Binomial distribution of "worlds" in MWI

[entropy1]

If we have a spin measurement with P(up)=0.5 en P(down)=0.5, this is equivalent to tossing a coin P(heads)=0.5 and P(tails)=0.5.

The probability of having five heads and five tails out of ten tosses is the binomial: $\binom{10}{5}(0.5)^5(0.5)^5$. So the same would hold for the spin measurement as given.

If we consider MWI to this respect, the probability of getting into (experience) a "world" in which we measure n(up) times spin up out of N measurements, would be the binomial $\binom{N}{n(up)}(0.5)^{n(up)}(0.5)^{N-n(up)}$.

So the distribution of the worlds with n(up) spin-ups out of N measurements would be the binomial distribution. All "worlds" are actually realized, and all have probability 1. For every "world" with spin-up, there is a "world" with spin-down and vice-versa. In world A, probability for up=1, in world B, probability for down=1; overall the probability is (A+B)/2. For other probabilities of the spin measurement, the reasoning is the same. In the case of two eigenvalues, like spin, the "worlds" with n(up) would always have the binomial probability distribution.

This contradicts the Born rule, because what happens in MWI is independent of probabilities of outcomes.

Would that be correct?

 

[andrewkirk]

For every "world" with spin-up, there is a "world" with spin-down and vice-versa. In world A, probability for up=1, in world B, probability for down=1; overall the probability is (A+B)/2

I think you need to elaborate on what you mean by this. Earlier you described a statistic n as the number of up spins observed. That is a meaningful statistic. But I don't understand what you are referring to when you write 'probability for up = 1'. You appear to no longer be referring to the number of up spins observed.

Two things to bear in mind here:

1. The probabilities for systems containing multiple identical particles are not binomial, because the binomial distribution assumes items are distinguishable.

2. In a MWI context, the Born-rule probabilities are epistemological, not ontological. That is, they tell us the probability of observing a certain outcome, given what we know about the system. We cannot validly use information we do not know, to modify the probabilities. It appeared to me that you might be doing that in your set-up, but I could not make sure as the description lacked detail.

 

[PeterDonis]

1. The probabilities for systems containing multiple identical particles are not binomial, because the binomial distribution assumes items are distinguishable.

While this is true, I don't think it applies to the scenario described in the OP; that scenario appears to me to be about repeated measurements on identically prepared single particles, not single measurements of systems containing multiple particles.

 

[PeterDonis]

All "worlds" are actually realized

Yes.

and all have probability 1.

This seems to be a common statement in the context of the MWI, but I don't think it's correct. This meaning of the term "probability" is not applicable in the context of the MWI, because the MWI is completely deterministic. In that completely deterministic context, the only thing that could meaningfully be described as having "probability 1" is the entire wave function after measurement.

In world A, probability for up=1, in world B, probability for down=1; overall the probability is (A+B)/2.

This does not seem valid to me, in the light of the comments I've made above.

 

[entropy1]

This meaning of the term "probability" is not applicable in the context of the MWI, because the MWI is completely deterministic.

My two cents:

There is a difference between quantum level and classic level, as is between subjectivity and objectivity. With respect to MWI, while the wavefunction on the quantum level encompasses all outcomes on the objective level, we observe an outcome that then has probability 1 (it has been actualized) on the subjective level which is also the macro level.

The outcomes in the different "worlds" have probability 1 on the subjective level, regardless the probability given by the Born rule, so the only difference between the worlds is the measurement outcomes, n(up) and n(down).

So, do you claim that MWI is a non-probabilistic view? What role does the Born rule then have in that view?

While this is true, I don't think it applies to the scenario described in the OP; that scenario appears to me to be about repeated measurements on identically prepared single particles,

To illuminate my application of the Binomial distribution concerning, that would be necessary I think, yes. But the essence of MWI in this matter seems to be to me that all worlds get realized by each measurement.

It would come to my own theories, which I don't want to share here. My knowledge about this is far too limited.

 

[PeterDonis]

There is a difference between quantum level and classic level, as is between subjectivity and objectivity.

This depends on which interpretation of QM you adopt.

we observe an outcome that then has probability 1 (it has been actualized)

No, we don't. As I've already said, "probability 1" is meaningless in this context. If you continue to make this assertion, this thread will be closed.

do you claim that MWI is a non-probabilistic view?

The MWI is completely deterministic, as I've already said. From the MWI point of view, the only role "probability" plays is epistemic--the MWI proponent has to explain why it appears to us that predicting the results of QM experiments involves probabilities.

What role does the Born rule then have in that view?

The MWI proponent has to explain why our observations of QM measurements appear to obey the Born rule. But the rule plays no role in the actual dynamics of the wave function in the MWI.

the essence of MWI in this matter seems to be to me that all worlds get realized by each measurement

Yes.

 

[entropy1]

The MWI is completely deterministic, as I've already said. From the MWI point of view, the only role "probability" plays is epistemic--the MWI proponent has to explain why it appears to us that predicting the results of QM experiments involves probabilities.

So my (not so well explained) view is that if the MWI is the hypothesis, that (in my view) the Born rule is superfluous, like you also seem to say.

But the rule plays no role in the actual dynamics of the wave function in the MWI.

After further investigation I even think that this may be in conflict with interpretations that use the Born rule. Until the Born rule is, as you say, explained within the MWI.

 

[PeterDonis]

if the MWI is the hypothesis, that (in my view) the Born rule is superfluous

I even think that this may be in conflict with interpretations that use the Born rule. Until the Born rule is, as you say, explained within the MWI

This is indeed a common criticism of the MWI.