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The many-worlds interpretation of quantum mechanics has always been explained to me at a high level using examples of binary events (e.g. atom either did or did not decay at any given moment in time), which leads to a conceptually clean idea of "branching" into two distinct universes. But how does branching work if you have, say, a 70% probability of something happening (e.g. measuring the spin of an electron having gone through an SG apparatus oriented at an arbitrary angle)?

Do you say that 7 universes got to spin up and 3 that got spin down (to account for the 70% probability)? Does that mean those 7 universes are in-every-way identical copies of each other? But then what if you had something with a 71.87154819...% chance of happening? You would need an uncountably infinite number of branches to be able to represent arbitrarily precise probability ratios, and then subsets of those branches would contain uncountably infinite universes that are 100% degenerate and identical to each other. Is this what the standard many worlds interpretation assumes?

If not this, then what? You can't say that there's a 70% chance of universe A happening and a 30% chance of universe B happening if you're saying both happen. How does the many-worlds interpretation put a "weighting function" on different branches or outcomes?

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  • $\begingroup$ Comments are not for extended discussion; this conversation has been moved to chat. $\endgroup$ – David Z Mar 18 '20 at 3:32
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Binary branching is just a simplification to make it easier to explain without math. The actual math is very simple, and can handle unequal probabilities.

At the simplest level, a branching occurs when you can write the wavefunction as a sum $$|\psi \rangle = |\psi_1 \rangle + |\psi_2 \rangle$$ where $|\psi_1 \rangle$ and $|\psi_2 \rangle$ are orthogonal and decohered, i.e. that there is no reasonable physical process that can make them overlap again. In this case we colloquially describe the two terms as "worlds" or "branches", and the probability of being in each one is the norm $\langle \psi_i | \psi_i \rangle$, which can be an arbitrary number between zero and one. The same logic goes for branching into more than two "worlds" at once, and repeated branching: you just get a sum of many terms, and the probability of each one is its norm.


After some comments, I get the feeling you really want a discussion of where the probability in the many worlds interpretation "comes from". Again, this is a very subjective and debatable thing, but my favorite take on it is "self-locating uncertainty".

Suppose that somebody kidnaps you, blindfolds you, and takes you somewhere in Uzbekistan. When you come to your senses, are you closer to Samarkand than Tashkent? You don't know for sure, so you can only answer in terms of probabilities. This is self-locating uncertainty: you're certainly in a definite place, and it's not like there are many copies of you running around, but there's probability nonetheless. You can use a variety of information to help. For example, if you weight by area, about 85% of the country is closer to Samarkand. (But this doesn't mean there are $85$ copies of you near Samarkand and $15$ copies of you near Tashkent!) But if you weight by population, substantially more of the population is closer to Tashkent, because it's the capital. Of course, which weighting is the correct choice depends on how the kidnappers set things up.

Now, suppose that after the spin of a particle is measured by a device, the state is $$|\psi \rangle = \sqrt{0.85} |\text{spin up measured} \rangle + \sqrt{0.15} |\text{spin down measured} \rangle.$$ You are living in one and only one branch of the wavefunction, but until you look at what the device is reading, you don't know which. At best, you can assign probabilities. The core assumption of many worlds is that the correct choice of probability (i.e. the choice that corresponds to what you actually observe, when averaged over many measurements) is to take the coefficient of each branch and take its norm squared, i.e. to assign an 85% chance to observing spin up.

If you ask where this assumption comes from, it's a perfectly legitimate question! However, the point is, there's no principle that says the probabilities have to be equal across branches. That's like saying every day must have a 50% chance of rain because it can either be rainy or not.

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    $\begingroup$ Hmm, I still do not understand what it means to say "the probability of being in 'branch' i is X%" if it's not a frequentist interpretation of how many branches have the features of i vs. number of branches that do not. From a "static multiverse" perspective, where there is not a sentient being solving the schrodinger equation, what about the multiverse structure leads to the observations that most subjective observers would observe X%? How is that probability physically enforced? $\endgroup$ – Sean49 Mar 16 '20 at 5:47
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    $\begingroup$ @Sean49 Great, that's the fundamental sticking point of many worlds. Every interpretation of quantum mechanics has at least one unsatisfying and unintuitive point (as it must, because if it were all natural and intuitive, it'd just be classical mechanics, not quantum mechanics). In many worlds, that point is that you have to put in this probability rule by hand (the "probability of being in each branch"). $\endgroup$ – knzhou Mar 16 '20 at 5:52
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    $\begingroup$ @Sean49 There are attempts to derive this rule from other principles (usually, you start by justifying 50/50 for equal branching somehow, then argue how it should scale based on the norms), but I've never found an example where those principles were any simpler than the rule they were cooked up to derive. $\endgroup$ – knzhou Mar 16 '20 at 5:55
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    $\begingroup$ "orthogonal and decohered" — if they are decohered, then it wouldn't make sense to write $\psi$ as a state vector and to sum them $\psi_1$ and $\psi_2$ as such: it should be a density operator, and $\psi_1$ and $\psi_2$ should enter into it as independent density operators $|\psi_1\rangle\langle\psi_1|$ and $|\psi_2\rangle\langle\psi_2|$. $\endgroup$ – Ruslan Mar 16 '20 at 10:21
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    $\begingroup$ @Ruslan Well, I mean, we are discussing the interpretation of quantum mechanics, so my only response is that your statement is... up to interpretation. $\endgroup$ – knzhou Mar 16 '20 at 19:26
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You would need an uncountably infinite number of branches to be able to represent arbitrarily precise probability ratios, and then subsets of those branches would contain uncountably infinite universes that are 100% degenerate and identical to each other. 

What's the problem with having an infinite number of branches? David Deutsch, a leading modern proponent of the Many Worlds interpretation, proposes that scenario in his popular book, The Fabric of Reality. In this picture, the universe started with an infinite number of parallel branches or strands, and at each quantum decision various subsets of those strands diverge, with all of the strands in any given bunch being 100% identical.

This version of MWI gets around an objection that many people have of MWI: at each quantum decision it seems that a whole new universe (or many new universes) needs to be created for the new branch (or branches), and that sounds like a flagrant defiance of the conservation of energy. Deutsch's scheme shifts that problem to the moment of initial creation at the Big Bang.

Personally, I'm not a huge fan of the Many Worlds interpretation, but Deutsch's version is my favourite flavour of MWI. To paraphrase Niels Bohr, it's a crazy theory, but I'm not sure if it's crazy enough to be true. ;)

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    $\begingroup$ Thanks! I had not heard of this version of MWI before and it is the easiest for me to understand and makes sense to me (though I am still going to see if I can understand the other versions being discussed i this thread too) $\endgroup$ – Sean49 Mar 16 '20 at 19:56
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    $\begingroup$ came to say exactly this. To clarify, when you say "diverge", it's not that new branches are created, it's that the already-existing branches are no longer identical. In this interpretation, all quantum mechanics is deterministic, rather than probabilistic within a given universe, but there is no measurement you can make to tell which univsere you are in, so the probabilistic phenomenology of QM is preserved $\endgroup$ – thegreatemu Mar 16 '20 at 20:26
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    $\begingroup$ Personally this just feels like a completely equivalent rephrasing -- like replacing the equation $4 = 2 + 2$ with $2 + 2 = 4$. The idea that there are universes violently exploding into existence like mini-Big Bangs whenever a measurement happens is not true in any interpretation -- it's just an unfortunate misinterpretation of a handwavy metaphor used in popsci. $\endgroup$ – knzhou Mar 17 '20 at 4:30
  • $\begingroup$ @knzhou Indeed! Deutsch's version isn't a different interpretation, it's still standard MWI. I like it exactly because it doesn't make people think that new branches must be sprouting / exploding into existence at every quantum decision. $\endgroup$ – PM 2Ring Mar 17 '20 at 4:55
  • $\begingroup$ This idea of branching can be reasoned about with Branching Time Temporal Logic. One consequence of this model is that is can always be possible that a coin will flip heads at the next step, but in no universe will the coin always flip heads. $\endgroup$ – gmatht Mar 17 '20 at 6:27
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I am no expert on the many-worlds interpretation, but I always thought that if the wavefunction is $\sqrt{7/10}\psi_1+\sqrt{3/10}\psi_2$ then there is (in this interpretation) one universe in which the wavefunction collapses to $\psi_1$, one universe in which the wave function collapses to $\psi_2$, and you have a 70% chance of branching into the former and a 30% chance of branching into the latter.

I don't think the Born probabilities have anything to do with the number of branches. There is no notion that branching into each branch is equally likely so you have to have 7 $\psi_1$ branches and 3 $\psi_2$ branches.

Instead, the number of branches is simply the number of possible outcomes for a measurement of the observable, which in this case is 2. For a more complicated superposition it could be more, but the number of branches is still unrelated to the branching probability.

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    $\begingroup$ But this interpretation implies that the observer is somehow some meta thing, distinct from the multiverse, that either branches here vs. branches there with some probability. My (incorrect?) understanding is that the multiverse is all there is, so the probabilities have to arise from the branching structures themselves. It's not like there are consciousnesses surfing through a background branched structure, splitting different ways with different probabilities, right? I don't think many worlds interpretation puts consciousness on any kind of special footing? $\endgroup$ – Sean49 Mar 16 '20 at 5:49
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    $\begingroup$ @Sean49 Nobody really knows what consciousness is. Some quantum interpretations do put consciousness on a special footing, eg the von Neumann–Wigner interpretation, aka the "consciousness causes collapse" interpretation. But generally QM interpretations try to avoid the question of consciousness. Typically, an adherent of MWI would say that with a 30:70 split, the you in the 30% branch feels just as real as the you in the 70% branch does. $\endgroup$ – PM 2Ring Mar 16 '20 at 9:52
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    $\begingroup$ @Sean49 You’ve gotten the same answer from two people, and both answers have gotten plenty of upvotes and no downvotes. As far as I am concerned, you simply have a misunderstanding of MWI. BTW, I don’t consider consciousness to play any role in the branching. $\endgroup$ – G. Smith Mar 16 '20 at 17:19
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    $\begingroup$ @G.Smith Yes I am confident that the answers I have received (including yours) are correct, and I am just having difficulty understanding them. My response was not meant to be "here is why you're wrong" but more "here's how I interpret that in my head; can you please help me identify my point of confusion and correct it?" I am trying to understand why sampling a random trajectory through the "multiverse" will on average lead to the observed probabilities, if the number of branches at each branching happens based on # of outcomes, not their probabilities. $\endgroup$ – Sean49 Mar 16 '20 at 19:42
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    $\begingroup$ @G.Smith: I think what's confusing is that the "many worlds interpretation" is commonly presented (in popular science) as meaning that we always take every branch: that is, every world actually exists. So "the probability of taking a branch" seems like it should be 100%. $\endgroup$ – ruakh Mar 16 '20 at 22:18
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Well, first of all, "branching" into different "universes" is a simplification, but I'll put that aside as that's a rather difficult area to explain. A gross simplification is that whenever an experiment can have two different results, each result exists, and the degree to which each exists is proportional to their probability.

So, let's say you have an experiment in which the Copenhagen interpretation says that you have a 70% chance of measuring a particle as spin up, and 30% chance of measuring spin down.

What MWI says is that the initial state can be decomposed into a quantum state in which the particle is spin up, and a quantum state in which the particle is spin down. As the states evolve, they interact with the rest of the universe, and so the states encompass the entire universe, including the experimenter. The first state ends up describing a universe in which the experimenter observes a spin up participle, and the second one a universe where the experimenter observes a spin down particle.

MWI also says that the number 0.7 is associated to the first state, and to the second, 0.3. These numbers are sometimes referred to as "measures", "weights", or "probabilities". Now, where things get a bit fuzzy is what these numbers "actually" are. The simple answer is that they are "probabilities"; the first number says that the "probability" of finding yourself in the first state in 70%. But that just raises the question of what "probability" means. There's no physical "thing" that the word "probability" refers to. One can say that it refers to long term trends; if we had 1000 experiments with "essentially the same" setup (whatever that means), then we should expect to find ourselves observing spin up around 700. More precisely, observing spin up 700 times is the most likely result. But that just defines it in terms of probability again.

Classical physics says "If you do $X$, then $Y$ will happen". Copenhagen interpretation says "If you do $X$, then $Y_1$ will happen with probability $p_1$, and $Y_2$ will happen with probability $p_2$". MWI says "If you $X$, then there's this quantity 'measure' that has a value of $p_1$ for $Y_1$ and $p_2$ for $Y_2$".

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