I've recently stumbled upon a semiclassical approximation to quantum field theory that I've never heard of and have a hard time understanding. Consider the Hamiltonian,
\begin{equation} 
H = \frac{c}{ M ^2  } \bar{\psi} \gamma _\mu \psi \bar{\chi} \gamma ^\mu \chi 
\end{equation} 
Now the claim is that we can approximation how the $ \chi $ particle will behave in a medium of $ \psi $ particles with the following Hamiltonian,
\begin{equation} 
H = \frac{ c }{ M ^2 }  \left\{ \int \,d^3p f ( E _p  ) \left\langle \psi ( p , s ) \right| \bar{\psi} \gamma ^\mu \psi \left| \psi  ( p, s ) \right\rangle  \right\} \bar{\chi} \gamma _\mu \chi 
\end{equation} 
where $ f ( E _p  ) $ describes the distribution of $ \psi $ field in the medium. This should apparently be an obvious step but I am having trouble wrapping my head around it. How can I justify this step. This is quite different then integrating out the $ \psi $ fields.

**Note:** Though the context likely isn't very important I stumbled upon this step in a derivation of the MSW effect for neutrino oscillations in matter that can be found [here][1]


  [1]: http://arxiv.org/abs/1308.1029