content_image/300605.jpg content_image/300606.jpg content_image/300607.jpg content_image/300608.jpg content_image/300609.jpg content_image/300610.jpg
Using the E-L equations ( ∂$_{α}$∂ L /∂∂$_{α}$A$_{β}$ − ∂ L /∂A$_{β}$ = 0), we find that 

<img src='content_image/300605.jpg'>

<img src='content_image/300606.jpg'>

where we get from the second to the third line be choosing to be in the Lorentz gauge ( ∂$_{α}$A α = 0). 

For the Beta decay problem, following Peskin 6.1 (pgs. 177-180) [251], our external current takes the form 

where, for the trajectory of our particle, we take 

We will work with the Fourier transform of the pertinent quantities: 

<img src='content_image/300607.jpg'>

<img src='content_image/300608.jpg'>

<img src='content_image/300609.jpg'>

<img src='content_image/300610.jpg'>

