# On the Ward Identity in QED

I am reading P&S, particularly Chapter 5.5. The authors are trying to derive an expression for the Ward identity (not formally, but still). They claim that the amplitude describing a photon emission (stripped off from the polarization vector of the respective photon) in any QED process can be written as $$\mathcal{M}^{\mu}(k)=\int d^4x e^{ik\cdot x}\langle f|j^{\mu}(x)|i\rangle$$ and then they dot product $$\mathcal{M}^{\mu}(k)$$ with the photon momentum $$k_{\mu}$$ to demonstrate that the Ward identity holds, $$k_{\mu}\mathcal{M}^{\mu}(k)=0$$ exploitting the fact that the current $$j^{\mu}(x)$$ is conserved (provided that classical equations of motion hold at the quantum level). My questions are the following:

1. Is $$\mathcal{M}^{\mu}(k)$$ a Feynman amplitude (i.e. the scattering amplitude with the momentum conserving $$\delta$$ function being stripped off) or a scattering amplitude? (My guess is that $$\mathcal{M}^{\mu}(k)$$ is a scattering amplitude! )

2. In deriving the Ward identity, we basically had to substitute $$k$$ with $$-i\partial$$ and then to exploit the fact that the four-divergence of the expression $$e^{ik\cdot x} \langle f|j^{\mu}(x)|i\rangle$$ vanishes inside the spacetime integral (otherwise we can not reduce $$k_{\mu}\mathcal{M}^{\mu}(k)$$ to a single integral $$\int d^4x e^{ik\cdot x}\langle f|\partial_{\mu}j^{\mu}(x)|i\rangle$$, that vanishes due to the divergence of the current being zero. Does this imply that the spinor fields vanish at the asymptotic regions in space? Or does the vanishing of the four-divergence occur because of something entirely different?

Any help will be appreciated.

• What is a "Feynman amplitude"? And how do you think it differs from a scattering amplitude?
– hft
Jun 29 at 20:26