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Let the system formed by particle(microscopic or macroscopic)- environment coupling be described by $|\phi><\phi| $ . In decoherence approach, to retrieve the classical properties, one focuses on $Tr_a (|\phi><\phi|)$. 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.

  1. 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 $|\Phi>$, 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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  • $\begingroup$ If you don't get an answer here, it may be worth trying in the quantum computing stackexchange $\endgroup$ – KF Gauss Jun 23 at 17:32
  • $\begingroup$ It's not clear to me in what sense you expect to recover classical physics, or whether you're expecting too much. Decoherence only produces classical results in the sense that it prevents pointer states from interfering. It doesn't allow us to measure both x and p of an electron classically, violating the Heisenberg uncertainty principle. $\endgroup$ – Ben Crowell Jun 24 at 3:24
  • $\begingroup$ @Ben Crowell, I wasn't expecting decoherence to produce the conclusion that under appropriate conditions, an electron could have well defined position and momentum. I suppose it shouldn't if it is consistent with experimental facts. But on the other hand, I was of the impression that decoherence could lead to the conclusion that a macroscopic body(By which I mean bodies that are macro-realistic and subject to non-invasive measurability of properties) should have a well defined position and momentum values, due to the interaction with environment, thereby deriving classicality. Can it? Does it? $\endgroup$ – GlaDos Jun 24 at 3:37
  • $\begingroup$ @LarryDavid: I was of the impression that decoherence could lead to the conclusion that a macroscopic body [...] should have a well defined position and momentum values I don't think this has anything to do with decoherence. A macroscopic body is subject to the Heisenberg uncertainty principle just as an electron is. It's just that the HUP is generally not a bound that matters to us when we talk about a macroscopic object, because position and momentum are uncertain for classical reasons. E.g., maybe I can measure the position of a dog to $\pm 0.1$ m and the momentum to $\pm 0.1$ kg.m/s. $\endgroup$ – Ben Crowell Jun 24 at 11:21
  • $\begingroup$ @Ben Crowell, I didn't understand what you meant by " uncertain for classical reasons " . Further, although HUP applies to all systems, I assume there's nothing in principle ( if not for the environmental noise) preventing a macroscopic body to have superposition in position basis or momentum basis in it's centre of mass degree of freedom. $\endgroup$ – GlaDos Jun 24 at 13:08
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As an example of simultaneous approximate decoherence with respect to the bases associated with two non-commuting observables, consider position and momentum.

Decoherence tends to diagonalize the density matrix in the position basis because interactions are local in that basis. However, the kinetic terms compete with this, so decoherence in the position basis is never perfect. Over time, these two effects tend to continually balance each other, so that positions and momenta both have approximately well-defined time-dependent values (for macroscopic purposes). For momenta, this occurs simply because the momentum observable is (proportional to) the time-derivative of the position observable, so if positions are continually being approximately localized, then so are the momenta.

Once positions and momenta both have approximately well-defined values, the energy does, too, because the energy (at least in one broad class of models) is expressed in terms of position and momentum observables.

One way to approach the calculations is to start with estimates of the decoherence rate and asymptotic coherence length in the position basis, as in

(Tegmark's paper is actually critiquing a modification of quantum theory, but calculations using ordinary quantum theory are included for comparison.) Once we have those quantities, we can compare them to the rate of dispersion due to the kinetic term and make estimates about where the balance lies. That's only a back-of-the-envelope approach, but it has the virtue of building intuition.

Does it work for all environment-system interactions and for all initial states? Probably not, but I suppose it depends on what you mean by "all." In any quantum system involving various "objects" (particles, fields, or whatever) interacting with each other, those objects will tend to become entangled. However, with a complex pattern of arbitrary not-necessarily-local interactions in the Hamiltonian, it's not clear to me that any given list of observables would consistently approach well-defined values for all initial states. I don't know of any "theorems" with that level of generality, though. Maybe some interesting results in that department are just waiting to be discovered. However, if we restrict attention to conventional quantum field theory in which the Hamiltonian is local in space, then approximate decoherence in the position basis (and, via time-dependence, the momentum basis) will probably tend to dominate.

That assumes, of course, that we're talking about a theory with one or more characteristic spatial scales built into it, as in QED or QCD. If we consider a conformal field theory instead, as in the CFT side of the AdS/CFT correspondence, then my argument fails. That's good, because otherwise the AdS/CFT correspondence wouldn't be viable as a theory of quantum gravity in the higher-dimensional AdS spacetime. For that reason, a study of decoherence in CFT would probably be very interesting.

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