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So in the many worlds theory my understanding is for each measurement outcome there is a world. But this does not make sense (in my opinion). Why? Because the number of worlds where outcome A is seen is an integer and the total number of worlds is also an integer. But the probability which is the ratio of these 2 numbers can be an irrational number!

I suspect there is some additional structure the many-worlders invokes? Can someone enlighten me?

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    "Because the number of worlds where outcome A is seen is an integer and the total number of worlds is also an integer. But the probability which is the ratio of these 2 numbers can be an irrational number!" I'm not at all sure I understand the problem you're posing. Mathematically the ratio between two integers cannot possibly turn out to be an irrational number -- that's what "rational" and "irrational" mean, definitionally, in number theory. Are you saying it is possible for this ratio to be irrational? If so, how? If not, are you suggesting that's a problem for the theory? If so, why? Commented Apr 8, 2022 at 14:58
  • Yes the problem I am posing is how does the many world interpretation explain an irrational number being the probability? When by the born rule this is a possibility Commented Apr 8, 2022 at 15:07
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    It likely has to do with the whole-part relation of 3D worlds to the whole wave. MWI has whole-part determination where the whole determines the parts and not the other way around (at least in wave function monism). Because of no traditional separability into parts, I’d hazard a guess this is where oddities like this arise. Hopefully someone can be more specific.
    – J Kusin
    Commented Apr 8, 2022 at 15:17
  • The function that assigns a number to the amount of worlds is different from the function that assigns a number to the local probabilities. Even if they were the same function, with sufficiently different inputs, it might have rational and irrational outputs (e.g. if it were the square root function, it could take 4 and give a rational number, 2 and an irrational one). On top of all that, though, my impression was that the total number of MWI worlds was potentially equal to the Continuum, where the set of all irrationals resides. Commented Apr 8, 2022 at 15:31
  • I don't know of any versions of the Everett interpretation that try to derive probabilities in a frequentist way from a finite integer number of worlds. There are some versions of MWI that try to derive subjective probabilities rather than frequentist ones using some version of decision theory, and p. 14 of the seemingly frequentist approach by Rubin here mentions a continuous infinity of copies of any given observer-state.
    – Hypnosifl
    Commented Apr 8, 2022 at 21:29

3 Answers 3

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There a a couple of problems here. First, the number of worlds need not be an integer; it only needs to be a cardinal, i.e. the number of worlds does not need to be finite. The is important because quantum mechanics generally deals with infinite dimensional spaces, where some measurements have infinitely many possible outcomes. Second, the number of worlds corresponding to a particular outcome does not necessarily tell you the probability of that outcome, because probability theory assigns probabilities to events based on measure, not cardinality. To illustrate what I mean here, consider a weighted, six-sided die. There are only 6 possible outcomes of a roll, but they do not all have probability 1/6, nor do their probabilities need to be rational.

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Infinite precision doesn't exist physically. Since irrationals require infinite precision they aren't physical. So this is not a problem.

The main problem in my opinion is what Wallace calls the metaphysical and incredulity problem - that there are supposedly a vast number of worlds. He solves it by avoiding it. Whoch is not a solution in my opinion.

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Say that the square of the superposition gives 1/pi(phi1)^2+(1-1/pi)phi2^2. So the chances are real numbers (no rationals). How does a wavefunction arrive in such a state in the first place? Well, you can posit it, like I did now, calculate it in advance, or prepare it in advance. But to come to know it you gotta have a lot of them and do measurements on them.

Let me first address what it means if the chances are not equal. If the chance is, say 1/10 that after a measurement the state is phi1 and 9/10 it's phi2. How does a state after a measurement differ from a state after a measurement on a fifty-fifty state? In both cases you end up with a phi1 state and a phi2 state. Two worlds in the multitude of worlds. What's the difference?

The answer lays in the fact that only after a lot of measurements the distribution becomes visible. If you perform the measurement over and over again there will be 90% of all worlds with the state of which the chance is 9/10, phi2, and 10% of the worlds will have a state phi1. The greater the number of measurements, the better the approximation, which indeed can also be used to find out the coefficients of the wavefunction (instead of calculating or assuming.

Same for real coefficients. After one measurement there is no difference. Only after infinite measurements. But ho! Isn't the number of universes still an integer then? How can the be coefficients like 1/pi and (1-1/pi)? Can a coin have these chances? Well if you keep throwing and throwing it will turn out that the chances will not be 1/2-1/2 exactly. Still the number of dices thrown is an integer. Take it from there.

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