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I want to show that the scattering amplitude of Compton scattering at the lowest order:

$$i(-ie)^2u(p)\left[\frac{\gamma_\mu (\gamma^ap_a + \gamma^aq_a + m)\gamma_\nu }{(p+q)^2 - m^2}+ \frac{\gamma_\mu (\gamma^ap_a - \gamma^aq'_a + m)\gamma_\nu}{(p-q')^2-m^2}\right] \bar{u}(p') \epsilon^\mu_{in}\epsilon^\nu_{out}$$

remains unchanged upon the polarisation shift:

$$\epsilon^\mu_{in} \rightarrow \epsilon^\mu_{in}+\alpha \, k^\mu$$

where k is the incoming photon's momentum. I tried playing with the expressions but I can see at all how this new term is to vanish.

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    $\begingroup$ Hi Piotr. Please note that exercise-solving questions are typically considered off-topic here, unless they ask about a concept rather than a calculation. In its current state (v1), your post is simply stating that you don't know how to solve this particular problem, and as such, it falls under the homework-and-exercises policy, i.e., it's off-topic. $\endgroup$
    – AccidentalFourierTransform
    May 25, 2017 at 17:19
  • $\begingroup$ I read the meta and I'm not sure what the problem is. Is it that I didn't supply the homework tag? Or that I'm asking about a calculation? If it's the latter, then how is this different from asking about the concept of the longitudinally-polarized photon not influencing the QED amplitudes? Would that be better? But then, is it not better to put that sort of question in some context? Sooner or later, to make the explanation concrete, one has to transition into mathematics as that's where the solution is - I'm looking to understand how the QED formalism deals with longitudinal photons. $\endgroup$
    – Piotr
    May 25, 2017 at 20:16

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The result is guaranteed to be unchanged from the Ward Identity, $k_\mu\mathcal{M}^\mu=0$, which is a result of gauge invariance.

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  • $\begingroup$ Consulting the textbook, the derivation of Ward identity seems to be a little beyond my current level. Does there exist a more pedestrian way of showing this? I've not seen much of the path integral approach to QFT. $\endgroup$
    – Piotr
    May 25, 2017 at 21:29

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