Yes, back when I studied this topic seriously, this was an issue. E.g., if you toss a quantum coin that has a 1/3 chance of coming up heads and 2/3 chance of coming up tails, this seems to result in only two "worlds". And if there are two worlds, why are the observed probabilities then not .5/.5 rather than .3333/.6667?

I didn't mention this in my OP because (1) that would have been something of a deep-dive for a summary post, and (2) there were ideas being floated about to solve this problem back when I was studying this, but I don't know how these ideas ultimately panned out.

I'm surely curious as to what the current best ideas are about this issue.

At some point this debate becomes a bit too confusing for me. All I can report is that the experts fretted over this.

The relevant probabilities are not derived by number of "worlds". Pick some particular moment and correlated history, look backwards (what is recorded in the current "configuration") at experiments, and one should see the proper statistics appearing in the "vast majority" of experiences.

However, there will be plenty of experimenters who see wrong statistics. Everett predicts this with certainty. There is a "world", according to this, that just split from the moment I am writing this, in which all future experiments have spin up coming up 100% of the time from that moment on. Over time, we all end up correlated with this as the experimenters report their fantastical findings.

If they truly believe in Everett's theory, they would accept that they just happen to be in the branch where this happens. In Bohmian mechanics, they would say something else is going on. The odds of seeing something like that in Bohmian mechanics are so vastly, incomprehensibly small, that it is more likely to see cracked eggs reassembling themselves from random thermal motions. But in Everett, it happens with certainty to some universe.

This is the difference. Bohmian mechanics can be readily falsified based on statistical outcomes of experiments. Perhaps not with 100% certainty, but certainly with 100% practical certainty. Everett can never be falsified based on statistics. It could be falsified if something that was supposed to happen with a literal 100% certainty failed to happen, but with anything statistical, it simply can't because the theory says it does happen.

One could modify the theory to cut out the "outlier" worlds. This is, in some sense, what GRW with a mass density ontology does.

In the Bohm Interpretation, that coin could always come up heads too, but again with a vanishingly small probability.

So they seem equivalent experimentally to me. (And to the experts who have written entire books on the subject.)

Max Tegmark came up with a way to experimentally determine if the Everett Interpretation is correct. (I believe it was Tegmark who came up with this.) It has a high cost for the experimenter, though!

What you do, is rig a gun to a fair quantum coin, so when you pull the trigger, the gun fires 50% of the time. Now shoot yourself in the head with it many, many times. If you end up surviving many rounds of this, you can be pretty darn certain that the Everett Interpretation is correct.

Never mind the billions of other versions of yourself that you murdered to discover the truth!

Hence, in the Everett Interpretation, if you shoot yourself using such a quantum gun, every time you pull the trigger, there will end up being one "world" in which the gun didn't go off and one world in which you put a bullet in your head.

One of the corollaries/interpretations of Sanov's theorem is that, generally speaking, when faced with an astonishingly improbable outcome (e.g. flipping 9,000 heads and 1.000 tails out of 10,000 independent coin flips), no statistical test can differentiate between "that improbable occurence with a fair coin" and "an unfair coin" - the fair coin, when it does something improbable, with have (with overwhelming probability) a specific tilted distribution that looks unfair.

TL;DR By grouping states together (human choice), certain arrangements seem more probable than others.

This is referencing the nature of the measure and the difficulty of deriving the Born rule. Why should the outcome of measurements be proportional to the square of the wave function? That's indeed a problem in MW.

But even if that's solved in some way, there's an even more foundational issue of how non-determinism can arise at all in a deterministic theory. The MW reply is that there is no non-determinism, but then has trouble explaining observed reality - which is not a good position for a theory to be in.

Is the mathematical formalism actually more unwieldy, or is it just this interpretation of what it's doing? The 'select a single world' sounds like taking some of the conceptual framework from Everett and trying to paste it on here, rather than giving the Bohm interpretation its own conceptual framework which could possibly be more elegant?

Once we settle on stuff with position, then we have to ask how that stuff changes. One option is that we specify accelerations; that's Newton's way.

Another option is to specify velocities. That is the Bohm way. Specifically, the velocity is derived from the wave function of quantum mechanics. It is basically the derivative of the wave function, normalized and made real. Done. You can see a simple derivation of the equations here: https://en.wikipedia.org/wiki/De_Broglie–Bohm_theory#Derivat... The first one derives Bohm's equation very quickly and simply from the starting wave relation equations of Einstein and de Broglie.

The complication is that the wave function is a function of the configurations of all the particles. Thus, to get the velocity, one technically needs the positions of all of the particles of the universe. Practically, one only needs the positions of entangled particles, but still. It is a non-trivial setup though the most basic, natural setup one could possibly have given particles and a wave function. Also, the particle positions do not influence the wave function evolution. This is rather unusual.

For the Dirac wave function, it is even simpler. That object directly gives the velocity of the particles; no derivative needed.

Contrast this with MW which says, I guess, that the fundamental thing that we are concerned with is the wave function. It does not seem that particles exist in any meaningful sense in that theory. It is as if we are machines able to track a singular aspect of the change of an abstract vector in Hilbert space. It is not clear why this vector is often represented as a function on configuration space when there is no configuration of stuff. The theory essentially says that actual reality is nothing at all like what we perceive. I would think an honest account of a theory which only has a wave function would be to formulate the theory on an abstract Hilbert space and derive configuration space and, thus, configurations from that. Not very likely, by the way.

Reality may be deceptive, but I certainly prefer to start with theories in which our experience is explained in a pretty simple and straightforward way: it looks like stuff is over there because there is stuff over there. We have singular experiences because a single experience is what actually happens.

Also note that in Bohmian mechanics, we specify the initial wave function and the initial particles and then evolve the system using differential equations. All of the operator stuff, collapse, etc., comes out of that evolution; we don't need to have any special considerations about them. The quantum formalism becomes analogous to thermodynamics, not a fundamental theory, but a useful practical one replacing the individual evolutions with some useful shorthand. In MW, there is this question of how to model an experimental situation. Where are the measurement operators coming from? What is a subsystem? In BM, these things arise essentially by conditioning on the configuration of the environment. In certain situations, this will give rise to roughly an isolated system evolving according to its own Bohmian dynamics. The measurement interaction is then represented by an operator, or its generalization, depending. But all of that emerges from the basic differential equations evolving the universe.

It is not clear to me how easy it is to do that kind of analysis in MW. After all, there is no singular experience to break it down to, there is no subsystem, there is no definitive experiment being done. It is not really clear how one would falsify a theory which, more or less, assumes everything happens.

This assertion does not make sense to me. The Everett Interpretation and the Bohm Interpretation are experimentally indistinguishable from each other, as I understand things. Consequently, there is no mystery at all with the Everett Interpretation as to why things appear to us the way that they do.

Since the Everett Interpretation is a significantly simpler theory than Bohm's, we should prefer it due to Occam's Razor. On the other hand, since they are experimentally indistinguishable from each other, we can never scientifically assert which of the two is correct, no matter how much evidence we have.

The "indistinguishable" part happens because, according to Everett, there is some version of the experimenter that will have the same experience as the single experimenter in the Bohmian world.

This is not simpler. I have no reason to believe that there are infinitely many copies of me out there. Everett's theory says that there are. Fine. I can't disprove it. I also can't disprove that every instant of my experience is being carefully orchestrated by a thousand angels. It is experimentally indistinguishable from any theory you care to posit.

But I prefer theories where my actual experience is supposed to be a reasonable reflection of reality. I experience a single me and therefore I would prefer a theory in which there is a single me. Bohmian mechanics provides that and in a completely natural and reasonable way.

Everett categorically disputes my experience as being reflective of reality. There are infinitely many copies of me and my experience of being singular is an illusion. I can't dissuade people from embracing that, but it certainly strikes me as peculiar.

Also, in terms of experiments, Everett has infinitely many copies of the universe where all of the statistics of the experiments come out wrong. There are infinitely many that come out right. Is that experimentally indistinguishable? I don't know. Kind of a strange question in the context of "most everything happens".

And that's fine, but the idea that this is simpler than a theory which says my experience is a reasonable reflection of reality, is not. Occam's razor is not about number of equations, it is about what is simplest. I experience a single "me". A theory which supports that experience directly and obviously is simpler than a theory which does not.

This is particularly true when the "extra" equations are simple and obviously a part of the other equations.

The reason why this all gets so convoluted is simply because it exposes how much we rely on terms and concepts that are defined very fuzzily, and often aren't even defined at all, but just accepted for granted as if everyone means the same by them. And then it turns out that we don't, which should really come as no surprise.

It would be nice to say that there’s a 50% chance that I measured 0, but how exactly do you get that in a rigorous way from the state vector above?

To make everything complicated, the answer should not treat the experimenter part of the universe specially.

As for why 50%, why not the Born rule? Or are you asking how we derive the Born rule?

Not true in a many-worlds model. The “I measured” part is intended to account for the environment, at least initially.

> Or are you asking how we derive the Born rule?

More or less. In a many-worlds interpretation, there is no Born rule per se. I’m saying it’s not entirely trivial to recover the statistics that the Born rule would give.