgroup$ – apt45 Dec 30 '15 at 14:20

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