I How Does the Many-Worlds Interpretation Handle Expectation Values?

[name123]

I may have misunderstood the expectation value, but if not then with the Copenhagen Interpretation it is easy to understand the expectation value for a wave function. It is just based on the probability of each event. If there were 4 possible events, and the probability of the event having a value of 1 was 0.1 and the probability of the event having a value of 2 was 0.2 and the probability of the event having a value of 3 was 0.3 the probability of the event having a value of 4 was 0.4 then (as I understand it) the expected value would be 0.1(1) + 0.2(2) + 0.3(3) + 0.4(4) = 3. The same as the expected value in probability theory for a ten sided dice which had one face with 1, two faces with 2, three faces with 3, and 4 faces with 4. So the theoretical basis would be clear (it is the same basis as with a dice).

But with the MWI the probability of each event would be 1, as each event happens. So what is the theoretical basis of applying the Born Rule in such a theory? For the expectation value for example. Because without one, it seems to me like an ad hoc application to explain why we expect to observe some results more frequently than others, and for the conservation of energy.

 

[Demystifier]

The origin of Born rule in MWI is indeed a subtle issue. Even among the MWI adherents there is no consensus how to explain the Born rule.

 

[name123]

The origin of Born rule in MWI is indeed a subtle issue. Even among the MWI adherents there is no consensus how to explain the Born rule.

Which sounds like it has been added ad hoc, and then people have tried to create a retrospective story to justify it.

 

[PeterDonis]

Which sounds like it has been added ad hoc

No QM interpretation has to "add" the Born Rule; it's part of the basic math of QM.

The issue for interpretations like the MWI is explaining why it works, since on the face of it it seems like it shouldn't if the MWI were true.

 

[name123]

When you state that it is part of the "basic math of QM" are you talking about the expectation value?

If so then perhaps look at the original post of this thread. The expectation value for the wave function seems to be based on the expected value in probability theory. Where each outcome is multiplied by the probability of that outcome.

The Born Rule giving the probability in the Copenhagen interpretation explains why the expectation value for the wave function involves the Born Rule for each outcome. If the probability had been imagined to be different, then that different probability would have been used. The same as with the expected value in probability theory.

Why with MWI wasn't the probability in that interpretation expected to give the expectation value? In other words why did they feel that the Copenhagen Interpretation proponents were wrong to think that the expectation value would involve the probability for each event? Why wasn't it expected to follow probability theory?

 

[PeterDonis]

When you state that it is part of the "basic math of QM" are you talking about the expectation value?

As I explicitly said in the post, I was talking about the Born Rule. But computing expectation values is also part of the basic math of QM.

However, interpreting expectation values as a weighted average of measurement results, with the weighting factor being the probability of observing the result, is not part of the basic math of QM. That is interpretation dependent. The Copenhagen interpretation, as you say, interprets expectation values as I have just described, but the MWI does not. It can't, because the weighting factors attached to each possible measurement result cannot be interpreted as probabilities in the usual way, since all of the results are realized, not just one.

The issue of how to interpret those weighting factors in the MWI is related to the issue I mentioned before, of how to explain why the Born Rule works in the MWI.