Many Worlds and probabilities

In this paper: https://arxiv.org/pdf/1405.7907.pdf Sean Carroll and Charles Sebens argue that the Born rule for deriving probabilities from the wavefunction can be derived from the Many Worlds Interpretation of Quantum Mechanics.

To quote the paper:

Quote:

A longstanding puzzle in the Many-Worlds or Everett approach to quantum mechanics (EQM) [2, 3] is the origin of the Born Rule: the probability of finding a post-measurement system in an eigenstate |ai of an observable A, given that the system is prepared in state |ψi, is given by |ha|ψi|2. Here we summarize and discuss the resolution of this problem that we recently developed [1], in which the Born Rule is argued to be the uniquely rational way of dealing with the self-locating uncertainty that inevitably accompanies branching of the wave function. A similar approach has been advocated by Vaidman [4]; our formal manipulations closely parallel those of Zurek

I'm having a hard time understanding the reasoning, and was wondering if it makes any sense to anyone else. I find myself stuck on what they call a naive view, that the probabilities that any individual observer should expect, if the many worlds interpretation were true, should be evenly distributed between the number of worlds.

Specifically, if I measure the spin of an electron, under many worlds what happens is that I become entangled with the electron. I am in a super-position of states where the electron is spin up and and I measure it up, and the electron is spin down and I measure it down. There are two "worlds", one in which the electron is up and the other in which it is down. Because of decoherence these worlds can no longer interact with each other.

So far so good. Given these two worlds, both of which happen, it's not the case that there was, say, a 50% chance that the electron would be up and 50% chance that it would be down. Instead there's 100% chance that it will be both. Rather there are two me's. Okay.

Now, the next step in the argument is that there's a moment after the measurement, after the worlds split, in which I don't know which world I'm in. I can now say, "well, I don't know which world I'm in, but I can assign some probability to which one", and that gives us the probabilities associated with QM.

Here's where I get to the issue. The "naive" view is that since there are two worlds, both of which occur, and I don't know which one I'm in, I should assign a 50/50 chance to each. But, we can prepare the electron in any state we like. I can prepare it in an initial state with a detector aligned at any angle I like relative to the detector I'm using to measure it's spin now. So the probability distribution can be anything I like from 0 to 1 of getting spin up or spin down. And yet, if the probability is 10% up and 90% down, there are still only two worlds, one associated with up and the other with down. Where do those probabilities come from under many worlds?

To quote the paper again:

Quote:

The main idea we use is that of self-locating uncertainty [14]: the condition of an observer who knows that the environment they experience occurs multiple times in the universe, but doesn’t know which example they are actually experiencing. We argue that such a predicament inevitably occurs in EQM, during the “post-measurement/pre-observation” period between when the wave function branches due to decoherence (measurement) and when the observer registers the affect of the branching (observation). A naive analysis might indicate that, in such a situation, each branch should be given equal likelihood; here we demonstrate that a more careful treatment leads us inevitably to the Born Rule for probabilities.

Another quote making the same point:

Quote:

Naively, the combination of indifference over indistinguishable circumstances and selflocating uncertainty when wave functions branch is a disaster for EQM, rather than a way forward. Consider a case in which the amplitudes are unequal for two branches: Edit by Roboramma: the forum didn't parse this equation correctly, so I've ommited it, please click the link to see, the point of interest is that there are probabilities of 1/3 and 2/3 to the separate outcomes(10) The conditions of the two observers would seem to be indistinguishable from the inside; there is no way they can “feel” the influence of the amplitudes multiplying their branches of the wave function. Therefore, one might be tempted to conclude that Elga’s principle of indifference implies that probabilities in EQM should be calculated by branch-counting rather than by the Born Rule – every branch should be given equal weight, regardless of its amplitude. In this case, equation (10), that means assigning equal 50/50 probability to up and down even though the branch weights are unequal. This would be empirically disastrous, as real quantum measurements don’t work that way. We will now proceed to show why such reasoning is incorrect, and in fact a proper treatment of self-locating uncertainty leads directly to the empirically desirable conclusion. In Section 6 we generalize our result to cases where there is both classical and quantum self-locating uncertainty, as in the cosmological multiverse.

So, how do they derive the Born Rule rather than giving each branch equal weight?

I'm sad to say that part is unclear to me.

Here's a little diagram that seems to sum up the argument:

Quote:

Figure 1: A schematic representation of the setup behind our derivation of the Born Rule. The states |Ψ1i and |Ψ2i are on the left and right, respectively. Factors denote the observer, the spin, the coin, and the rest of the environment. Thin diagonal lines connecting the spin and coin represent entanglement within different branches of the wave function. The horizontal/vertical boxes made from dotted/dashed lines show two different ways of carving out the “Observer+System” subsystem from the “Environment.” The ESP implies that the probability of the system being in a particular state is independent of the state of the environment. Applying that rule to both the spin and coin systems implies the Born Rule as the uniquely rational way of assigning credences

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