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P&S write in Section 7.4 on page 238:

We will prove the Ward-Takahashi identity order by order in $\alpha$……The identity is generally not true for individual Feynman diagrams; we must sum over the diagrams for $\mathcal{M}(k)$ at any given order.

I am troubled for this "at given order", here are my thoughts:

  1. I thought, what P&S really proved in section 7.4, is for "any particular diagram". Because their logic is to handle insertion of photons in external line and internal loop separately. Also, P&S write on page 242:

Summing over all insertion points for any particular diagram, we obtain and $$\begin{aligned} k_\mu \mathcal{M}^\mu\left(k ; p_1 \cdots p_n ; q_1 \cdots q_n\right)=e \sum_i & {\left[\mathcal{M}_0\left(p_1 \cdots p_n ; q_1 \cdots\left(q_i-k\right) \cdots\right)\right.} \\ &\left.-\mathcal{M}_0\left(p_1 \cdots\left(p_i+k\right) \cdots ; q_1 \cdots q_n\right)\right] . \end{aligned} \tag{7.68}$$ enter image description here

So would P & S proved Ward-Takahashi identity to a given order?

  1. On P&S book page 186, they also used the Ward Identity to specify the electron vertex formal factor; $$\Gamma^\mu=\gamma^\mu \cdot A+\left(p^{\prime \mu}+p^\mu\right) \cdot B+\left(p^{\prime \mu}-p^\mu\right) \cdot C . \tag{6.31}$$ where $q_{\mu}\Gamma^{\mu}=0$. However, it seems that this formal factor is up to full order, why in this situation Ward Identity also works?
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1 Answer 1

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1,Note that although the diagram identity below 7.68 is true for any particular diagrams of the shaded circle, to apply LSZ reduction formula to extract S matrix you need to sum up all contributions at a given order then you get 7.68, that's why Ward-Takahashi identity holds only "at given order", which means you need to sum all the diagrams that contribute to the amplitude at that order. 2, Since Ward-Takahashi identity holds order by order, it also holds up to full order.

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