content_image/323504.jpg content_image/323505.jpg content_image/323506.jpg content_image/323511.jpg
external field “tickling” the charge, we find that 

<img src='content_image/323504.jpg'>

In order to make contact with the classical limit we assume a soft photon: | ⃗ k | ≪ | ⃗ p ′ − ⃗ p | . Then we can take 

and we can ignore the / k in the numerators. 

Using Eq. (E.11) we have that Eq. (F.82) becomes 

<img src='content_image/323505.jpg'>

<img src='content_image/323506.jpg'>

Summing over polarization states, integrating over phase space, and taking the conditional probability yields 

<img src='content_image/323511.jpg'>

For massless QED we have that in a diagram that satisfies the Ward iden- tity ∑ λ ϵ λ $_{µ}$ϵ ∗ λ ν = − g$_{µν}$ . Then we have that the energy radiated, to lowest order, 

