(I am importing this question that I asked in Physics stack exchange)
Let the system formed by particle(microscopic or macroscopic)- environment coupling be described by $|ϕ\rangle\langleϕ|$ .
In decoherence approach, to retrieve the classical properties, one focuses on $Tr_a(|ϕ\rangle\langleϕ|)$. Where the subscript 'a' denotes a particular basis spanning the Hilbert space of the environment. The amplitudes associated with the states in this reduced density matrix are then interpreted as classical probabilities. I have a single questions with this approach in relation to the idea of retrieving classical dynamics.
As for the particle is concerned, this consideration is with reference to only one of the observable associated with said particle. i.e. In the state $|Φ\rangle$, the entanglement involves the eigenstates of an observable of the environment on the one hand and the eigenstates of an observable of the particle on the other hand, and when one says that the particle's quantum coherence is approximately lost, it is only with reference to the eigenstates of the latter observable characterizing the particle. But for the claim 'classical physics has been retrieved' to be validated, one would expect a similar loss of coherence for all the physical quantities associated with the said particle. Which means, not just it's position(say) must be shown to apparently become classical, but also it's momentum, internal energy and other classical physical quantities that are associated with that particle. Is such a calculation possible within the decoherence approach? If so, how does one proceed? And does it work in the broadest generality possible(That is for all environment-system interaction and for all initial states)?