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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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That is only true if you take a rough look at the number of neutrinos. Actually, a small difference is found when comparing the number of neutrinos during the night (going through the earth) and the day (without having to cross the earth). This difference is mainly due to the high sensitivity of the detectors on electron neutrinos. When interacting with ...



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