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A short course on Bohm's theory.

  1. Feb 13, 2005 #1

    vanesch

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    I would like to invite people who want to study Bohm's theory in this thread. The aim is to *study* the theory, because people who (think they) are confortable with quantum theory (such as me) sometimes don't know much about Bohm's formulation of it (such as me!). As there are some proponents of this theory around (ttn ?) maybe they can enlighten us.
    DrChinese posted an interesting link already: http://plato.stanford.edu/entries/qm-bohm/

    I have in front of me: "Quantum Theory" from David Bohm, his textbook on QM. However, it dates from 1951 (I have a Dover reprint) and seems to be on the Copenhagen theory ! Maybe it was the completion of this book which pushed Bohm not to be content with this approach ?

    So, professor, I'm listening !

    cheers,
    Patrick.
     
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  3. Feb 13, 2005 #2

    vanesch

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    I did some reading up on Bohm (thanks to DrChinese's link).

    If I understand well the premisses of Bohmian mechanics, it goes as follows:

    There is a wavefunction on configuration space psi(q1,...qN) which will evolve exactly as dictated in conventional quantum theory, with the schroedinger equation. So, in Hilbert space notation, we work in the "configuration basis" (the position basis) |q1,...qN>.
    Next, it is postulated that there is a real configuration, given by the variables {Q1...QN} which is the "real" state of the system, but of which we have to POSTULATE that initially, this configuration has a probability of |psi(q1...qN)|^2 to be equal to {q1,...qN} in a statistical mechanics way.

    It is further postulated that the evolution equation in time of these values Q1..QN are given by:

    dQk/dt = j_k/rho

    with j_k and rho the probability current and density in configuration space, as given by standard QM.

    Given the evolution equation for the {Q1...QN} it is clear that, considering the conservation equation d rho/dt + div j = 0, at ANY time the initial population of Q1...QN will remain distributed according to psi* psi.

    So it is as if, in the position basis (or more generally, the configuration basis) we have a wave function psi evolving purely unitary, and we consider a kind of "token" that indicates WHICH one is now the real one. But because of our initial uncertainty on where was the token initially, we end up having a statistical distribution in position basis which is the same as the one given by the Born rule upon position measurement (or "configuration measurement").

    I take it that Bohm somehow also postulates that every measurement, at the end of the day, is a position measurement of something (the pointer states?).
    (in the above, I use "position" and "configuration" mixed ; it is clear that there is a preferred basis in Bohm's theory, and it is the one I'm talking about).

    Do I have it right to think that the "initial wavefunction" is part of the state description, but that the distribution of the initial {Q1...QN} is simply due to lack of knowledge on my part ?

    If so, I hit a conceptual difficulty.

    Imagine I do a 2-slit experiment. So I have an initial wavefunction which describes the two slits, and I'm supposed to accept an initial ignorance distribution of {X} corresponding to psi* psi.

    After measurement of the impact, however, I can track back exactly through WHICH slit my particle went, because knowing the initial psi, there is no limit to the accuracy by which I can reconstruct the trajectories. So AFTER the measurement of X (which, by itself, can be made arbitrarily precise) I can reconstruct in the past, the track back.
    But that would mean that with the information I gained now, I KNOW the initial value of {X} and it is not distributed anymore as psi* psi, but psi should now be replaced with a deltafunction on X. However, that will not give me the right final density.

    The problem seems to be that the information obtained by the measurement changes the ignorance I have about my initial distribution of {Q1...QN}, in that it picks out one single value. And then my nice relationship between psi* psi and my probability distributions don't go anymore.

    How is this handled by Bohm defenders ?

    The second conceptual difficulty I have goes as follows.
    Imagine a closed system, including me, but within a container that doesn't allow for any external influence. This system can be described by a huge number of particles, so its configuration space is enormous, but with a finite number of q_i.
    Now, imagine I do a spin measurement, and I observe it. According to unitary QM (which is shared with Bohm), the overall wavefunction is now in a superposition:
    |me+> |spin+> + |me->|spin->

    I take it that, given that Bohm postulates unitary QM everywhere (I like that :approve:) we arrive at such a wave function in configuration space.
    But now I've SEEN that it was spin+. I'm now going to do an experiment with the spin of that particle. In a "collapse" thing (Copenhagen), the fact that I've seen this spin + would indicate that the new wave function is simply |me>|spin+>. But we cannot do that in Bohmian mechanics, which postulates strict unitary evolution.
    So what will be the distribution now of the "tokens", if I consider this as an initial situation ? It should be 50/50.
    And if I do a NEW measurement of spin ? I should get 50% chance of having spin+ and 50% chance of having spin -.
    But that cannot be ! I know it is spin + (and a second measurement will confirm this). So OR after this first measurement, the link between my psi* psi and my distribution of tokens is broken (my wave function says: 50-50, and I know 100-0) which is in contrast with one of Bohm's postulates, OR my wavefunction has to collapse, with the same problems as in Copenhagen.

    How does a Bohmian respond to this ?

    cheers,
    Patrick.
     
    Last edited: Feb 13, 2005
  4. Feb 13, 2005 #3

    ttn

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    That article by Sheldon Goldstein is a great place to start. Highly recommended.

    There was also a very good paper in AmJPhys about 6 months ago written by Roderich Tumulka. It's a "dialogue" between someone asking questions about Bohm's theory and someone who mostly answers them. Very readable and very illuminating on some typical FAQs. Maybe the paper is on arxiv??? .... Yes! Here it is:

    http://www.arxiv.org/abs/quant-ph/0408113



    That's it exactly. Bohm came up with his hidden variable theory in 1952 -- just after thinking through the Copenhagen approach in great detail while writing that 1951 book.
     
  5. Feb 13, 2005 #4

    vanesch

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    Yeah, nice. But I'm now more and more convinced that my objection holds: even if we have initially taken for granted that, in a closed system in quantum state psi, the true configuration is unknown with a distribution given by |psi|^2, and even if I accept without any problem that the way Bohm's mechanics is build, this stays so during unitary evolution, there IS a clash from the moment you make a measurement:
    the wavefunction continues to evolve in the same way as if it weren't a measurement, but the measurement HAS increased your knowledge and HAS changed the "ignorance" probability distribution of the "true configuration" ; so after a measurement, it is NOT TRUE ANYMORE that the ignorance of your state is still given by |psi|^2 EXCEPT if you now collapse psi.
    You can try to weasel out in open systems, by somehow postulating conditional wavefunctions and so on, but it won't work in a closed system. And in fact these arguments are JUST AS BAD as the claims that DECOHERENCE SOLVES THE MEASUREMENT PROBLEM. This last claim is NOT true in QM, and it is just as untrue in Bohmian mechanics.
    The only thing which is achieved in both is that for all practical purposes, the LOCAL density matrix diagonalises in the pointer states (in Bohm: configuration variables). BUT THE STATISTICAL SELECTION OF ONE is just as much a difficulty in Bohm as it is in decoherence.

    cheers,
    Patrick.
     
  6. Feb 13, 2005 #5

    ttn

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    Which what is the real what? You mean something like: which branch of the wf is real? I guess that's OK, but it's more precise to talk about a specific point in the configuration space being the actual configuration of the particles. That way you don't get into any trouble when it's not clear what basis to use for the wf (and hence "how many branches" there are, etc.).


    Yup.


    Is this really an extra postulate? Can you think of an example of a measurement of something that isn't, in fact, a measurement of the position of something?



    Yes, plain old ordinary classical-stat-mech style uncertainty.

    I'm with you up until the last part. Yes, you can "retrodict" the trajectory of the particle, e.g., infer from the spot where it lands which slit it went through. So... you can "beat the uncertainty principle" in the past, but not in the future. I'm not sure what you're worried about with your last sentence, though. You find out where the particle landed, and then infer back to where it was at the beginning. Now you want to say: "aha, if you pretend you knew all along that that's where it started, you'd have an initial delta function probability distribution and hence predict that it lands with certainty at this one particular spot on the screen! But that's crazy since that's impossible according to QM." Or something like that, yes? Well, it's not crazy at all! The new, delta-function initial probability distribution does indeed predict that the particle will land at some one particular spot on the screen later -- the very spot where it does in fact land! So what's the problem??

    Maybe you're worried about what happens next, i.e., after the measurement, we somehow have better-than-\psi^2 knowledge of the probability. But that's not true. Looked at from the perspecitve of regular quantum theory, you did a position measurement, so the wave function collapsed to something like a delta function. So there is no contradiction between knowing where the particle is now and having the probability be given by |\psi|^2.

    Of course in Bohm's theory there's no actual collapse, no separate dynamical process. Collapse is something theorists do when they find it convenient (as I think Bell said). But I'm sure we'll get to that issue soon enough...


    I still don't see what the problem is. It's true that, when you go back and reconstruct that past event (the particle starting near the slits and then landing ... right there) you'll know more than you could have known watching it "live". But so what? This extra knowledge doesn't spoil the later/subsequent use of P ~ |\psi|^2, and, as far as the past reconstruction is concerned, it merely ensures that the thing you know happened, actually happens. No problem!


    Schroedinger's cat!

    OK, so that means the two branches of the wf are widely separated in the configuration space (the atoms in the needle of your measuring instrument are here rather than there, all the electrons in your brain that stored your memory of seeing the needle point a certain way are in different places than they would have been otherwise, etc...). And one of them contains the actual configuration point. That is, one or the other of those two things actually happened. You said you saw that it came out spin+. OK, so that's the one that *happened*.

    Also (and here's where the "collapse" comes in), the dynamics is local in configuration space. So picture the two branches of the wave function as two "lumps" of nonzero value at two widely separated points in the config space. The equation that governs the subsequent evolution of the "particle position" (in the configuration space, so... shorthand for the positions of all the particles in the system) gives the velocity in terms of the value of the wf *where the particle is sitting*. So that means, whichever "lump" of the wf doesn't contain the particle, is going to have *no effect* on the dynamics of the particle, now or ever (unless those two lumps are made to overlap at some point, but decoherence shows how extremely unlikely that is). OK? So you can simply decide to *ignore* that empty part of the wf and live life from now on as if that empty part had been "collapsed away". Of course, it hasn't *really* been collapsed away, but so long as it doesn't influence the motions of the particles, who cares? You can just ignore it. Anyway, that's how the collapse works in Bohm's theory. But now back to the cat...


    You can do it, for the reasons I outlined above.


    No way! If you just arbitrarily strip off the part of the wf refering to you and pretend that the state is |spin+> + |spin->, *then* you could say that there ought to be a 50/50 chance of getting spin-. But that isn't the state! Leave the "Patrick" (or was it a cat?) factors in the wf and then perform an additional spin measurement. You'll of course find that you always get spin+, since you already *told* us that you got spin+ the first time -- i.e., we already *know* that the actual configuration point after the first measurement is in the spin+ (and "patrick believes spin+") branch of the wf. Make sense? :smile:

    OK, I'm sure that will raise some additional questions/thoughts....
     
  7. Feb 13, 2005 #6

    ttn

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    Yeah, I think what's throwing you here is that you're not being consistent about what you mean by "closed system." If you're including yourself and all your favorite lab equipment inside the system, then you have to include the factors for all those things in the wf you write down. And if you don't want to include all that junk in your "system", well, good luck *measuring* the state of anything while keeping the system closed!


    I don't agree. The actual configurations (i.e., definite particle trajectories) add to decoherence just what is needed to solve the measurement problem. Decoherence shows that the different "lumps" of the wf in the configuration space get to be non-overlapping and it shows that under ordinary circumstances the lumps *stay* well-separated -- i.e., you pretty much cannot ever get them to overlap ("interfere") again. Well, in ordinary QM, you still need a collapse postulate to pick one of these as real. But not so in Bohmian mechanics -- the actual configuration is already there, and has been there all along. So given that there already is one of those lumps that is picked out as "special" (in the sense that it contains the actual configuration) decoherence is what justifies tossing out all those now pointless (dynamically impotent) lumps. So Bohmians do collapse the wf -- only, they are able to say that it is merely something physicists do for convenience, rather than some kind of unitarity-violating physical process.



    No, I think it's no trouble at all in Bohm.
     
  8. Feb 13, 2005 #7

    vanesch

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    No, I'm talking about the entangled state
    1/sqrt(2) {|me+>|spin+> + |me->|spin->}

    I think you agree with me that this is the wavefunction after measurement, right ?
    Now, I KNOW that the "token" is in the first branch, but the wavefunction is still the above superposition, which is the "initial state" to continue with. And it was postulated that the "initial uncertainty of the token" must go like |psi|^2, so according to this view, the token should be with 50% chance in the first branch, and with 50% chance in the second.

    If I take as initial state
    1/sqrt(2) {|me+> |spin+> + |me->|spin->} and I do AGAIN a spin measurement, I will find a state:
    1/sqrt(2){|me++>|spin+> + |me-->|spin->} through unitary evolution, and I was supposed to take as initial distribution of tokens a 50-50 distribution (remember: I had to start out with an initial distribution equal to |psi|^2 in order for Bohm's theory to be on par with QM).
    So I now have the paradoxical situation that AFTER the second measurement, I should have 50 % chance to have measured twice + and 50 % chance to have measured twice -.
    But this cannot be true ! After the first measurement, I ALREADY KNEW that the token was in the first branch.
    So for the second measurement, I'm apparently NOT allowed anymore to take the function "psi" and accord a probability of the initial distribution of {Q1,...} according to |psi|^2 !

    So, depending on what I know, the initial uncertainty on the {Q1...} variables can be given by a probability distribution of |psi|^2, or it cannot, depending ? But then why should we assume initially a distribution according to |psi|^2 ? If sometimes it isn't ?

    cheers,
    Patrick.
     
  9. Feb 13, 2005 #8

    vanesch

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    Yes, I want to include it all. It is possible. After all, in a closed room you can do measurements, no ? Or is that prohibited ? Is the result of a measurement dependent on the fact that the door is open ?

    EDIT: I should maybe add that the |me> states in the previous message are "the rest of the room, including me, and my measurement apparatus".

    I understand the argument about the separation in configuration space. It is the same as the high dimensionality in decoherence, and comes down that we can consider individual EVOLUTION of the branches without - for all practical purposes - any interference from the other branches.

    But the thing I'm having difficulties with is that, in order for Bohm to get the right results out, he has to POSTULATE that the initial distribution of {Q} must be equal to |psi|^2.
    Clearly, it is not always the case, for instance in the trivial case I showed, when I (part of the closed room) KNOW already in which branch the token must be. But if it is not always the case, then how fundamental is this ? And if it is not a fundamental thing that the initial distribution of the {Q} is given by |psi|^2, then why should it be so in the first place ?

    cheers,
    patrick.
     
    Last edited: Feb 13, 2005
  10. Feb 13, 2005 #9

    vanesch

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    The color of light ?

    cheers,
    Patrick.
     
  11. Feb 13, 2005 #10

    vanesch

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    Yes, but on the condition that you keep the wavefunction to be the original wavefunction, and do not replace it with a deltafunction. Because if you do so, your trajectories will change, and your particle will NOT land where it landed, and from which you tracked back.

    cheers,
    Patrick.
     
  12. Feb 13, 2005 #11

    ttn

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    Sure.


    Sure, nothing *forces* you to throw away the now-dynamically-irrelevant parts of the wf. So in principle you are free to believe that there is a 50% chance for each of the two branches to "contain the token." But then that contradicts what you said in the original setup of this example -- namely, that you learned that the outcome was *in fact* spin+. But if you want to carry along the empty part of the wf in your calculations, you're free to do so.


    So what's the problem? If -- AS YOU CLAIMED BEFORE! -- the "token" was in the + branch before the second measurement, it will remain there after the second measurement too. Of course, if you pretend you don't know where the token is, you will say there's a 50/50 chance for +/-. What's the problem? :tongue2:


    Look, either you somehow knew that the actual configuration after the first measurement was +, or you didn't know that. If you didn't, there's no problem saying there's a 50/50 chance after another measurement, obviously. If you did, then you know that the chances *aren't* actually 50/50 because you have more information about the actual configuration. You evolve both the wf and the "token location" forward in time, and you find (duh) that the wf is |me++>|spin+> + |me-->|spin-> AND THAT THE TOKEN IS IN THE FIRST TERM. To claim that there is really a 50/50 chance that the second outcome will be -, you have to *contradict* what you said earlier -- namely, that the token was in the first term.

    Is your point just that this means the probability isn't always psi^2? i.e., How come it's possible sometimes to know the location of the "token" with better precision than |\psi|^2? Is that what you're after?

    I think you just have to remember that the "location of the token" suffers merely "regular old classical type" uncertainty here. So if you learn something about it, you should incorporate that knowledge into your future predictions. The |\psi|^2 law isn't like a fundamental law of nature, it's just something that seems to describe how particles seem empirically to be distributed within their wfs. If you learn more, use it (typically by tossing out now-irrelevant parts of the wf).


    I'm not entirely sure why you'd want to. Doing so would literally contradict what you claim to have just learned. It's like saying: flip a coin; there's a 50% chance for heads; you see it land tails; can't I still say there's a 50% chance of heads? Well, you can say that, sure. Maybe what you mean is that there's a 50% chance of heads for the next flip or something. But if what you mean is that there's a 50% chance that the tails staring you in the face is really a heads, I'm not sure what would possess you to say that.

    But in principle you can do this -- in your words, you are certainly "allowed" to (say, ignore what you learned and) thus use the full

    |spin+>|me+>+|spin->|me->

    state. But I don't think any contradictions will arise from this -- if you don't later *renege* on the claim that the "token" is in the ++ part of the config space! For example, if you do this, you'll find yourself in the "me++" state later, never in the "me--" state...


    The question of where the psi^2 distribution *comes from* in Bohmian mechanics is an important one. Let's just leave it for another message...
     
  13. Feb 13, 2005 #12

    vanesch

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    Just to make a resume of my "objections" (which are more nitpicking I suppose):

    The link between the "initial probability distribution of the token" and |psi|^2 is not 100% clear. If starting from scratch, you take them equal, but if not starting from scratch and already knowing some stuff, you assign other probabilities to your initial situation.

    On one hand this is not so dramatic. After all, it is a bit like in statistical mechanics: you can propose a certain probability distribution of the microstate phase space ; but on knowledge of say, the position of one molecule, this changes the probabilities, without affecting macroscopic parameters such as entropy or temperature.
    However, what is a bit disturbing is that the link between the initial distribution of {Q} and |psi|^2 seems to be broken ; so it is not axiomatic. So then it should be _demonstrated_ that this distribution must be given by |psi|^2 somehow. But I can imagine that this can be done (imposing, maybe, certain conditions on the initial state of the universe).

    Once |psi|^2 and the distribution of tokens are completely separate, another issue comes in. It is the fact that the wavefunction is an essential part of the description of the system. So it is not because, in the wave function:
    |me+>|spin+> + |me->|spin->, I know that the token is with the first branch, that the second, empty one looses its meaning. It exists, but is token-less. So we cannot deny it. We can deny it for all practical purposes, because of the size of the configuration space, and the lack of overlap, except in EPR like situations. This is identical with MWI in fact, except that there now is a "token mechanism" that implements the Born rule if somehow, we apply the right initial conditions.

    But I now have a problem with an EPR setup.

    Imagine I have 2 entangled electrons, in spin states |z+>|z-> - |z->|z+>

    Imagine I take the second particle on a spaceship and take it 100 lightyears from here, while the first particle remains in a container on earth.
    500 years later, the descendants of the travellers decide on whether they will do an x-spin measurement or a z-spin measurement with a stern-gerlach machine.
    About around the same time, on earth, a similar decision is made.

    How does this translate in the Bohmian formulation ?

    I would think that locally on earth, whatever do the travellers, this will not influence the measurement on earth, because the wavefunction is unaffected by the measurement results: it continues to evolve unitarily, doesn't it.
    But then we are in Bell Locality conditions, aren't we ?

    So how do we obtain the QM results ?

    Could you elaborate on such an EPR situation in Bohm ?

    cheers,
    Patrick.
     
  14. Feb 13, 2005 #13

    ttn

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    Not very fundamental at all.


    There are several schools of thought. Bohm himself speculated that there might be some sub-quantum randomness that would make the particles jiggle their way into the psi^2 distribution as a kind of "maximum entropy" equilibrium state. Valentini wrote some pretty cool papers on this in the 1990's and showed that you can formulate Bohm's idea clearly. Others, however, prefer a different approach based on the fact that psi^2 is the only natural candidate for a probability density because it is the only distribution which will remain true if it is true at some one time. This suggests it's not crazy to think that "god" distributed the "token" according to psi^2 at the beginning of the universe; then it can be shown that subsystems will obey it now.

    And there are some other ideas out there too for thinking about where this comes from. I personally haven't spent a lot of time worrying about them, though. I think once you recognize the non-fundamentality of the born rule in Bohm's theory, it doesn't much matter "why" the born rule is true. A perfectly good answer is just to say: the born rule is based purely on experiment. Given the deterministic dynamics of Bohm, we just have to find out *from experiment* how particles are sprinkled into their guiding waves. And psi^2 has always done the trick perfectly.

    cheers,
    patrick.[/QUOTE]
     
  15. Feb 13, 2005 #14

    ttn

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    How do you measure that exactly? With a prism? Or were you thinking of direct perception of the color with human eyeballs?
     
  16. Feb 13, 2005 #15

    vanesch

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    Ok, that's the point. But then, if it isn't a fundamental law of nature, one should explain me why I should take THAT as an initial distribution when I'm ignorant. But I don't claim that it cannot be done. Only, this somehow has to come out of the dynamics now and cannot be postulated.

    cheers,
    Patrick.
     
  17. Feb 13, 2005 #16

    ttn

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    Oh, OK, I get your point.

    This is a good illustration of the relative fundamentality of the wave function and the psi^2 probability distribution. In normal situations, the psi^2 prob dist seems to be the correct "best guess". (Abnormal situations here evidently include ones in which you have some additional information that you could use to winnow down the probability distribution in some way -- in which case, you ought to use it.) But, in principle, there's no reason why say god can't know the exact locations of all the particles. Then he'd be able to predict with certainty the outcomes of all sorts of experiments that we can only guess at with born rule probabilities. But this knowledge of god's wouldn't change the wave functions. In particular, the wave functions wouldn't become delta functions just because for god P(x) = delta(x).

    So, when you are doing "retrodiction", you can be just like god. You can know exactly where the particle was 5 minutes ago -- i.e., you can have knowledge that is much better than psi^2. But that's just knowledge. It doesn't make the wf change.
     
  18. Feb 13, 2005 #17

    ttn

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    The Valentini papers I mentioned argue that you can derive this (in some sense) from the dynamics.

    But I don't think I agree with you that this is necessary. I mean, it would be cool if you could do it, no doubt. But it's not like it makes the theory wrong if you just have to accept the psi^2 rule as an additional postulate to make the predictions match experiment.
     
  19. Feb 13, 2005 #18

    vanesch

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    Yes, for instance. The reason is that Bohm takes it easy with "the preferred basis problem" by just saying that it has to be the position basis. Now, decoherence people go through a great many effort and come to the conclusion that for macroscopic systems, often, the "decoherent basis" is ... well, the position basis. At least, for charged or massive things.
    But this is different for the EM field. There, the coherent states (classical plane EM waves) are closer to the natural basis. Hence my point. Now, as we, human beings, are mainly made up of massive and charged matter, the position basis will probably do. And if you dissect my eyeball, you'll probably say that in the end, I'm measuring the position of potassium and sodium ions in my optical nerve cells. If I were a creature made out of photons, however, the result would be different.

    Now, I can still throw something at you. I guess that when you talk about configuration space, that this is in the Hamiltonian sense. Now, what stops me from applying a canonical transformation, and redefine Q as the old canonical momenta "p", and take this as my configuration space ?

    cheers,
    Patrick.
     
  20. Feb 13, 2005 #19

    ttn

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    Yes.


    An objection that some MWI people make against Bohm is the following: since all these empty branches of the wf are "really out there", what makes you think you're not *in* one of them? Isn't there a "me-" wave function factor out there somewhere in config space who just measured "spin-" for his particle and will get "spin-" again if he measures again and so forth. And since that story is exactly the same as the story for the corresponding + terms (the only difference being that the + terms have "the token" sitting in them, but...) what's the difference really? Maybe you only *think* that the + terms are real. After all, the guy in the - terms thinks he is real! etc.

    I don't have any great answer at the ready for that, so maybe I'm not doing alot for my case here by bringing it up. But it seemed to be the direction you were heading and I do think it's an interesting point.



    Bohm's theory is nonlocal. Sending one particle through some magnetic fields with gradients in some particular direction will cause the distant particle to deviate a bit from the path it would otherwise have taken.

    I don't understand what you mean exactly. What the travellers do (e.g., which axis they point their SG magnets along) *will* influence the particle on earth. Look at the guidance condition in Bohmian mechanics (v = ...). The velocity of each particle depends on the entire configuration! That's the nonlocal part of the mechanism. When one particle gets "kicked", any of its entangled bretheren will feel the kick!


    Bohm's theory violates Bell Locality!
     
  21. Feb 13, 2005 #20

    ttn

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    If you can figure out how to talk about things like measurement-device-pointers in the momentum basis, more power to you.
     
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