content_image/313455.jpg content_image/313461.jpg content_image/313456.jpg content_image/313460.jpg content_image/313462.jpg
The occupation number n k is given by the Bose-Einstein distribution 

<img src='content_image/313455.jpg'>

where we are neglecting the chemical potential. Substituting this into Eq. (A.10) gives 

The 1 2 term that appears in Eq. (A.10) is not present in Eq. (A.12) as it can be removed by one of the following: 

− Having n$_{ω}$$_{k}$ ≫ 1 2 

− Subtracting the vacuum 

− Performing normal ordering 

Eq. (A.12) can be expressed as 

The proof is as follows. The definition of the J term is 

where for our derivation we define 

<img src='content_image/313461.jpg'>

<img src='content_image/313456.jpg'>

<img src='content_image/313460.jpg'>

<img src='content_image/313462.jpg'>

