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No paradox in the EPR paradox

  1. Dec 6, 2005 #1
    I don't see why the EPR thought experiment would ever be concieved as a way to demonstrate anything against QM...

    (you can find an explanation of the EPR thought experiment here: http://en.wikipedia.org/wiki/EPR_paradox)

    Ok before Alice measures anything P(+z)=0.5 and P(+x)=0.5 for both electrons.

    Let's say Alice measures the spin in the z diretion and finds it to be +... this means Bob will find his electron's z spin to be - upon measurement, and that P(+x) remains = 0.5 for both electrons. No paradox if we look at it that way right?...

    Ok well imagine that instead of measuring the z spin of his electron he instead measures the x spin of his electron and finds it to be -ve. The way I think about Heisenberg's uncertainty principle is that he has affected the electron in such a way that he can no longer be certain of its z spin, and P(+z)=0.5 for his electron only, (just as when you measure an electron's z spin then measure x spin, you can no longer be sure of the electron's z spin) while Alice's electron remains in a state of +ve z spin and P(+x)=0.5.

    As far as I can see there is no problem...

    Learned views on this would be much appreciated.
    Last edited: Dec 6, 2005
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  3. Dec 6, 2005 #2
    As far as I understand (which has about 0.5 chance of being correct), EPR's argument is that Alice's electron's z spin can be measured precisely, and its x spin can be inferred precisely from Bob's measurement, thus violating Heisenberg's uncertainty principle. The fact that the two measurements can influence each other faster than light is the heart of the paradox.
    Wai Wong (QM newbie)
  4. Dec 7, 2005 #3


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    No, it's not a matter of violating Heisenberg's uncertainty relation. The whole point of EPR was the question of "hidden variables". In the experiment we have two electrons aligned so that they have opposite spins. As you say, before we measure them each has probabilty 0.5 of having positive spin. Is this a matter of the two electrons actually having no specific spin at that moment or does one of the electrons actually already have spin +, the other -, but we just don't happen to know which?
    In the EPR experiment, you allow the two electrons to separate and go apart, perhaps some very large distance. Now measure the spin of one of the electrons. The instant we measure the spin of one electron, we instantly know the spin of the other., If it were true that neither electron has any specific spin until it is measured, then when our experiment forces one electron's wave function to "collapse", the same thing happens instantly to the other electron, perhaps in another galaxy by this time!
    Einstein argued that that showed that the wave function is just a matter of our lack of knowledge- that an electron really does have a definite spin at any time- we just don't know what it is until we measure it.
  5. Dec 7, 2005 #4


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    Einstein's main objection to "orthodox quantum theory" (which means, essentially, the standard theory plus Bohr's completeness doctrine) was that the completeness doctrine rendered the theory non-local (i.e., apparently inconsistent with relativity's prohibition on superluminal causation). For Einstein, we simply have to choose either to reject the completeness doctrine (and, he hoped and believed, construct some kind of local hidden variable theory) or reject locality as a criterion (and accept a non-local theory). In other words, Einstein's point was that the completeness doctrine comes at a price, and the price is non-locality.

    However, it should be noted that the actual text of the EPR paper was written by Podolsky, and Einstein was extremely annoyed at how poorly it came out. So the EPR paper itself (not to mention the abundant secondary literature, most of which is quite bad) is not a good place to look to understand Einstein's real worries/objections/arguments against the quantum theory.

    A good place to start might be this article
  6. Dec 7, 2005 #5
    Oh I see now, thankyou. Hmmm.

    Isn´t it a pointless discussion? I mean, either of the two ways of looking at the issue could be correct right? I mean if Alice measures +ve spin you can argue that the electrons started out as A = +ve, B = -ve; or you can argue that she collapsed the superposition of states "A=+ve, B=-ve" and "A = -ve, B = +ve" into the state "A = +ve, B = -ve" right? I mean you´re never going to be able to prove one way or the other, no? (I personally prefer to think that the particles started out in one or the other state and we didn't know which until we measured, I mean its obvious QM is not a complete theory anyway hehehe: no gravity, singularities all over the place... and further you can´t say a theory violating locality is preferable to a theory with an unknown hidden variable.)

    (About the paper on the Einstein boxes: I prefer to talk about the EPR experiment; the concept of inserting a partition is vague.)
    Last edited: Dec 7, 2005
  7. Dec 7, 2005 #6


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    It would have been pointless if Bell hadn't come along. The actual results are inconsistent with the idea that Alice had (x+ and z-) or (x+ and z+) simultaneously. If you advance that idea, you run afoul of Bell's Theorem. All measurements will serve to convince you that there is only one non-commuting observable at a time.
  8. Dec 7, 2005 #7
    So according to your view of Bell's results Einstein was correct?

    However I read from: http://en.wikipedia.org/wiki/EPR_paradox


    "In 1964, John Bell showed that the predictions of quantum mechanics in the EPR thought experiment are actually slightly different from the predictions of a very broad class of hidden variable theories."

    Doesn't your view of Bell's results contradict Wikipedia's views?

    No offence but I see Wikipedia as more trustworthy an information source than yourself =)

    As far as I interpret the Wikipedia quote, it means that no-one has come up with a hidden variable theory that stands up to Bell's inequality test. This does not however mean that it is proven that there is no possible hidden variable theory that could stand up to Bell's inequality test. And so we are left with the choice: unkown hidden variable theory, or violated locality... and I don't see how one is better than the other, except that I feel more comfortable with the former.
    Last edited: Dec 7, 2005
  9. Dec 7, 2005 #8


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    No offense taken, although my web site is actually referenced by that particular page (and I didn't do it). :smile: Actually, my comments had precisely the opposite meaning and of course are fully in keeping with Wikipedia and other sources.

    EPR was not correct in the belief that there was simultaneous reality to non-commuting variables - called realism. Realism is often used interchangeably with hidden variables (HV). HV theories make predictions which are different than QM in some cases. In tests, the predictions of QM hold up and therefore the predictions of HV theories do not.

    Please note that this result applies to the entire "local" class of HV theories, which are now considered not viable.
  10. Dec 7, 2005 #9
    lol, oh sorry, I guess I misunderstood what you wrote. :blushing: Fancy that a reference I use to undermine your post actually references you! xD jeje :tongue2:

    So the Wikipedia reference is now outdated and all local theories are ruled out? :bugeye: Even the possibilty of a succesful local theory is ruled out? wow that's BIG news, and I'm extremely dissapointed. This means the Copenhagen interpretation is correct and that now I have to actually believe in superposed wavefuntions and non-local particles.. :yuck: yuck yuck yuck :( :cry: Please tell me it's not true!! (or confirm if you must and I will resign myself to the ugly truth... although actually I think I'll adopt the many worlds interpretation... still... gloom)
    Last edited: Dec 7, 2005
  11. Dec 7, 2005 #10


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    Wikipedia is right, it is simply a bit hard to understand. I am working to try to improve the language there but it is a long process. I have made a lot of progress on the Bell's Theorem page but not yet gone far on the EPR Paradox.

    I would do at least one of the following:

    a) throw out at all theories that do not allow for non-local wave function collapse i.e. that superposition does not extend to space-like separated regions.


    b) throw out all realistic theories i.e. where there IS simultaneous reality to non-commuting observables.

    You will find those of us around who might push a) over b) or vice versa, but the above is safe. Pick one. :tongue:
  12. Dec 7, 2005 #11


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    It's worse than you suggest, because Copenhagen QM is also non-local (and in precisely the same sense that Bell showed HV theories must be non-local). So the real conclusion of this whole development is that there is no viable local theory, period. (And note, it isn't just that nobody's found one yet: Bell's theorem allows us to know that nobody will ever find one.) Non-locality is a fact of nature.

    *That* is indeed BIG news. But it actually opens the door to something more reasonable if you're bothered by the craziness of Copenhagen quantum theory (and it sounds like you are). Specifically, it means that non-local HV theories are very much viable. And it isn't just a scholastic exercise: such a non-local HV theory actually exists, actually works, and (if you ask me) makes a heck of a lot more sense than Copenhagen QM. For more information, look here:

  13. Dec 7, 2005 #12


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    Ok, Vanesch, I think this one's ripe for you...
  14. Dec 8, 2005 #13


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    How many times do I have to act as the Guardian of the Faith :rofl:

    Non-locality is not a fact of nature. It is a fact of nature if you insist upon the unique existence of remote measurement results (which, I admit, may sound like a very reasonable hypothesis, but is totally oposed to the basic axioms of quantum theory).

    There *does* exist a theory (or an interpretation of quantum theory) that is both local, and in agreement with the observed violations of Bell inequalities, and that is a many-worlds view of quantum theory.

    The price to pay is of course that remote measurement results are not unique and well-determined. But we knew that already ! From the moment we accepted the superposition in quantum theory, we accepted that the electron was both above and below the proton in the hydrogen atom (not, as is often said, that it is "smeared out"). If you consistently apply that basic axiom to EVERYTHING, then yes, the multimeter both reads 5V and 3V. The strangeness doesn't appear here, it was built in from the start. Why do you not see a multimeter showing 3V and 5V ? Well, because there are two "yous" (your bodystate is also in a superposition), one who sees the 3V and the other who sees the 5V.

    Now, this can sound crazy (even to me :-) and all you want, but it is *nothing else but the strict application of the axioms of quantum theory to everything*, and not just to microscopic particles.

    All the interpretational problems come about because people "find this too crazy" and then go and mess with the formalism because they DON'T want to apply it according to the axioms to macroscopic systems, and invent extra tricks such as introducing a non-local collapse, giving definite and unique status to remote measurement results and all that, and then run into a lot of paradoxes.

    What Bell showed us is that *if you want to play these tricks* and at some point you want to LEAVE the quantum formalism and plunge into a classical formalism, with DEFINITE remote measurement outcomes, then you have no choice but to adopt non-local interactions - and as such kick out relativity (SR and GR).

    So my point is, as things stand today:
    1) we think that relativity is right (so we have to stick to locality)
    2) we think that quantum theory is right (so we should stick to its axioms)

    there is no real problem, because both together give you without any problem the EPR results in a local way... BUT we have to accept that there are "several Alices and Bobs" in superposition ; in other words, many worlds (in one flavor or another).

    To this one should add one big caveat. We don't know yet how to deal with quantum theory and *general* relativity. This may need a change of the picture. But as we don't know yet - by far - in what direction that will go, let's stick to what's known to work today and let's build a coherent picture of it.
  15. Dec 8, 2005 #14


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    Yes, my apologies. When I said that non-locality was a fact of nature, I meant only that non-locality is a fact of nature *according to normal standards of scientific proof*. (Here's roughly what I mean by normal: when a scientist does an experiment using some apparatus and, at the end of the experiment, the apparatus gives some kind of output, the scientist is entitled to believe that what he is seeing -- the output of the apparatus -- really exists. In particular, when a physicist shoots a photon toward a polarizer and uses a detector behind the polarizer to determine whether the photon passed or was absorbed, he is entitled to believe that the detector either definitely does or definitely doesn't "fire". That, after all, is what the physicist literally *sees*. If you accept that, then non-locality is a fact of nature.)

    The real issue here, by the way, is the hierarchy of knowledge. As an empiricist, I believe that all knowledge is ultimately based on perception. (This is what distinguishes science from mysticism.) It's true, in some sense, that the Bell experiments permit us to make a decision: either we can side with some extremely abstract principle of quantum theory (unitary evolution), or we can side with what we literally see with our eyes. To me, this is no choice at all. The whole empiricist approach tells us that the "extremely abstract" is *always* shaky relative to what we see with our eyes. So *of course* we should believe that experiments have definite outcomes -- we *see* that they do.

    I'm not sure that's true. MWI explains the observed violations away. It teaches that what we erroneously took as experiments showing violations of the inequalities, are actually something very different. Our beliefs that the individual runs of the experiment had definite outcomes (the detectors either fired or didn't) are, according to MWI, *false*, and hence so are all the fancy averages and correlation coefficients we calculate on the basis of the individual results. According to MWI, all of this (and much more) is a big delusion.

    Outcomes having definite measurements is uniquely part of "classical physics", eh?

    No, they allow you to explain those results away -- to tell a story in which those results didn't happen in anything like the way we *thought* they did based on what we actually saw.

    Ah, good advice! :smile: In particular, let's stick to the idea that you can believe what you see with your eyes. Surely the success of empirical science argues that that's the most basic thing that's "known to work today".
  16. Dec 8, 2005 #15


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    Ok, let's beef this out :biggrin:


    The problem is, that if you want to interpret a theory, you should STICK to it, and not make shortcuts to make things come out the way you'd like them to come out.

    But let's continue...

    Well, we only see that with OUR eyes. So we shouldn't - a priori - trust what the *other guy's eyes* have seen.

    This is a too distorted view of MWI. Imagine we look upon the thing from Alice's point of view. Then, ONCE she learns about Bob's outcomes, these outcomes are real, and existing and everything you want, to her. There's somewhere another Bob in another branch, which she didn't see (and will never see), who observed the _other_ results. But all this doesn't mean that Alice's outcomes are *illusional*. The results are really there in her branch ; only, that branch only makes sense when she interacted with Bob's information carrier (for instance, Bob himself) - before, the branch didn't exist, and as such, from Alice's point of view, Bob's definite result didn't exist. That's no problem because she didn't observe it herself.
    And this is what circumvents Bell's theorem: there IS all right direct influence of the measurement of Alice on the outcome at Bob, but not when the remote measurements have been performed (which, for Alice, do not have definite outcomes yet, and about which she didn't learn anything), but only when Alice LEARNED about them (and hence created as such, the branch in which these results make sense).

    *remote* measurements having definite outcomes is indeed, a classical concept. In quantum theory (in the MWI version), outcomes of measurements are a relative concept, relative to an observer ; and it is only relative to an observer that a set of measurements make sense. He can "retrodict" remote outcomes of measurements of other things which look like observers, but they only really make sense when he learns about it himself.

    But local outcomes of measurements, with respect to an observer, DO make sense - they are not illusions. This is the required mental shift to accept MWI - and as such never to be bothered again by EPR situations :-).

    This is not true. Because WHEN we see the results, they (relative to us) DO exist. Only, we extrapolate back in time that Bob's results already existed BEFORE we learned about them. And it is this *extrapolation* back in time which gives us all the EPR problems.

    Well, I agree with that: we should believe what we see with our OWN EYES. EPR gives problems when we also believe what others told us they saw with THEIR eyes, even when we had no means to see them with our OWN eyes.

    Note that MWI has not been invented to circumvent EPR ! It is there because it is what the *formalism* of quantum theory, systematically applied to everything, tells us, is going on.
  17. Dec 8, 2005 #16


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    We've been around on this so many times here, I'm already dizzy. So I'll make a few comments, and then give you the last word. Then we can let other people assess the two views and decide for themselves.

    It's not a question of how I'd like it to come out. The point is: if sticking to a theory involves letting go of basically everything you ever thought was true (including everything you ever perceived) perhaps the price is too high. (My view is: the price is definitely too high.)

    There are different flavors of MWI, so you're right that maybe what I said involved a distortion of your favorite flavor. But I don't agree that even your view fails to make basically all of Alice's knowledge illusional. She comes to believe that Bob's measurement had a definite outcome. In actual fact, according to MWI's portrait of the state of the physical world at that instant, this belief is false: Bob's measurement did *not* have a definite outcome. To say that it did is surely to deny the essence of MWI. To say that it didn't is, frankly, to let go of the kind of rational common sense that science is based on.

    You make it sound like when Alice learns about Bob's results, the wave function collapses and suddenly (retroactively) Bob's measurement comes to have a definite result (which before it didn't). But that implication is wrong. According to MWI, the wf never collapses. Bob's measurement never had and never will have a definite result -- Alice's belief to the contrary notwithstanding.

    Sure, if you define "relative to us they do exist" to mean nothing but "we erroneously believe that they exist"!

    If it weren't for the claim that MWI saved locality, I submit that *nobody* would take MWI even remotely seriously. It's just too stupid/crazy/la-la-land to even consider *science* unless there's some very powerful argument in its favor. I concede that saving locality is in the ballpark of such an answer. But I still come down on the side of saying: the price is *way* too high.
  18. Dec 8, 2005 #17


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    It's so good to learn at last that the sun really does go around the earth. For with my own eyes I saw it set in the West last night and rise in the East this morning. Heliocentrism is just a fairy tale! And don't get me started on that round earth fiction! Who could believe people on the other side hanging upside down! And in spite of local hills and dales I can see the earth is globally flat as far as my own eyes can reach!
  19. Dec 8, 2005 #18


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    Golly, I kept waiting for you to come to my rescue in the other thread on Realism/Locality when I was saying all that stuff about non-local WF collapse, and you never did... :tongue2:
  20. Dec 8, 2005 #19


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    Oh, you're not already giving up, are you ? :shy:

    What I'm trying to make you see is that you do NOT have to let go everything that you ever thought was true (only partly, the things you _reasonably expected to be true but weren't aware of_).
    It is LESS "Alice in Wonderland" than you seem to imply. (ok, it is MORE Alice in Wonderland than Newtonian physics, agreed).

    Her true knowledge is NOT illusional. Her _extrapolations_ are, up to a point, and not even.

    Now, why would she believe that ? I agree that in a classical picture, this is correct. But why would it be fundamentally a requirement to believe that somewhere remote, something had a definite outcome before you learned about it ?

    More, in fact it is _correct_ that there is *A* Bob which had a definite outcome (and ANOTHER Bob which had another, definite outcome). Alice simply doesn't know yet with *which* Bob she will end up in her branch, once she entangles with it. And then, if she extrapolates back in time, it is correct that *this* Bob had the definite outcome he told her later.
    The only true illusion Alice can have, is that there is only ONE Bob. But she can use that as a useful working hypothesis, because all she will ever learn from the one and only Bob that will be in her branch, will be consistent with having only one Bob around.

    I don't agree with the last statement. Of course it would be EASIER if we could deny those other Bobs to exist. But it is no strict requirement for science to be based upon this. Imagine for a moment MWI to be true.

    No, I didn't. Things only happen to Alice that way: she has now a locally observable Bob state in her branch which is completely consistent with what would have happened if Bob had a definite result. In fact *this state of Bob DID have a definite result* (and his twin in the other branch had also a definite, opposite result). But the whole point is that this branch, with Alice and Bob in local contact and with memory states which are completely compatible with a backwards projection in time of a definite measurement outcome, ONLY CAME INTO EXISTANCE when Alice met Bob after the measurements.

    So the wf never collapses of course, but _from the point of view of Alice_ everything happens AS IF the wf collapses into the branch in which she is (because she will only ever be interacting with other things in that branch).

    Well, if a theory shows you that "relative to you, things have to be such and so" and things appear to us "such and so" then for me, that's good enough. I agree that it would be somehow more confortable if the only ontology around WAS exactly what is "such and so", but if that ontology can produce an appearance for me for things to be "such and so" that's all I can reasonably demand, no ?

    The locality issue is not the essence of MWI. It is taking the superposition principle seriously all the way which is the basis of MWI. And Bohmians, no matter how they deny it, ALSO have that problem, because the unitarily evolving wavefunction *is just as much part of the Bohmian ontology as it is in MWI*. You ALSO have parts of the wavefunction which look like dinosaurs running around on earth right at the place where I have my office, simply because a small quantum effect long ago deviated the trajectory of the asteroid who killed them. You don't have particle positions corresponding to them, but their "ghost" branches in the wavefunction are nevertheless still present in the wavefunction you have, which is part of the Bohmian ontology. You need to postulate that human beings are only consciously aware of particle positions and not of the wavefunction in order to save the observed part of the world in your view, and the "unobserved" ontology (the wavefunction) is just as extended in BM than it is in MWI.

    If you call MWI "crazy" because an entire part of the ontology is not perceived, then BM is just as crazy. Only, on top of that, you've kicked out relativity.

    I really don't see what is so terribly "crazy" for me to claim that you are only aware of ONE term in the wavefunction, while it should be perfectly "normal" for you to claim that there is a wavefunction (the same one) we are NOT aware of at all, and particle positions we are aware of. In both cases, there is an external ontology which is vastly more complex than what we consciously experience.
  21. Dec 8, 2005 #20
    May you could also put Hardy's paradox forward...?
  22. Dec 8, 2005 #21
    But then you say:
    ??Don’t you see these as a contradictions?

    Dividing results in to an unknown & uncertain number of different MW’s and landing in one of those does not “saved locality” it collapses the WF of superposition that’s all. Just claiming that it’s local within that “new world” doesn’t make it so. It’s still a HUP coin flip as to moving from the original world to select (though wf collapses) which newly formed uncertain world you landed in.

    If EPR-Bell has given us anything it should be the knowledge that only one thing is true.
    Either 1)QM-Superposition (essentially Bohr completeness)
    OR 2)Locally Causal spatially separated states
  23. Dec 8, 2005 #22


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    I can't tell if you're even serious, but if so, it is not true that you literally perceive the sun going around the earth or the flatness of the earth as a whole. What you perceive is that the sun moves relative to the earth. That is true, and any scientific theory which disputed *that* would be correctly dismissed as preposterous. Likewise, what you perceive in regard to the earth is that it is not noticably curved on km-type length scales. (Well, until the 1960's when humans were able to perceive the earth as a whole by being further away than usual.) And again, any scientific theory which denied this would be laughed offstage. To say that the earth is round (with a radius of thousands of km) does not in any way contradict what you see with your eyes. Indeed, it is only by putting together a vast collection of stuff people saw with their eyes that you come to discover that, on this larger length scale, the earth is round (or that the planets orbit the sun, etc.). Surely you don't think that one gets to higher-level knowledge (like scientific theories) by *denying* the evidence of the senses (in favor of what? revelations? mystical insights?) The fact is, there just isn't any other source for valid knowledge. That is the meaning of empiricism, and it is fundamental to science.
  24. Dec 8, 2005 #23
    Dear Dr. Chinese, thanyou kindly for your posts. Further I appreciate the wikipedia EPR page very much =)

    On the subject of whether to chose a or b... I find a) preposterous, and do not understand why Einstein had difficulty in grasping option b). In fact I readily accept b). I explain:

    An electrons spin is up or down in only one spatial dimension at a time. If you measure the electrons spin in z, then in dimension x the electrons spin is neither up nor down. It is not that it is unknown whether the spin is up or down in x, it simply that it is neither of the two. Neither is it true that there is some strange unconcievable physical superposition of up and down in z. It simply does not make physical sense to talk about the electron having a certain spin in any other spatial dimension than in the one you have measured.

    If you understand why electron spin in certain spatial dimensions is a non-commuting observable, then you will clearly see that there is no simultaneous reality of the spin of an electron in more than one spatial dimension; WITHOUT having to invoke some wierd, unintuitive, unconvievable explanation. In fact it seems pretty obvious to me now that b) is the only viable physical interpretation of the the EPR problem that I have ever been presented with. Its just very difficult to explain... in that sense its odd... its simple and obvious yet so difficult to put into words... it is to do with the way QM spin is defined. QM spin in the context of a spatial dimension is either spin half up or spin half down, there is nothing else to it. And if it is up or down in one spatial dimention, it is neither up nor down in any of the other spatial dimensions. Draw a diagram with an arrow and define that vector as one of the spatial dimensions. You will see that it is pointing neither up nor down in any of the other spatial dimensions. It is absurdly simple.

    Now what about before you measure the spin? Well since the spatial dimension in which you measure the spin is physically defined ONLY when you make the actual measurement (at least as far as the electron is concerned), it doesn't make any physical sense to talk about the electron having a definte spin in any of the spatial dimensions.

    And what are we to make of the mathematical equations describing superposition of states? well thats just it they are purely mathematical, purely abstract and have NO physical interpretation. They are inherent to the QM model.

    I strongly suspect this kind of interpretation applies to all non-commuting observables.

    As a last word: Dr. Chinese, thankyou very much for making me chose between those two options. I have for the first time truly understood non-cummuting observables and QM uncertainty. Funny thing is I'll probably get a torrent of posts in this thread telling me how wrong I am xDD hehehe
    Last edited: Dec 8, 2005
  25. Dec 8, 2005 #24


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    If I helped in any way, you are very welcome. Any choice you make in the interpretations, I think is good. I too choose to believe that non-commuting observables can only have a single value at a time.
  26. Dec 8, 2005 #25
    I have looked at this interesting article, and I have the following dumb question (I assume ttn is the author of the article). Unlike EPR, Einstein's boxes relate to one particle. Is the Bohm's theory, when limited to one particle, necessarily non-local? Of course, if you consider the non-relativistic Bohm's theory, it is non-local "by definition", just because it is not lorentz-invariant and allows arbitrarily high velocities. If you consider the Bohm's theory based on the Klein-Gordon equation, you can say that it is non-local because the 4-vector of current may be space-like. But what if you consider the Bohm's theory based on the Dirac's equation?
    Another dumb(er) question about Einstein's boxes. If we consider them in one dimension, then, no matter how far they are from each other and no matter how high the potential barrier between them, the particle will regularly tunnel from one box to the other and back (if, say, the potential outside of the boxes is higher than the energy of the particle, otherwise the particle can tunnel out of the boxes). How does this affect the treatment of this thought experiment? (in a more realistic case of three dimensions the particle can just tunnel out of the boxes).
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