Two questions regarding decoherence 1) preferred pointer basis: Decoherence explains why a quantum object interacting with a measurement device plus environemt results in a "classical state" of the measurement device. What is the explanation for a "preferred pointer basis"and why does the "classicalization" i.e. "localization" takes place in position space (why not momentum space)? 2) double slit experiment: In a double slit experiment with single quantum objects passing through the slits these quantum objects appear as localized dots at a screen behind the slits. This localization is caused by decoherence due to interaction with the environment after interference took place. But the individual quantum objects interact with environment already when passing through the slits; so why does localization not take place earlier (which would mean that no interference pattern does appear)?
You seem to be assuming that decoherence solves the measurement problem and describes the full way to a classical description of the system. This is not the case, specifically the pointer basis is not specified. But there are also other problem. Decoherence only describes the mean behavior of ensembles and not single state histories, so it's also not capable of describing the single system measurement outcomes we are observing. I personally believe that decoherence is overrated as a tool for understanding the macroscopic and open system limit of quantum theory. So I don't believe you will find answers to your question within this framework. Cheers, Jazz
No, I don't. I know. Does that mean that there is no explanation for a pointer basis? where it comes from? construction of the device? Any idea regarding question 2?
This might be helpful; it describes how an apparatus which is designed classically to measure a certain observable ends up in quantum mechanics to have a pointer basis relevant to that observable. Basically, it has to with the way in which the apparatus is interacting with the particle being measured; the symmetries of this interaction lead to the appropriate pointer basis.
The preferred basis of decoherence is the basis consisting of the eigenstates of an operator commuting with the Hamiltonian. (That's because such states are also Hamiltonian eigenstates, so do not change with time.) Decoherence usually takes place when the interaction is strong, i.e., when the interaction part of the Hamiltonian V(x) is dominant. But V(x) commutes with x, so in the approximation in which the Hamiltonian can be approximated by V(x), the preferred basis can be approximated by the basis |x>. It doesn't take place earlier because a good double slit experiment is carefully designed such that all interactions before the screen are sufficiently weak, so that their effect can be neglected.
Good point; sounds reasonable It would be interesting to see the results of a not so carefully designed double slit experiment ;-)
If there was a strong decoherence-producing interaction at the slits, then there would be no interference between waves coming from different slits. Consequently, the distribution on the screen would look like a distribution of classical particles coming from two slits, without the interference fringes.
Schlosshauer's book on decoherence mentions a double-slit experiment that was repeated many times with different air pressures. As they kept increasing the pressure, the pattern showed less interference. If I remember correctly, the particles in that experiment were C70 molecules (like buckyballs with 10 extra atoms). The old article "Pointer basis of quantum apparatus: Into what mixture does the wave packet collapse?" by Zurek is probably a good place to start. http://astrophysics.wfis.uni.lodz.pl/100yrs/pdf/14/030.pdf. I don't remember the details, since it's been years since I read it, but think it explains why the pointer states are eigenstates of an operator that commutes with the interaction part of the Hamiltonian. I think this turns your question 1 into a question about why all the interesting interaction terms involve functions of x rather than some other observable.
A thought experiment in Ghirardi's "Sneaking a look at God's Cards" purports to provide a means of empirically distinguishing between actual wavefunction collapse and decoherence. (In fact Ghirardi apparently makes a bolder claim, that this is an empirical test of the Copenhagen interpretation!). Here's how it works: if A is the observable whose eigenstates form the pointer basis of an apparatus, Ghirardi proposes to perform a measurement on an observable Z of the apparatus which is incompatible with A. Does anyone know whether such an experiment has been performed? In practice our apparatus has a position pointer basis, because we have to read off the position of the pointer, so we would have to somehow perform a momentum measurement of the apparatus pointer or something.
That's the thought experiment I keep referencing to! As far as I know, no experiment like it has been performed.
This is a very vague explanation, and cannot be correct. First: How can you tell given a Hamiltonian, whether the interaction part dominates? Second: As H always commute with H, the pointer basis should be the basis of eigenstates of H. But it usually isn't. So your explanation misses the question which operator commuting with H is the relevant operator. Just postulating that it is V(x) is no better than postulating the preferred basis itself. Third: The relevant Hamitonian is the Hamiltonian of system plus environment, in which case the interaction is not just V(x) but V(x_1,...,x_N) with huge N. But the pointer basis is a basis of the small system. The real explanation is something like that: The relevant pointer basis exists (and then is often diagonal in position) only if the system is macroscopic, in the sense that one can approximate it well enough by going to the thermodynamic limit. Decoherence doesn't work without such a limit, which is explicit or implicit in all decoherence models that start with the environment included. In this limit, 3D position typically plays a distinguished role as the slowest modes of the whole system (including the environment) are parameterized by observable mean fields with a position argument. This is the hydrodynamic regime that we observe macroscopically; the fast modes are averaged over, hence are unobservable except with extremely sensitive equipment. Thus, typically, whatever is observed is localized in position. To see some of the mathematics involved, look at: Section 2C of the article by Herbert Spohn in Rev. Mod. Phys. 52, 569–615 (1980) contains the hydrodynamic limit of a simple system. (The general case is not very well understood.) A paper by Davies, Comm. Math. Phys. 39 (1974), 91-110 (available e.g. at http://tristan.benoist.free.fr/Stage/ ) shows rigorously how in a simple situation a pointer basis and the associated POVM arise as a Markovian limit (which is a form of the thermodynamic limit). This was done long before the modern terminology of decoherence attracted the attention to this sort of phenomenon. Actually the screen localizes most photons arriving - they get stuck at the screen. Only a few photons pass the screen (as there are two slits where screen interaction with the photons is virtually vanishing) - these are bilocalized by the screen, and spread out biradially after the screen, thereby producing the interference pattern.
Neumaier, what do you disagree with in the following paper by Zurek? http://astrophysics.wfis.uni.lodz.pl/100yrs/pdf/14/030.pdf
Already the abstract is vague. ''The operator that commutes with the apparatus-environment interaction Hamiltonian'' is not well-defined. There are many such operators, for example the all zero operator, whose eigenvectors are completely arbitrary. This ambiguity is not resolved in the main text. It appears again in (3.4), where the more detailed form is given - the operator Pi there is quite arbitrary, apart from the fact that it is composed of pointer basis dyads. In fact, rather than making this arbitrary conclusion form (3.3), the fact that (3,3) must hold for arbitrary coefficients pi means that it must hold for each dyad alone, see the discussion immediately before (3.3). This implies that the pointer variables are eigenvectors of the interaction part H_{AE} of the Hamiltonian. Note that it is also assumed - in (iii) at the top of the column containing (3.3) - that the pointer basis also consists of eigenvectors of H_A. Thus one needs common eigenvectors of H_A and H_{AE}. And one needs a pointer basis - therefore many of these eigenvectors should exist. It seems to me that this requires that H_A and H_{AE} commute. But this excludes the case where H_A is purely kinetic and H_{AE} purely potential - the situation that resembles most the somewhat garbled setting discussed by demystifier. Thus Zurek's main result - stated at the bottom of the column containing (3.6) - has the following more precise statement: A necessary condition for having an apparatus is that H_A and H_{AE} commute. In this case, the pointer basis is the basis of H_A consisting of joint eigenvectors of H_A and H_{AE}, projected to the apparatus Hilbert space.