Semiclassical approximation in Quantum Field Theory

Question

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

Answer 1

I see this as a partial trace on the $\psi$ space, that is :

Let $H$ be the hamiltonian applying on the $(\psi, \chi)$ space, and $H_{eff}$ the reduced hamiltonian applying on the $\chi$ space. Take some density matrix $\rho$ applying on the $\psi$ space.

Then : $H_{eff} = Tr_\psi(\rho H)$

Here one is taking $\rho = \int d^3p f(E_p) |\psi(p,s)\rangle \langle \psi(p,s)|$

Answer 2

This looks to me like essentially a mean-field approximation. One is replacing $\psi$ with its expectation, so one is treating $\psi$ classically and $\chi$ quantumly. The back-action of $\chi$ on $\psi$ is ignored.