I must admit I struggle with the MWI Born rule derivations based on rational credence. I don't see why proving that one ought to assign credence in such and such a way is sufficient to prove that that's the way nature is. It feels too much like deriving an "is" from an "ought", although in a slightly different way than what Hume objected to!
> While it has been claimed that Born's law can be derived from the many-worlds interpretation, the existing proofs have been criticized as circular.
So it seems you aren't alone in feeling this way.
Edit: Forgot the link - https://en.wikipedia.org/wiki/Born_rule
The latter article is more recent at 2005 but as far as I can tell Carroll's self-locating uncertainty ideas weren't introduced until around 2014.
The TL;DR is that the critics are correct. Deriving the Born rule begs the question because it makes an unjustified assumption (branching indifference) and also introduces an "invisible pink unicorn", i.e. a concept that, according to the theory, has physical significance but cannot be measured. In the case of MWI that concept is branch weights.
[UPDATE] The critique I wrote is based on Wallace (https://arxiv.org/abs/0906.2718). Carroll's argument appears to be somewhat different. I'm just now working my way through his paper (https://arxiv.org/abs/1405.7577) but I'd be very surprised if it did not also have some untenable assumption hidden in there somewhere.
I have previously come across Deutsch's formulation in terms of information flow , along with what seemed like a very strong criticism from Wallace and Christopher Timpson  that his model was not gauge-invariant.
When scientists say Many Worlds, do they actually mean Worlds as physical 'parallel universes' that pop up into existence?
Or they only mean that those are probable outcomes of our measurement (probable histories) that never actually happened (only one of them happened - the one we end up being it)?
I think in general people accept that there can be a wave function for a cat with the states alive and dead. Cats are much more complicated than that. Presumably they have memory and inside that memory can be things like perhaps the result of a electron spin meansurement experiment. One state of the wavefunction of the cat might have the memory that the electron result was spin up. Another state of the cat might have the memory that the electron result was spin down.
This scenario above is a cat watching a electron spin measurement. Afterwards, the electron does not collapse into a single state upon being observed. Instead, it is still in two states but it is correlated with the memory of the cat (or entangled with the cat).
The belief, at least as I see it and I assume others believe this too, is that the cat has a conscienceness for each state of the wave function. So there is a "conscienceness" or "cat" that thinks the electron was measured as spin up. And then there is another that thinks that electron was measured as spin down. This is where the term many worlds comes from, the fact that there are two "consciences" (well, many consciences). I can see why people would think that part is weird. But how else should it work?
I guess it all comes down to what is the experience of a person (or cat) having a wave function and being in multiple state at the same time (just like all other objects in quantum mechanics). We are not external observers to the world, we are a part of it and we have a wave function too. Or, it would be more correct to say we are a part of the wave function of the system. There is not a separate wave function for each thing. There is just one wave function.
How exactly would they interfere?
In textbook/Copenhagen interpretation, we say that when the electron passes through the slits the wavefunction has not yet been measured. But with the detector in place, you collapse the wavefunction to an eigenvector of position. By the uncertainty principle, the momentum uncertainty is now very high, so the electron shoots off in a random direction and cannot be expected to follow any pattern.
In MWI, we start with the interference pattern. But adding the detector does not "collapse" anything - instead it introduces lots of degrees of freedom. The particle passes through both the left and right slits, but the degrees of freedom ensure there is no fixed relative phase delta between these possibilities. The detector turns waves into "static noise", and so any interference pattern is lost. Both options are realized, but cannot detect the other possibility: distinct Worlds.
The key point is that MWI has no special role for measurement. The left-slit and right-slit worlds interfere, but with the detector in place the pattern is destroyed: the interference is uncoordinated and averages to zero, so undetectable. Whereas in textbook/Cophenhagen QM we say that the measurement results in a single outcome.
I had a discussion with someone on HN about this a while ago and realised that "world" or "split" isn't necessarily synonymous with "superposition". Rather, I believe that a split occurs when the superposition entangles with the environment and causes sufficient decoherence that there is negligible probability of (measurable) interference.
In principle you could imagine a thought experiment where you had a super-powered quantum machine which finely could control the quantum state of a large, isolated room. In that situation you could imagine someone in the room conducting an electron spin measurement and looking at the outcome, before the machine enacts a reversal of the room's quantum wave-function, thus causing the two copies of the person to interfere. If we ever reach that level of technology, it will be fascinating to see how the interpretation debate progresses.
Backing up, the problem is that you have these different approaches—depending on how you count them you could maybe group them broadly and say there are 3-4, or as large as a dozen or two—that are all mathematically identical. They all predict the Born rule, but suggest different ways that Nature really fundamentally would act to produce that rule.
Since they are mathematically identical, it is provable that there is no way to choose between them. As a result, the one which steps the most out of peoples’ ways to enable experimental results, has better “genes” for its own reproduction in the publication of papers. And that has just been the Copenhagen interpretation: you the experimentalist have a soul and when that soul measures the world the nice unitary evolution of the world comes crashing down with probabilities given by the Born rule. Contrast with pilot-wave theories where you have to work out a whole separate equation that doesn’t do anything which makes any further observational impact.
The basic issue that we are facing is that while the notion of souls seems laughable for fashionable sciencey people, it also seems in some distressing way inevitable. You take Many Worlds for instance, you admit the reality of every single possibility of the entire universe as a much broader multiverse. An equation, the Schrödinger equation, essentially works over, say, a Planck time to create a vector field on top of this, saying “This instant is followed by that instant is followed by that instant.”
In the middle of that, what uniquely qualifies my experience here and now as I know such experience must exist? I do not perceive a multiverse; I perceive a changing universe. And MWI says “well actually there are a million yous, frozen in time, all perceiving changing universes. Your experience of motion through time is actually kind of a lie.” This is not a unique problem to QM; it happened much earlier with special relativity where we discovered that you are actually a rope of worldlines thrusting through a static four-dimensional Lorentzian manifold, every part of that rope being presumably in a separate conscious state, perceiving itself as moving through time but we “on the outside” can see that there is no unique present defined such that it can wash over all of the ropes simultaneously. MWI just happens to facilitate the same basic “unrolling of time” because it has already unrolled all of possibility-space. And to fix it you need something—I’m calling it a soul, you can get fancy—which “zips along the worldline” and contains my conscious experience, or you need to argue that my experience is an illusion, or you need some “universe-soul” to act like a coherent “present moment” for all of us, or the like. It all kind of sucks.
There is a nice perspective sitting in the middle of this due to Andreas O. Tell , and it is pleasantly agnostic while still doing something like what MWI is doing to try and derive the Born rule from normal wavefunction evolution. In brief, he says: “use the state-matrix formalism for QM, and take a completely agnostic view of what the cosmos is and how it behaves. Still if you have a local information-processing system which is embedded in the cosmos and changing, then it receives information and must update its model of the universe. Its model of the universe must necessarily come down to a list of wavefunctions with a list of weightings, but there is a freedom-of-perspective which allows you to choose the wavefunction with the highest weight as “the” one that you think the system is in. New data just forces these weights to cross in size, causing the Born rule when you try to determine whether those weights will cross over and the system will be in the new state.”
In some sense then we can live in a very Copenhageny world where we are changing data-processors uniquely present in some space and time, but we can use Schrödinger evolution to derive the Born rule just the same as the many-worlds interpretation does, but rather than committing to its plethora of different universes we might be able to just remain non-committal about what is in the rest of the universe, beyond what I see in it.
I actually believe that this idea has been rediscovered in it essence at least half a dozen times, going back as early as (at least) the late 90s:
I think it's arguable that even Everett himself held this view, and there is some evidence that Schrödinger held it as well but didn't have the courage to admit it (because he didn't have the benefit of the Aspect experiment to support it).
> Undetectable to the observer, different alternate realities can fight for becoming the dominant one, at least over a short period of time. This effect appears to be highly unsettling and not really greatly preferable to the world-splitting n the Everett interpretation
Tell himself is indeed not working in the field anymore—this preprint was submitted to at least one journal as I understand but was not accepted before the grant was up or so and he could no longer keep pushing for publication; instead he went into acoustic signal processing with a friend, and they both started a company called SoundTheory now. Something about trying to maximize the information “punch” of music to your brain makes it sound better or reduces background noise or something.
The paper is still interesting on its own merits though. I mean, it’s interesting to me; of course your mileage may vary.
And, what is the view of the cosmos that is not difficult? Would that be some instance of wavefunction "collapse"?
There's a great paper by van Kampen that points out that what we call a measurement and collapse of the wave function is entangling a quantum state to an irreversible statistical process. Measurement and wave function collapse are limit cases of that that are useful approximations when doing calculations.
: http://www.johnboccio.com/research/quantum/notes/vankampen.p... (a version sans paywall!)