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First, the classical and semiclassical adjectives are not quite synonyma. "Semiclassical" means a treatment of a quantum system whose part is described classically, and another part quantum mechanically. Fields may be classical, particle positions' inside the fields quantum mechanical; metric field may be classical and other matter fields are quantum ...


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There are results that are mathematically rigorous concerning the semiclassical limit of quantum theories. It is in fact an ongoing and interesting theme of research in mathematical physics. However you need to be rather well versed in analysis to understand the results. The bibliography is quite huge, but I would like to mention the following (some quite ...


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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 ...


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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.



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