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$$q_\mu \Gamma^\mu (p + q, p) = S^{-1}(p+q) - S^(p)^{-1}.$$

Often the 'Ward-identity' is stated as, if one have a physical process involving an external photon with the amplitude

$$M = \epsilon_\mu M^\mu$$

then

$$q_\mu M^\mu = 0$$

if q is the momentum of the external photon. One can argue on that the latter identity is true because of current conservation, but can one show that it follows from the Ward-Takahashi identity above? If so how?