content_image/295581.jpg content_image/295582.jpg content_image/295583.jpg
With two transverse polarizations ⃗ ϵ$_{λ}$ ( ⃗ k ) we have that 

<img src='content_image/295581.jpg'>

This leads us finally to 

<img src='content_image/295582.jpg'>

We would now like to show that one may replace the three vector quantities with their corresponding four vectors, then take ∑ ϵ$_{µ}$ϵ ∗ ν → − g$_{µν}$ . If we orient k µ = ( k, 0 , 0 , k ) µ along the z -direction, then we may take our two transverse polarizations as ϵ µ 1 = (0 , 1 , 0 , 0) µ and ϵ µ 2 = (0 , 0 , 1 , 0) $^{µ}$. It is now clear that we may freely take the three vectors to four, as the zeroth components of the polarization vectors is zero. To turn the sum over polarizations into a metric we need that the Ward identity be satisfied, k$_{µ}$j µ = k$_{µ}$ A µ = 0. Then, with our k oriented along z , 

<img src='content_image/295583.jpg'>

