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I'm thinking that the Many worlds interpretation trivializes the Born rule. Suppose we take an electron's spin's state vector such that the probability of spin up is 0.6 and that of spin down is 0.4. We take 100 such identical systems and perform measurements on them one after one.

According to the Many worlds interpretation, a total of $2^{100}$ worlds will be created, each corresponding to the decohered branches of the wavefunction. We can draw it like a tree where there are $2^{100}$ child nodes. Each of these nodes has a unique history leading up to the parent.

But, which of these histories follow the Born rule? There would inevitably be some histories for which the number of Spin up measurements was $60$, and the number of Spin down measurements was $40$. The number of these histories will be $C(100, 60)$

These particular histories trivially just happen to follow the Born rule. There is no preference given to the Born rule in the wavefunction evolution. This would mean that the Born rule is not really a law of physics. We just happen to belong to one of the histories which has followed the Born rule so far. Does the Many worlds interpretation really say this?

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  • $\begingroup$ "But, which of these histories follow the Born rule? There would inevitably be some histories for which the number of Spin up measurements was 60 , and..." What measurements? Once you make your measurements, the outcome is determined in the other world, rendering measurement meaningless? Or at least ignorance or the outcome? I'm not sure about this since I haven't studied interpretation in detail, but is the wave function collapse part of MWI or something else? If not, maybe a more important question would be one surrounding the validity of wave function collapse and not the Born rule? $\endgroup$
    – joseph h
    Commented May 3, 2023 at 5:26
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    $\begingroup$ You are prodding a known pain point of MWI. In fact, this is a pain point generic to all interpretations that use decoherence. It is not yet clear how these interpretations would get the Born rule, or give a satisfactory resolution to this probability problem. $\endgroup$ Commented May 3, 2023 at 5:31
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    $\begingroup$ @josephh The wavefunction is not taken to collapse in MWI. $\endgroup$ Commented May 3, 2023 at 5:33
  • $\begingroup$ @naturallyInconsistent Okay. So there is no satisfactory answer to this question? and it should be one of collapse? $\endgroup$
    – joseph h
    Commented May 3, 2023 at 5:34
  • $\begingroup$ @naturallyInconsistent Ahhh. I see. Thanks. $\endgroup$
    – joseph h
    Commented May 3, 2023 at 5:35

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"According to the Many worlds interpretation, a total of $2^{100}$ worlds will be created, each corresponding to the decohered branches of the wavefunction."

That's a simplification. In the Many Worlds interpretation, when the observer and the electron interact, they both evolve jointly into a superposition of orthogonal states. So you don't just have to count all the possible outcomes for the electron state, but also all the possible outcome states for all the $>10^{23}$ atoms of the observer. So say each observation results in a split into $10^{23}$ joint electron-observer states, in about 60% of them the electron spin is 'up', and 40% of them the electron spin is 'down'. Then subsequent observations split the cases further, into $(10^{23})^{100}$ cases. But the distribution of the total number of 'up' observations among those states has a peak around 60. The vast majority of worlds follow the Born rule, only a much smaller number (but still very big) don't.

When we decompose a state $s$ into a sum of orthogonal states $s_1+s_2+s_3+\ldots$, their magnitudes combine like orthogonal vectors. $|s|^2=|s_1|^2+|s_2|^2+|s_3|^2+\ldots$. If each individual substate is assumed to have roughly equal magnitude, then the number of joint substates divides in proportion to the squared magnitudes of the states being observed.

Followers of MWI believe that, unlike other interpretations, the Born rule can be explained from more basic principles, rather than having to be assumed as an axiom. But the details of exactly how it works are widely considered to be still a work in progress. Proposals are somewhat fuzzy and hand-wavy, and still controversial.

Everett argued in his thesis that it had been explained, although his argument amounts to saying that because squared magnitudes were the only measure on state space that could conserve probability, it was the appropriate one to use. (To comment on what MWI says or doesn't say, it's well worth reading his thesis first.) You can argue about whether or not he was right, but MWI certainly doesn't propose a trivial "we just happen to belong to one of the histories which has followed the Born rule so far" explanation of the Born rule.

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    $\begingroup$ I don't understand. Why does the number of universes have to do with the number of atoms? The $10^{23}$ atoms have a joint wavefunction $\psi$. After measurement, we get two universes which are the decohered branches : $|\psi _{up} , up\rangle$ and $|\psi _{down}, down\rangle$. And if only two branches are created, then the Born rule does not get any preference as a law of the universe, but rather it just happens to be trivially true? $\endgroup$
    – Ryder Rude
    Commented May 3, 2023 at 14:00
  • $\begingroup$ @RyderRude I think the choice of $10^{23}$, pretty close to Avogadro's number, was either unintentional or a mistake. The central point is still just that the number of universes created per measurement is much bigger than two. $\endgroup$
    – AXensen
    Commented May 3, 2023 at 20:09
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The Born rule states that if you have a state $|\psi\rangle$ and you measure an observable with eigenstates $|i\rangle$, the probability of getting the $j$th state is $|\langle j|\psi\rangle|^2$. Regardless of what interpretation of quantum theory you adopt, that statement is entirely consistent with getting a set of observations whose relative frequencies don't match the Born rule probability, just as it is possible to flip a fair coin and get 100 heads in a row.

In most intepretations of quantum theory, the Born rule is treated as a postulate. In the MWI the Born rule has been explained in a couple of different ways.

One explanation uses decision theory. The probability of an outcome is a function of the state that satisfies particular properties required by decision theory, like if a person uses those probabilities you can't make him take a series of bets that will make him lower the expectation value of his winnings, the probability of an outcome can't be changed by a later measurement and some other assumptions:

https://arxiv.org/abs/0906.2718

https://arxiv.org/abs/quant-ph/0303050

https://arxiv.org/abs/quant-ph/9906015

https://arxiv.org/abs/1508.02048

There is another approach to deriving the probability rule called envariance, which uses properties of the state of a system that remain unchanged after interaction with the environment:

https://arxiv.org/abs/quant-ph/0405161

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  • $\begingroup$ But in case of MWI, there is no indication that the Born rule is given any preference. It's just that some histories will trivially happen to follow the Born rule. In the other interpretations of QM, the Born rule is preferable as the best "predictor" of future, even if the observations don't exactly align with it. In MWI, how can any probability rule be given any preference? $\endgroup$
    – Ryder Rude
    Commented May 3, 2023 at 8:21
  • $\begingroup$ @RyderRude The relative state has more structure than just being a collection of parallel universes. For example, if outcomes have equal amplitudes you can permute them without changing the state and this is part of the explanation of why they have the same probability. You could read the papers linked and ask specific questions about their contents. $\endgroup$
    – alanf
    Commented May 3, 2023 at 8:57
  • $\begingroup$ The Born rule is not a postulate. It is a description of observations. I don't know why people find it so hard to believe that the founders of quantum mechanics actually knew atomic physics. What else would they have been talking about? Heisenberg, Bohr, Schroedinger, Einstein, Planck etc. had a century's worth of unexplained observations that needed a theoretical explanation. $\endgroup$ Commented May 3, 2023 at 20:14
  • $\begingroup$ @FlatterMann dictionary.cambridge.org/dictionary/english/postulate $\endgroup$
    – alanf
    Commented May 3, 2023 at 21:23
  • $\begingroup$ Axiomatizing scientific theory blindly interferes IMHO with a proper understanding of nature. "Laws of nature" are human shorthand for large numbers of observations and ignoring the details of the actual observations leads to gaps in understanding. The SE, for instance, allows us to calculate atomic energy levels. What we are observing, however, are the energy differences between these levels. Why? Because the way we are observing atoms is by scattering light off them, which is not exactly what the unperturbed SE describes. That's why the S-matrix was invented shortly after the SE. $\endgroup$ Commented May 3, 2023 at 21:44
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The Born rule relates quantum amplitudes (complex numbers in the closed unit disc) to classical probabilities (real numbers in $[0,1]$). That doesn't actually help with interpretation, because the meaning of classical probabilities is no clearer than the meaning of quantum amplitudes.

What does it mean to say that a coin has a 0.6 probability of coming up heads? It doesn't mean that if you flip it 100 times you'll get 60 heads. According to classical probability calculus, the chance of getting 60 heads in that experiment is just $0.6^{60} 0.4^{40} C(100,60) \approx 0.08$. Incidentally, what does that mean? It doesn't mean that if you repeat the 100-flip experiment 100 times, you'll get 60 heads eight times.

You can prove that in the limit of infinitely many tosses, the ratio of heads to tosses approaches a limit of 0.6 – or more precisely, that the probability that it doesn't approach that limit is exactly 0. You can prove the same thing about amplitudes of worlds in the many-worlds picture, without any reference to classical probabilities or the Born rule. And that's about the best you can do.

In practice, we deal with this by assuming that sufficiently unlikely scenarios don't happen – for example, that the very large amount of evidence we've gathered in favor of the existence of electrons is due to electrons actually existing, and not to a statistical fluke, even though the chance it's a fluke isn't precisely zero. For that purpose, you can equally well assume that the classical probability isn't within $\epsilon$ of zero or that the quantum amplitude isn't within $\sqrt\epsilon$ of zero.

I'm not suggesting that this is an acceptable state of affairs, only that the Born rule doesn't make it better. There's no evidence that classical probability underlies quantum mechanics any more than Newtonian mechanics underlies special relativity. Older ideas about the nature of the world aren't more sensible, just more familiar.

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  • $\begingroup$ A probability is the limit of a frequency for an infinite number of repetitions of the same experiment. There is no great mystery here. That the law of large numbers works is an observation. Also not a mystery, simply the scientific method applied to scenarios for which no useful microscopic description exists. The Born rule tells us that the irreversible dynamics of "measurement" is not the same as the reversible dynamics of the isolated quantum system. Why is it a problem that "QM system + CM measurement" is different from "QM system alone"? $\endgroup$ Commented May 3, 2023 at 20:18
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    $\begingroup$ @FlatterMann The first half of your comment is similar to what I wrote in the answer. I just disagree that there's no mystery to it. In the second half I think you're just saying that modeling measurements as quasiclassical is a useful approximation, which I agree with. A+B (where B has a temperature) is different from A alone, but A treated quantum-mechanically + B treated classically should be the same as A+B treated quantum-mechanically, or at least should approximate it well. $\endgroup$
    – benrg
    Commented May 3, 2023 at 21:22
  • $\begingroup$ To me it seems that the "observables" are dictated to us by nature in form of actual physical interactions between the quantum system and "an external system". The fundamental ones happen to be the conserved quantities of energy, momentum, angular momentum and charges. What von Neumann calls "a measurement" is, kind of, a generalized "analog" Fourier transform that happens before we irreversibly remove energy from the quantum system. For some measurements the "transformation" can approximated with a magnetic field or an optical grating but in general there is no "physical" implementation. $\endgroup$ Commented May 3, 2023 at 21:35
  • $\begingroup$ I understand this, but I think that the Born rule is not getting any preference as a "probability law" in case of MWI. The other interpretations postulate the Born rule as a law, which justifies the statement : "The relative frequences aproach $6:4$ as $n\rightarrow \infty$." But in case of MWI, this happens to be trivially true for some of the histories. That isn't even the majority of histories. The majority of histories will happen to prefer the 50-50 probability rule. These histories will be $C(n, \frac{n}{2})$ in count. $\endgroup$
    – Ryder Rude
    Commented May 4, 2023 at 1:37
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    $\begingroup$ @RyderRude A plurality of potential Born-rule measurement outcomes have 50% heads. You're assuming that's okay because those outcomes are "less likely" to be actualized, but you're in the middle of establishing what "less likely" means (in terms of repeated trials), so the argument is circular. Really the collapse picture assumes not just the Born rule but also some subtle and probably unformalizable assumptions about what the numbers it produces mean. You could draw the same conclusions about amplitudes if you were willing to make the same assumptions about amplitudes. $\endgroup$
    – benrg
    Commented May 4, 2023 at 3:07

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