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Many worlds question

  1. Apr 19, 2008 #1
    Suppose there is a quantum observable that can take two values, A or B. If a state is prepared that it is a superposition of the states A and B, then if I make an observation, according to many worlds interpretation, there will then be a superposition of two me's, one that observed A and one that observed B.

    But suppose the state prepared is

    [tex]|\psi\rangle = \sqrt{\frac{1}{3}}|A\rangle + \sqrt{\frac{2}{3}}|B\rangle[/tex]

    If I perform the experiment 1000 times, then there will be a superposition of 2^1000 me's, one for each possible set of results of the observations. But this can't be right. In such a case a typical "me" in the ensemble will have observed approximately 50% A's and 50% Bs. But the answer should be 1/3 and 2/3.

    The only way for the statistical nature of quantum mechanics to be due to such an all-possible-outcomes-happen picture is for the result of one an observation of the state above to result in 1 me that observed A and 2 me's that observed B. Then everything counts up correctly.

    Do I understand this right? So if we have instead

    [tex]|\psi\rangle = \sqrt{\frac{79}{162}}|A\rangle + \sqrt{\frac{83}{162}}|B\rangle[/tex]

    an experiment results in 79 me's that get A and 83's me that get B?

    If this is really the many-worlds answer to the statistical aspect of observations, then it is even kookier than I thought.
     
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  3. Apr 19, 2008 #2

    Hurkyl

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    Each "you" has a different amplitude associated to it. And if you compute the transition 'probabilities' from
    [tex]
    | \text{You haven't counted yet} \rangle \otimes | \psi \rangle^{\otimes 1000}
    [/tex]
    to
    [tex]
    | \text{You have counted yet n Bs} \rangle \otimes | \psi \rangle^{\otimes 1000}
    [/tex]

    you will find they are distributed according to the binomial distribution for 1000 trials with probability of success 2/3.
     
    Last edited: Apr 19, 2008
  4. Apr 19, 2008 #3
    I never knew you could mix amplitudes of two different quantum states D=
     
  5. Apr 19, 2008 #4
    I don't know what this means.

    The point to many-worlds--I thought--was to do away with the collapse of the wavefunction. Each potential outcome actually happens.

    In the usual intepretation, only one outcome happens, with a probability that is proportional to the amplitude's modulus squared.

    If each outcome actually happens, in what way is an amplitude "associated" with it.

    If every time we make one measurment, there is only one me in the resulting superposition that observes an A and one me that observes a B, the resulting distribution is necessarily symmetric with respect to As and Bs. After 1000 tries there will be one me that got all As, one me that got all Bs, 1000 me's that got 999 As + one B, 1000 me's that got 999 Bs + one A, etc.
     
  6. Apr 19, 2008 #5

    Hurkyl

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    The central tenet of MWI is that unitary evolution (e.g. evolution according to Schrödinger's equation) proceeds without exception.

    If we scale back to performing one experiment with a 'counting' device, then that means the state

    [tex]| \text{unmeasured} \rangle \otimes | \psi \rangle[/tex]

    evolves into

    [tex]\sqrt{ \frac{1}{3} } |\text{measured 0}, 0\rangle + \sqrt{ \frac{2}{3} } |\text{measured 1}, 1\rangle[/tex]

    and through thermodynamic interactions with the environment, it will further decohere into the mixed state

    Proportion 1/3 of |measured 0, 0> and proportion 2/3 of |measured 1, 1>.

    If we focus just on the measuring device, it is in the mixed state

    Proportion 1/3 of |measured 0> and proportion 2/3 of |measured 1>.


    Because the components of the mixed state cannot interact (except possibly through an extremely unusual interaction with the environment), it is not unreasonable to describe the state as if it has separated into two different 'worlds', one where the device measured 0, and the other where the device measured 1, and each 'world' does have a real number coefficient between 0 and 1 attached to it. (And the coefficients will add up to 1)
     
    Last edited: Apr 19, 2008
  7. Apr 19, 2008 #6
    Sure. The wave function never collapses and the amplitudes persist. So what then do the amplitudes have to do with the statistics?

    Let's take another example.

    [tex]|\psi\rangle = \sqrt{\frac{1}{1000}}|A\rangle + \sqrt{\frac{999}{1000}}|B\rangle[/tex]

    After two measurements, the state of the measuring device (which ultimately includes me) is

    [tex]\frac{1}{1000}|A,A\rangle + \frac{\sqrt{999}}{1000}}|A,B\rangle + \frac{\sqrt{999}}{1000}}|B,A\rangle + \frac{999}{1000}|B,B\rangle [/tex]

    Right? But what is the statistical significance of these amplitudes? The amplitude of 2 Bs is much larger than 2 As but there is still only one me that got two As and one that got two Bs. In what sense is 2 Bs likelier than two As?
     
  8. Apr 19, 2008 #7

    Ken G

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    Yes, you are describing the situation correctly, and interestingly you thereby show the flaw in MWI thinking.
    There's the flaw right there, in bold. Let's call your above stages an initial state, a first step, and a final step. How is the first step separable from the final one, so you can plunk a "further" in there? The whole way that the "measured associations" you cite in the first step are achieved is via the decoherence, that's exactly what the measuring apparatus does, so that's what a measurement is. Since the MWI approach has to break that single happening into two separate happenings, the first being unitary and the second breaking the unitariness (but only after the MWI step has already appeared, that's the point of it), it shows quite clearly that the MWI step is a fiction. In other words, your analysis is identically the same if you simply omit that first step. So why include it?
     
  9. Apr 19, 2008 #8

    Ken G

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    That's another flaw in MWI thinking. Hurkyl is certainly right that even though there is one "me" that got two As and one "me" that got two B's, the latter "me" is far more likely to be actualized in the final decohering step. So what this means is that "counting the me's" doesn't tell you anything at all, what matters is their amplitudes. But again, I point out that there is actually no "amplitude stage" anyone could point to or isolate in any way, the decoherence is already there, so the mixed state is already achieved-- the superposition stage is a fiction. So that's the other reason there's no point in "counting the me's". It's like, if I enter an olympic track meet, it makes no difference at all how many other racers are competing, one or a hundred. I have no chance of winning, either way.
     
  10. Apr 19, 2008 #9

    Hurkyl

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    There is the obvious significance that the amplitudes will decohere into a probability distribution on the possible outcomes (assuming you've written the state relative to the correct basis). But as to what the "meaning" of that probability?

    (aside: it might be useful to ponder the "meaning" of statistics as applied to pre-quantum physics, or to anything else related to 'reality'. One of the things I find aesthetically pleasing about QM is that some of the probabilities actually correspond to something)

    One thing you can check is that these probability distributions are, if you assume a mild continuity argument, consistent with the frequentist interpretation of probabilities. Consider a device that makes the following measurement:

    Output |0> if the proportion of B's is outside the range [1/3 - e, 1/3 + e]
    Output |1> if the proportion of B's is in the range [1/3 - e, 1/3 + e]

    For any positve e, in the limit of the number of trials approaching +infinity, the state under consideration device converges to the pure state |1>.
     
    Last edited: Apr 19, 2008
  11. Apr 19, 2008 #10

    Hurkyl

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    For pedagogical reasons, it seems clearer to describe a measurement that proceeded in two phases. And such a procedure is certainly applicable to certain situations -- e.g. a half-silvered mirror splitting a photon is an example of a 'coherent measurement', which we are demonstratably capable of erasing before it decoheres.


    I'm not exactly sure what that means -- if we ignore the information that escapes into the environment, decoherence collapses the state into a mixture of all possible outcomes, with coefficient equal to what we usually call the probability of that outcome.
     
    Last edited: Apr 19, 2008
  12. Apr 19, 2008 #11

    Ken G

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    And there's nothing wrong with MWI as mere pedagogy-- but that's not how it's argued.
    No, the fact that you can "quantum erase" the connection proves that there is no such state as |mirror sends 1, 1> unless you decohere it. You may as well claim there is a state in a two-slit experiment |slit sends photon left, left> and |slit sends photon right, right> without decohering those results. Can you cite any difference at all between that fictional state and just |left> and |right>?
    The scientific method involves things that are actualized, because the actualizing is how we do the tests that define science. Can you cite any counterexamples to support your idea that science can be done on mixed states without actualizing them?
     
  13. Apr 20, 2008 #12

    Hurkyl

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    Who said it was?

    Let's consider the very simplest 'coherent measurement': a CNOT gate, with one input initialized to |0>. Actually, let's not -- I believe the original poster's intent was to actually learn what MWI says, so derailing it into a debate about the "correct" metaphysics would not be appropriate here. A new thread on the topic would be fine though. (And if you do start one, and use the word 'actualize' in it, you really ought to try and give it some sort of definition)
     
  14. Apr 20, 2008 #13

    Ken G

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    Here's from Wiki: (http://mail.google.com/mail/#inbox/11969e65e7cad267)
    " Many-worlds denies the objective reality of wavefunction collapse."

    Here's from the Stanford Encyclopedia of Philosophy (http://plato.stanford.edu/entries/qm-manyworlds/):
    "The Many-Worlds Interpretation (MWI) is an approach to quantum mechanics according to which, in addition to the world we are aware of directly, there are many other similar worlds which exist in parallel at the same space and time. "

    Or there's this poll result from ( ):
    ""Political scientist" L David Raub reports a poll of 72 of the "leading cosmologists and other quantum field theorists" about the "Many-Worlds Interpretation" and gives the following response breakdown [T].

    1) "Yes, I think MWI is true" 58%
    2) "No, I don't accept MWI" 18%
    3) "Maybe it's true but I'm not yet convinced" 13%
    4) "I have no opinion one way or the other" 11%"

    and that article goes on to describe a key assumption of the MWI:"The metaphysical assumption: That the wavefunction does not merely encode all the information about an object, but has an observer-independent objective existence and actually is the object. For a non-relativistic N-particle system the wavefunction is a complex-valued field in a 3-N dimensional space."

    Those are just the first three Google hits. Need I go on? Note that none of those statements describe a pedagogy (which is a way to teach something, essentially a conceptual mnemonic). They are all quite clearly statements of a world view, and whenever I discuss MWI with anyone, it is usually quite clear that they also take it in that light. The Wiki hit says it "denies" things, which pedagogies do not do, they are expressed entirely constructively. The second hit uses the ontological word "exists", and being a philosophy text, it would know the difference between that and a pedagogy. The third hit goes so far as to cite "objective existence", this despite the well-known fact that it cannot be objectively demonstrated (which is why Everett originally termed it a meta-theory, a far more appropriate term than either "interpretation" or "pedagogy").
    Agreed. But the OPer might also be interested in whether the MWI is a theory, a metatheory, or a pedagogy. And since you mention it, the word "actualize" is trivial to define in a scientific context. This would work fine: a mixed state is "actualized" into an "outcome" whenever the results of an experiment are tabulated. Indeed, every scientific experiment ever done invoked the concept of actualization, there can be no more obvious condemnation of the MWI approach than that it seems to engender the idea that "actualization", which has always been a crucial step in science, is somehow a less central concept in science than "unitarity", which is a mathematical model.
     
    Last edited: Apr 20, 2008
  15. Apr 20, 2008 #14

    Hurkyl

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    You misunderstood -- when I said "who said it was", I was (rhetorically) asking who said that MWI was a mere pedagogy. Recall that when I invoked the term, I was justifying my choice of example to present.


    The MWI is, as its very name suggests, is primarily an interpretation -- a (metaphysical) mapping from the fundamental notions of the mathematical theory of quantum mechanics to "reality".

    Also MWI has apparently spawned a new round of philosophical discussion on the notion of nondeterminism -- and I have seen this discussion also called labelled as MWI, but I don't know if that is by scientists who use MWI, or by people eager to find reasons to dismiss MWI unfairly.


    If this is an accurate definition, then according to MWI, each possible outcome is 'actualized' in one or more components of the decohered state.


    The only thing MWI endangers in this respect is our confidence that we can make meaningful assertions about things that are unobservable. :tongue: The most commen alternative is to assert that some unobservable facet of reality does not exist... and the price for making that assertion is that you have to assume that quantum mechanics works only when we're not looking.
     
  16. Apr 20, 2008 #15

    Ken G

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    Certainly, that is just what the MWI would say, but it must still confront the clear fact that each scientist, doing science, experiences only one set of actualizations. That is my point, MWI does not represent a scientist doing science. As such, it cannot be claimed to arise from science-- it arises from philosophy. It arises from building an exoskeleton around science, mimicking the skeleton we observe when we interact with our classical instruments, but that cannot be tested by science, for purposes of sheer "mental aggrandizement". I would say that description applies equally well to all other efforts at external and untestable scaffold-building, including religion. But that's the stuff of another thread.
    How does MWI endanger that, it seems to me it is an effort to do gain precisely that confidence. To me, what MWI endangers is the idea that meaningful assertions come from what is observable, i.e., that scientific enlightenment stems from doing science as it has defined itself ever since it started actually doing something other than creating self-styled authorities on all things metaphysical.
    No, you only have to define quantum mechanics in a way that is restricted to something that is demonstrable using the scientific method.
     
    Last edited: Apr 20, 2008
  17. Apr 20, 2008 #16
    I'm not getting it, guys. But thanks for trying. - Todd
     
  18. Apr 20, 2008 #17

    Hurkyl

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    Until you provide an actual experiment where your assertions and the predictions of MWI differ, perform it, and observe the results disagree with MWI, you are a verbose hypocrit.


    Note that for the most obvious ways of empirically counting 'actualizations', MWI predicts that any experiment will certainly result in the answer 1. (e.g. repeatedly ask an honest scientist what answer he observed for a particular experiment)
     
  19. Apr 20, 2008 #18

    Ken G

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    OK, then getting back to the OP, the MWI is exactly the same as looking at a superposition state from the outside. So saying that we have "many worlds" when the scientist is part of the system is exactly that same as saying a photon in a double slit experiment is part of "many worlds" which we are observing from the outside. So you can reform your OP in exactly those terms-- set up an experiment where the photon has a 1/root(3) amplitude of going through one slit, and a root(2/3) amplitude of going through the other slit. Now "count the photon possibilities"-- you still have two "photon-worlds", one that is more likely than the other. So there's no inconsistencies there-- and there's also no need to think in terms of these separate "photon worlds". It's an empty model, the amplitudes are all you need, and all you use to check the predictions.
     
    Last edited: Apr 20, 2008
  20. Apr 20, 2008 #19

    Ken G

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    At no time did I claim the flaw of MWI is that it is contradicted by science, my claim is that it is simply not science. Is the existence of a supreme being contradicted by science? Would I also be a "verbose hypocrite" to claim that belief in a supreme being is not science, unless I had an experiment that proved it? That is precisely your logic here.
     
  21. Apr 20, 2008 #20

    Hurkyl

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    However,
    (1) you make assertions about the untestable
    (2) you criticize the consideration of untestable things
    thus satisfying the connotative and denotative meanings of the word "hypocrit".


    Incidentally, science is a subfield of philosophy, so you have no grounds for criticizing anything on the basis of it being philosophy. :wink: And besides, the metaphysical assignment of the notions in a mathematical theory with facets of reality is an integral part of science.
     
    Last edited: Apr 20, 2008
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