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Why MWI cannot explain the Born rule

  1. Dec 17, 2009 #1

    Demystifier

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    Here I propose a VERY SIMPLE and intuitive argument that MWI, with its MINIMAL set of assumptions, cannot explain the Born rule.

    The argument goes:

    The minimal set of assumptions defining MWI is:
    1. Psi is a solution of a linear deterministic equation.
    2. Psi represents an objectively real entity.

    Assume that the Born rule can be derived from the assumptions above. Then ANY system satisfying these assumptions must necessarily have the probabilistic interpretation defined by the Born rule. Therefore, in order to show that the Born rule cannot be derived from the assumptions above, it is sufficient to find one counterexample of a system that satisfies these assumptions but does not have the probabilistic interpretation. And it is very easy to find such an example; just take some appropriate wave equation from CLASSICAL mechanics of fluids. Q.E.D.

    Of course, this is just a rough idea for the argument. I'm sure it can be further refined, e.g. by replacing 1. above with something more specific and yet sufficiently general. Since the idea is so simple, I'm sure that many readers of this can contribute to further developments of the idea.

    Any suggestions? Comments? Objections?
     
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  3. Dec 17, 2009 #2
    This is also my opinion. You can also consider simulating a virtual quantum world on a classical computer. The classical computer simulates the world according to the Schrödinger equation, and then you can consider the internal virtual observers. Do they indeed have observations consistent with the Born rule? It is easy to see that the probabilities for their observations are uniformly distributed over the set of all possible experimental outcomes and therefore not given by the Born rule.
     
    Last edited: Dec 17, 2009
  4. Dec 17, 2009 #3

    Demystifier

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    Count Iblis, your idea to simulate the Schrodinger equation on a classical computer is even better than mine. Excellent!
     
  5. Dec 17, 2009 #4

    Fredrik

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    I don't think those are sufficient to define the MWI. The "worlds" in the MWI are just correlations between states of subsystems, so we can't expect to see any worlds unless we assume that the universe has subsystems.

    How would we define the "internal virtual observers" in the suggested computer simulation? Aren't they supposed to be subsystems?

    I still think that assuming that the Hilbert space of the universe is the tensor product of the Hilbert spaces of the subsystems is essentially equivalent to assuming the Born rule. (And I still don't have a complete proof of that).
     
  6. Dec 17, 2009 #5

    Demystifier

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    I don't think that it is fair to define "worlds" as correlations, because a correlation is, by definition, something that has to do with statistics and probability, whereas we want to DERIVE statistics and probabilities.

    Of course, you are allowed to postulate any axioms you want that will help you to incorporate the Born rule, but then the main virtue of MWI - the minimal set of axioms - is lost.
     
  7. Dec 17, 2009 #6

    Demystifier

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    Fredrik, consider a CLASSICAL wave psi(x1,x2,x3,t). You can think of 3 coordinates x1, x2, x3 as 3 space dimensions x, y, z. But you can also think of these 3 coordinates as 3 particles moving in 1 dimension. The mathematics is the same. Yet, in this second case you have a natural and intuitive definition of subsystems (each particle is another subsystem). So, how the words which say that 3 coordinates represent 3 particles, without changing the mathematics, imply that the classical wave above has a probabilistic interpretation?
     
  8. Dec 17, 2009 #7
    Lets begin from the very beginning.
    How you define the 'probability' in the determenistic theory?
    You can not apply Born rule 'as is' in MWI
     
  9. Dec 17, 2009 #8

    Demystifier

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    Through ignorance of some deterministic property.

    For example, in BM (which is the most classical interpretation of QM), the particle lives in one many-world branch only, but you just don't know which one. Therefore, your knowledge about the position of the particle can be described in terms of probability. Unfortunately, there is no analog of a Bohmian-particle-in-a-single-branch within pure MWI. That's why it seems reasonable to supplement MWI with an additional axiom of particles, which leads to BM. BM is nothing but MWI with one additional axiom that allows to explain the Born rule in a simple and intuitive way.
     
  10. Dec 17, 2009 #9
    Just because the probabilities always happen to be 1 doesn't mean they aren't probabilities, does it :tongue:?

    I'm actually serious here. I think it's easier for us to consider that deterministic theories are realistic while probabilistic theories are not. But do we have any more evidence for the reality of deterministic theories than probabilistic theories? EPR's criterion for reality requires determinism, but do we accept determinism as the ultimate test of reality? Logically it's fallacious to draw any such conclusions about the nature of reality.

    There is a tradition of associating determinism with reality, from both EPR and the classical paradigm. Deterministic theories really are just special cases of probabilistic theories though. The fact (I think - might need more research) that we can always replace a probabilistic function with a deterministic function regarding uncertain variables further deprives such differences of much meaning.

    I guess I'm not sure how this applies to MWI in particular. I think you make a good point by comparing current theories vying for "reality" to similar theories that make no such claim. Can we come up with a criterion for reality that doesn't lead to contradictions when applied to disparate theories? If not, and I suspect that we can't, then what justification do we have for claiming the reality of any theory?
     
  11. Dec 17, 2009 #10

    Fredrik

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    I think this is like starting with a manifold without a metric and trying to derive the physical significance of special relativistic inertial frames (which would be defined using the isometries of the Minkowski metric). It's impossible by definition, no argument necessary.

    I agree that it loses something significant, but I think Everett, Tegmark and others were just plain wrong when they said that the MWI can be defined this way.

    They can't. I'll go even further and say that no mathematical model can ever imply anything about the real world. We need something more than just mathematics. A theory of physics is defined by a set of axioms that tells us how to interpret the mathematics as predictions about probabilities of possible results of experiments. The Born rule is an essential part of the axiomatic structure of QM. Without it, the theory is crippled. It's not even a theory anymore.

    This model of 3 particles hasn't been endowed with axioms that turn it into a theory. It's not clear what mathematical objects represent observables, or how they can be measured.
     
    Last edited: Dec 17, 2009
  12. Dec 18, 2009 #11

    Demystifier

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    So we basically agree on that. :approve:

    I guess you allow also the deterministic theories as a special case, in which all probabilities are either 1 or 0.
     
  13. Dec 18, 2009 #12
    Suppose you start with a wavefunction that describes a repeatable experiment (say, an asymmetric coin toss) and an observer. The observer counts the number of heads-up outcomes. There's a decoherence factor involved, which has the effect of adding a uniformly distributed random phase to all observers after each toss.

    You evolve the system for 100 coin tosses, and you end up with a superposition of 101 observers who observed 0, 1, ..., 100 heads-up outcomes.

    I claim that you'll find that, however you define the measure on the Hilbert space, you'll observe that the outcome distribution will conform to the Born rule, i.e. the mean number of observed heads-ups is 100*|c1|^2/(|c1|^2+|c2|^2), where c1 and c2 are complex amplitudes for heads-up and tails-up transitions in each coin toss.
     
  14. Dec 18, 2009 #13

    Demystifier

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    Of course you will, this is the experimental fact. The issue is to find a theoretical EXPLANATION of this fact.
     
  15. Dec 18, 2009 #14
    This is a purely theoretical explanation. Write some code to simulate this using transitions described, and you'll get the expected outcome.

    My point is, Born rule naturally arises through a series of repeated experiments in presence of random decohering influence.
     
  16. Dec 18, 2009 #15
    I was thinking too about the 'measure of existence' vs the 'number of observers'

    Say, I go to work.
    In 1 branch I had successfully came to office. But there are 1000000 low probability branches, where I made something stupid – robbed the gas station, killed the cop, etc etc. So there is 1 high-intensity branch versus many low intensity branch. So there are more weird observers than normal ones.

    But then I had realized that reality is even more complicated. Single ‘normal’ branch is an illusion. I can not be aware of all macroscopic degrees of freedom of my body and even of my brain, because my consciousness have less degrees of freedom like computer chip can stored less information than is encoded in all matter it consists of.

    So and observer is not only a branch, but a huge ensemble of branches. And you actually don’t have a SINGLE basis, but rather a sum over ensemble of basis. How it is relevant to the Born rule? Not sure. But I think that before we talk about the ‘probability’ we must check if ‘the number of distinct observers’ approach is equivalent or not to ‘the measure of existence’ approach.
     
  17. Dec 18, 2009 #16
    Noturalu arises... to what observer?
    Say, I flip a coin 1000000 times.
    There are 2^10000000 observers:

    Observer 1: 0000....000000 (1000000 times)
    Observer X: 0101010101...0101011010101
    Observer Y: 101011010....1010101010011
    Observer (last): 1111111....111111 (1000000 times)

    Why observers X and Y are more 'natural' then 1 and last?
     
  18. Dec 18, 2009 #17

    Demystifier

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    Is there a paper where more details can be found?
     
    Last edited: Dec 18, 2009
  19. Dec 18, 2009 #18
    They are not. But there's only one observer who saw 0 heads and only one observer who saw 1000000 heads, and there's an enormous number of observers who saw 500000 heads.

    Furthermore, if you assign a factor of 0.6 * exp(i*random_phase) to all heads flips and a factor of 0.8 * exp(i*random_phase) to all tails flips, and you sum up things to construct meta-observers called "0 heads", "1 heads", ..., "1000000 heads", the meta-observer with the largest factor will be very close to the value predicted by the Born rule, 1000000 * (0.6*0.6) / (0.6*0.6+0.8*0.8) = 360000 heads.
     
  20. Dec 18, 2009 #19
    There probably is, and it's probably dated 1930 or so, and therefore I have no idea how to find it. If there isn't, feel free to write one and include me as a reference.
     
    Last edited: Dec 18, 2009
  21. Dec 18, 2009 #20

    Demystifier

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    I am both disappointed and satisfied with this answer. Disappointed for obvious reasons. Satisfied because my arguments still seem cogent to me. :smile:
     
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