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:
- 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
$$\begin{aligned}
k_\mu \mathcal{M}^\mu\left(k ; p_1 \cdots p_n ; q_1 \cdots q_n\right)=e \sum_i & {\left[\mathcal{M}_0\left(p_1 \cdots p_n ; q_1 \cdots\left(q_i-k\right) \cdots\right)\right.} \\
&\left.-\mathcal{M}_0\left(p_1 \cdots\left(p_i+k\right) \cdots ; q_1 \cdots q_n\right)\right] .
\end{aligned} \tag{7.68}$$
So would P & S proved Ward-Takahashi identity to a given order?
- 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?
Best Answer
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.