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I'm not asking for any derivation. What is probability related to in the MWI? Is related to the fraction of observers that see various outcomes? Or something more objective?

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In quantum theory, physicists commonly use a rule called the Born rule to calculate the probabilities of measurement outcomes. In the MWI there is an explanation of why the Born rule is the appropriate rule for calculating probabilties. This explanation derives the Born rule by saying that an observer who wanted to make rational decisions by a particular standard would use the Born rule to calculate expectation values of measurements. The standard in question is that you should want to rank all of the possible sets of bets on an experiment so that another better could not use your rule to make you lose money consistently.

See this paper:

https://arxiv.org/abs/quant-ph/0303050

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    $\begingroup$ It has been noted by many that David Wallace’s arguments are circular $\endgroup$ – user164839 Sep 21 '18 at 13:20
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    $\begingroup$ @Nov If you have a criticism, then you should link it. $\endgroup$ – alanf Sep 21 '18 at 15:04
  • $\begingroup$ @PM2Ring It's a typo. $\endgroup$ – alanf Sep 23 '18 at 16:10
  • $\begingroup$ that is very, very interesting $\endgroup$ – lurscher Sep 23 '18 at 16:35
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    $\begingroup$ Here is a reference where it is explicitly argued that Wallace/Deutsch decision theory approach is circular. I haven't combed the article so I'm not sure how good it is.. arxiv.org/pdf/quant-ph/0604191.pdf $\endgroup$ – jgerber Sep 24 '18 at 7:48
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See disclaimer at bottom*

I would argue that there is no such thing as probability in the many worlds interpretation.

There is a nice explanation of Hugh Everett's original intention with many worlds in the video here. The basic idea on this view is that many worlds (or pure wave mechanics) takes the Unitary evolution of the Schrodinger equation as a law of physics but rejects the Born rule and or any sort of collapse. In Barrett's terms, if you don't put a probability rule into your theory as an assumption then there is no way you can get probability out of the theory. This would be what he calls (as a slightly technical term) "magic".

There have been attempts to extract probabilities from the many worlds interpretation but I think Barrett's point would be that all these techniques are somehow sneaking probability in at some point. For example, if you introduce some sort of principle of indifference to "worlds" you are saying something like "all worlds happen with equal probability". Well, at that point you've now added in a new postulate to your theory and on Everett's view you've now deviated from pure wave mechanics.

It seems to me that any theory or interpretation which seeks to "derive the Born rule" or extract probabilities from the many worlds interpretation MUST be sneaking in some probability axiom into the theory somewhere.

edit: To directly answer the questions in the OP:

What is probability related to in the MWI?

Probability isn't related to anything in MWI. There is no probability in MWI.**

Is related to the fraction of observers that see various outcomes?

No. It is not possible to rigorously define "the fraction of observers".

Or something more objective?

No. like I said, there is no probability in MWI so it is not related to anything.

*This is just what I've gathered from my own reading. I don't work in this field so I'm not an expert and I wouldn't claim that the view I'm expressing here is in any sense mainstream. It may be even be counter to mainstream views in the foundations community, I don't know.

**I gave a quick answer here but I can't leave it unqualified. The reason there is no probability in MWI is that there are no outcomes in MWI. MWI doesn't tell us how to relate the mathematical formulas we see to our experience. As such MWI cannot make any prediction about our experience. This means MWI is a pretty bad physical theory. Something must be added to it to give an account of our experience.

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  • $\begingroup$ +1: A principle of indifference is adding a new principle; so even on the basis of being economical in terms of foundational axioms MWI by forgoing the collapse axioms isn't as economical as a first reading might suggest. $\endgroup$ – Mozibur Ullah Sep 23 '18 at 16:42
  • $\begingroup$ @BruceGreetham That's my point. Physicists should be able to describe the probability of radioactive decays. However, in my post, I argue that many worlds is unable to predict these probabilities. This means that physicists should find MW problematic. You should give -1 to many worlds and not my post. One can help MW by introducing a principle of indifference which allows you to come up with probabilities, but as Mozibur Ullah points out, this sacrifices the economical expedience which was hoped to be gained by adopting many worlds in the first place. $\endgroup$ – jgerber Sep 23 '18 at 21:39
  • $\begingroup$ Sorry! I didn't mean to be too harsh in comment. I was just trying to point in (in perhaps an overly tongue in cheek way) that you and I are on the same page in that we both criticize a theory that can't explain radioactive decay/quantum phenomena. I was just trying to point out that I agree with your criticism and deflecting your criticism of my post onto MWI itself. I've edited my post to directly address the questions in the text of the OP. $\endgroup$ – jgerber Sep 24 '18 at 4:38
  • $\begingroup$ @jgerber I have undone my -1. You misunderstood me: I did not say whether I thought MWI was good or bad. I have deleted my comments as they were unclear. $\endgroup$ – Bruce Greetham Sep 24 '18 at 7:26