Ok, how does one derive the Born rule from that idea?
(It being a problem with MWI is new to me, I am absolutely a physics amateur).
Here's three choices, none of them good:
1. Local Hidden Variables: When you open your eyes, you see which color your ball had all along! This is so obvious, but it's wrong for tricky reasons (Bell's Theorem) which I can't fit into the analogy; basically the ball cannot have "had a color all along."
2. Copenhagen: When you open your eyes, the ball just decides what color it is. The other balls will mysteriously agree, so there's not two blue or three red or whatever. How did they do that? Is that a real physical change, or just learning something new? (This is wavefunction collapse and it's very uncomfortable for obvious reasons.)
3. MWI: When you open your eyes, one version of you sees a red ball, and one version of you sees blue. Ok, but if there's red you and blue you, how can the red ball be "more probable?" Are there two separate "red ball" yous? Or maybe the red ball you is "more real?" (This is the relationship between the measure of the wavefunction and experimental probability, aka Born's Rule; MWI struggles to explain it, Copenhagen just postulates it).
You answered your own question!
Yes. There are multiple observers, they're cloned along with everything else.
All the "mystery" disappears as soon as you accept that.
It's like heliocentrism. Everything just "clicks" once you adjust you perspective.
Say you have an electron which is either up or down, a 2/3 percent chance to be up. The wavefunction looks like `2/3 * up + 1/3 * down`. In the MWI picture, when you measure the electron, you split into two worlds: how do you get 2/3 probability of "up" from two worlds?
You can say you split into three worlds, two of them identical. But that is not found in the Schrödinger equation (so now you have new postulates, which MWI hoped to avoid), and what if the probability were 1/pi, how many worlds would that be?
(E.g. Imagine all the universes created map to the range 0 to 1, and the first 2/3's come up red, and the last blue (or reverse that, or shuffle it up). The mapping onto 0 to 1 is merely for visualization purposes, and is thus arbitrary.)
"Many worlds" may arise naturally from the Schrödinger equation in this way: if you measure a system, the components of its wavefunction decohere, so that they no longer interfere. These components may be intuitively understood as "worlds."
We find ourselves in the world corresponding to the larger-measure component more often. But why? If your answer is "because more universes were created for larger measures," that is not found in the Schrödinger equation; it requires additional axioms, which is precisely what MWI hopes to avoid.
This is a misunderstanding. For simplicity’s sake, mwi people say that the universe “splits” into one child universe for each possibility- but that’s not what mwi actually says, just the pop science simplified layman version.
In actuality just as the quantum wave function is a continuous, analog wave there is a continuous analog sea of possible universes, and some have greater measure than others. And some are more correlated than others. The discorrelated ones appear to be separate from each other. But they’re all part of the same multiverse.
Think about it this way: we have no trouble conceiving of the universe having multiple possible futures, but only one past. That is not the case however. We also have multiple possible pasts, and in fact they’re not just possibilities but are real. Each future universe with this one as a past is real. Each past universe with this one as a future is equally real (that’s the mwi view of Schrödinger’s cat). Because of the arrow of entropy there will be more of the former than the latter.
Hope that helps..
What is the physical significance of "a universe with a greater measure than another?" What is the physical significance of the measure in MWI at all?
MWI view: There are many experimenters, in fact there is a continuum of them for each "run" of the experiment. The "setup" of each run that produces the mixed state in the bowl of marbles is splitting the whole universe (at the speed of light) and also splits the experimenters. Some of them are now in the "red" universe, and some of them are now in the "blue" universe and give matching answers. The percentage of them giving each answer is the same as in the situation above, but this percentage applies to even one run of the experiment.
However, the many experimenters and their percentages is not observable in a single experimental run from inside their universe. The experimenters cannot communicate, so after each run they only have 1 bit of information each (red or blue). They need to repeat the experiment to gain more than one bit of information.
However, now, their states are different. Some experimenters have a different history to the others. They aren't all the same any more! There's no "reset" button. This is the core of many of the "mysteries" and "apparent superluminal communication" in QM. There's no mystery. The state -- including the experimenters -- is splitting.
The classic "superluminal" communication isn't: Entangled opposite particles are simply saying that some universes have the (A,B) pair, others have the (B,A) pair. When you find out that you have the "A" particle, you instantly know that in your universe the other guy must have the "B" particle. There's no communication, instead the experiment is all about finding out one bit of information about "where you are" in the multiverse.
The objection is: what fraction are in red universes, and why? Why is it 2/3rds and not, say, 3/5ths or zero?
In QM we can compute these fractions using the Born rule, which says that the probability of each outcome is proportional to its measure in the wavefunction. This is natural to postulate in Copenhagen, where we interpret the wavefunction as a probability density. If the measure of spin-up is twice that of spin-down, it means that spin-up is twice as likely.
But in MWI, the wavefunction is definitely NOT a probability density. Both spin-up and spin-down are physically realized. So why should it matter that spin-up has a higher measure?
Where does the Born rule come from in MWI? Can it be derived from the Schrödinger equation? If so, how? If not, what additional axioms are needed?
This squaring in QM is basically saying that there are two probability distributions, not one. There isn't just the particle, there is the particle and the experimenter. Similarly, there's the particle and itself in interference experiments. Or the particle and the detector. There's always (at least) two required for an interaction. Hence a squaring.
It's saying that the observed and the observer are part of the same system. We're not Gods viewing squiggly cartoons on a page, like we like to draw.
We are the drawing.
My understanding is that there are two "red yous" for every "blue you"