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(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)?

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  • $\begingroup$ Is this equivalent to can it be in a basis state of every operator at the same time? Then the answer is no, because different operators (corresponding to different properties) have different basis states, and you can't decoher all of them at once. $\endgroup$ – Mahathi Vempati Jul 2 at 4:38
  • $\begingroup$ I am not sure if I got the meaning of your question. Anyway, this question is related to the observation that interactions usually leads to the suppression of interference among the eigenstates of one particular observable(the position observable for example), and leaves the other non commuting observable untouched.Given this state of affairs, how does decoherence account for the character of classical objects in which ALL observables(Or Physical quantities) associated with it have sharply defined values $\endgroup$ – GlaDos Jul 3 at 6:36

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