A In what sense does MWI fail to predict the Born Rule?

[DarMM]

If you don't mind akvadrako, I'm still mulling over your post regarding Vaidman's derivation, I want to read the paper closely again. So I'll just ask a simpler dumber question!

If a river splits into two branches, one twice as wide as the other, nobody questions that a random fish will more likely end up flowing down the wider branch. Even though it has the same branching structure as an equal divide.

How is this actually shown I guess is what I am asking. I don't see how the branch is twice as wide, unless it's because there are more copies of that branch.

EDIT: I should say I do of course see how it's "twice" as wide in the vector space sense, just I don't understand how more observers flow down it.

 

[bhobba]

They don't, but when they introduce the reduction of the wave function as an assumption they can incorporate the Born rule into that assumption. The difficulty for MWI is that MWI rejects any reduction postulate, so has to find the Born rule in unitary evolution.

Decoherent histories has the same issue. It resolves it simply - consistency replaces observation:
https://arxiv.org/pdf/gr-qc/9407040.pdf

You have mentioned many times by means of a coarse graining argument you can get the classical world from QM. Decoherent Histories builds this up from the concept of history which is simply a sequence of projections. In that interpretation QM is simply the stochastic theory of histories. The classical world, complete with the outcome of an observation emerges naturally without even introducing such concepts in the interpretation. In the view of Gell-Mann MW is simply decoherent histories where all the histories exist together in different worlds. He seemed to think the difference was more semantic than actual. The advantage of MW is just one wavefunction, the advantage of DH is its more commonsenseical.

Thanks
Bill

 

[DarMM]

This multiverse has the nice property that the probability of any sequence of coin flips is equal to the fraction of worlds where that coin flip sequence happens. Great.

Alright this is a bit clearer to me, is there a proof that the multiverse does in fact have this property?

That will be true, if we use the Born rule to weight possible worlds. But my point is that we developed QM within a single world, and what's important for us is that the Born rule works for repeated trials in our world. Why is it relevant to us what happens in other worlds?

So in essence the Born Rule is simply an accident, it happens to be the ratio we see. Why do we continuously see it hold across several experiments? I would imagine the answer is because it (approximately) holds in "most" worlds. This leads back to my first question above. Is there a proof that "most" worlds have a Born Rule obeying history?

 

[Stephen Tashi]

there are a number of interpretations of probability - decision theory is just one of them.

Interpreting Decision Theory might be a problem. In courses I have taken, Decision Theory involves making decisions given both a utility function and a probability model. Such an approach assumes probabilities are already defined.

The current Wikipedia says there are different types of decision theory. My understanding of the Decision Theory being applied to MWI is that it bases decisions on assigning "weights" to "branches" , without calling these "weights" probabilities. When it (supposedly) demonstrates that an observer in any (typical) world would always infer the same probabilities from an given type of experiment , it concludes that the common set of inferred probabilities is a function of the weights associated with the outcomes of the experiment -as given by the Born Rule.

This argument admits there are "maverick" worlds where experimenters infer the wrong probabilities. It considers these worlds unimportant. If it justifies their unimportance by saying they are "improbable" then it has become a circular argument which uses the concept of probability in order to define a concept of probability.

Actually, I prefer "history" in the sense of "recorded history". There is a macroscopic record of what has happened in previous experiments, and previous observations. Of course, we don't actually write down everything that happens and everything we see, and maybe we misremember, but I'm assuming that the only way we know what has happened in the past is because we have memories of it in the present, which is a fact about the present.

But in high-class theoretical arguments, isn't a "history" supposed be enough to define a unique "branch" up to the time the history is recorded? Papers about MWI used the terminology "memory sequence". People can have false memories. I don't think "memory sequences" are allowed to be false records.

Presumably, even if building a tunnel isn't something likely to be repeated, we can break it down into subevents that are repeatable: For example, metal striking stone. We can reason about the complex process in terms of the component events, right?

Yes, we can reason that way based on common experience, but is it explicit or implicit in any mathematical formulation of QM that such repeated component events exist?

 

[bhobba]

Interpreting Decision Theory might be a problem.

Some argue it is. But do you think we will reach a resolution and it will not devolve into philosophy? Specifically it is a type of Bayesian - that utility function can be objective or it can be subjective. In MW its a subjective weight a rational entity would assign to a world (history might be a better term) that rational entity would bet on experiencing. Of course just like the Baysian interpretation of probability that rational entity does not have to be there and is a minefield of all sorts of unresolved philosophical issues.

Thanks
Bill

 

[DarMM]

Some argue it is. But do you think we will reach a resolution and it will not devolve into philosophy? Specifically it is a type of Bayesian - that utility function can be objective or it can be subjective.

I don't know if we'll reach a resolution, but there are criticisms of the Decision Theory approach that aren't philosophical, such as those of Kent in "One World Versus Many: The Inadequacy of Everettian Accounts of Evolution, Probability, and Scientific Confirmation":

https://arxiv.org/abs/0905.0624

Now some of his objections are philosophical, but his criticisms of Wallace's axioms are mostly physical.

 

[Derek P]

Derek P said: ↑
Which is why I asked why.

Who says MW defines probability? Some make use of a certain version of it - decision theory - you can read about it - to derive the Born Rule.
[good stuff]
But its philosophical which we do not discuss here. That is the real issue with MW - it's philosophical basis is very arguable - but philosophy is not what we discuss here.

Quite so. I asked that people keep on topic in post 22. I'm still nursing a faint hope I'll get an answer to my question without needing "philosophy" other than a naive ontology.

 

[DarMM]

A world is superposition of a vast collection of microstates created through entanglement. They are decoherent and therefore add as the square root of the number of states.

The precise model is given by decoherence, quantum darwinism and other unitary dynamics. That's the same as all interpretations and it's complex, so I think it's right not to focus on it in regards to the Born rule.

Just another question, doesn't decoherence already require the Born rule, to permit tracing over the environment? Hence without the Born Rule, how do you show the state vector is of essentially Schmidt form to permit the clear branching structure without the Born Rule?

 

[akvadrako]

How is this actually shown I guess is what I am asking. I don't see how the branch is twice as wide, unless it's because there are more copies of that branch.

I can try but I don't think I can be more clear than the author. If you are interested, similar techniques are used in most of the other attempts so they might be enlightening.

To start, let's say nothing is assumed about the relation between two branches. So your example with $\sqrt{\frac{1}{3}} | \uparrow \rangle + \sqrt{\frac{2}{3}} | \downarrow \rangle$ isn't analysable yet. But after decomposing it into terms with with equal weights of $\sqrt{\frac{1}{3}}$, you can use the symmetry principle to assume branches of equal weight are equally likely and the probability can be calculated.

 

[bhobba]

Now some of his objections are philosophical, but his criticisms of Wallace's axioms are mostly physical.

I don't know about mostly but yes some are physical. All however are arguable. Wallace give an account in his book at least on some.

I am not advocating one view or the other.

I want to point out its relation to decoherent histories and both have similar unresolved and arguable issues. The peculiar issue to MW is in what way can the Born rule be given meaning and/or derived. This has been argued ad-infinitum with no resolution. It will not be resolved here. If people wish to pursue it I think its counterproductive to the forums aims. There are texts and papers on it. Specific questions can be asked and that's fine - beyond that it won't go anywhere IMHO.

The answer to the asked question is MW is a deterministic theory - there are issues in applying conventional notions of probability to such. Various views exist - and I have given papers and a text. Read the paper and texts then formulate answerable questions.

Thanks
Bill

 

[DarMM]

Okay fair enough, thanks for the discussion everybody, I think I need to read a bit more and return with more specific questions.

 

[bhobba]

Okay fair enough, thanks for the discussion everybody, I think I need to read a bit more and return with more specific questions.

Look into Gleason.

Thanks
Bill

 

[DarMM]

Look into Gleason.

Thanks
Bill

I know the proof of Gleason's theorem, but it has never genuinely helped me comprehend the MWI arguments as it comes from a very different direction, Wallace argues that his proof is a separate line of argumentation to Gleason. I think I need to read Wallace's book in full perhaps.

 

[A. Neumaier]

In the "typical" history, the relative frequency for heads is 0.5. In an atypical history, maybe the relative frequency for heads is 0.7.

But these are histories that can be many times generated and compared, so that one can tell what is typical. For ''worlds'', this is impossible - so calling a world typical or atypical if only one is known is kind of weird.

In the many-worlds interpretation, there will be many versions of my notebook.

There are all imaginable versions of your notebook, and calling some of them typical is prejudice based on the few notebooks in the single world you have access to.

 

[A. Neumaier]

There will be "worlds" where those expected frequencies occur, and worlds where they don't. The first type of world will be happy with quantum mechanics, and the other type will not be happy with it.

So you call the ones conforming to the laws observed in our world typical, because they are similar to our worlds, and worlds that do not, atypical. I believe the correct word for the notion you have in mind is not ''typical'' but ''like the one we actually observe''. Thus it is tautological that in these worlds we observe Born's rule.

I would imagine the answer is because it (approximately) holds in "most" worlds.

The problem is that ''most'' has no meaning unless you have a means to actually estimate the numbers. With access to only a single world, it is as if we had been shown only the result a single throw of a die (say, 2), without recourse to the mechanism of generating it (involving symmetry and labels 1,...,6), and have to deduce from that the probability law for casting dice.

 

[A. Neumaier]

in what sense does MWI fail to predict the Born Rule?

The real question is: In what sense does MWI predict Born's rule? I cannot see any coherent argument.

 

[stevendaryl]

So you call the ones conforming to the laws observed in our world typical, because they are similar to our worlds, and worlds that do not, atypical.

No, that's not what I was saying. A world is typical or not relative to a proposed law of physics depending on whether the relative frequencies in that world correspond to the probabilities derived from that theory.

 

[A. Neumaier]

No, that's not what I was saying. A world is typical or not relative to a proposed law of physics depending on whether the relative frequencies in that world correspond to the probabilities derived from that theory.

Yes, this is exactly what I was paraphrasing.

You know exactly one world and what is typical there. Based on this you propose a law of which you know that it holds in this particular world to some approximation. Then you postulate that among all other worlds, those are typical that behave like the single example you know of. It is no surprise that this way of proceeding predicts, no matter with which world you start, that this world satisfies the laws you started with.

If the world under consideration permitted only experiments where the result is always 0.707, this would be the observed law of physics. Now you might propose for arbitrary worlds the law of physics that the result is a constant $p\in [0,1]$. Then a world would be called typical if $p\approx 0.707$. To make you theory predictive in the MWI sense, you just need to postulate in addition that the worlds are distributed such that $p^2$ is uniformly distributed in [0,1] and that our world is randomly drawn from it. Now - abrakadabra - you can find an easy derivation of the law of physics in our particular world by doing elementary statistics:
The given world represents the mean value of all possible worlds. By improving the statistics of the proposed collection of worlds, i.e., by requiring that these worlds are even more like the given world, one can make the probability of finding the given world overwhelmingly large.

Of course ''possible'' and ''probable'' is only according to your prejudiced assumptions about what constitutes a possible world. Nothing at all depends on any of these worlds actually existing, or on the true existence or nonexistence of many other worlds that are mostly completely different from the given world.

Thus you assumed what you wanted to derive/explain, just disguised in a lot of mystery about alternative worlds.

 

[DarMM]

I suppose I have one specific question.

We can roughly divide C*-algebras into a few classes based on a two axes of distinction. The first being commutative and non-commutative. The second being the factor type:
$$I_{n},\quad I_{\infty},\quad II_{n},\quad II_{\infty},\quad III_{\lambda}$$

Discrete Classical Probability theory is commutative $I_{n}$, qubit systems are non-commutative $I_{n}$, normal non-relativistic QM is Type $I_{\infty}$. Type $II$ algebras, depending on if commutative or non-commutative, are classical or quantum statistical mechanics (in the thermodynamic limit).
Non-Commutative Type $III_{1}$ has been shown by Fredenhagen and Longo to be the algebra involved in QFT, making QFT in a sense the most general probability theory possible.

Given that Many-Worlds keeps the same formalism as Quantum Mechanics, I was wondering if in the literature there is a discussion of the reasons the theory is isomorphic to a probability theory (and in QFT's case, the unique most general one) given that it is deterministic? It seems odd to me that in a theory that is about the deterministic evolution of the universe, with probability only arising from local subjective viewpoints, the deterministic evolution would have the mathematical form of a probability theory. Is this discussed anywhere? (I've looked, but people here might have better knowledge)

 

[Stephen Tashi]

I think it is a very awkward attempt to reinstate the idea of probability as a real property of the system. I don't know why anyone would want to do this as MWI seems to predict observation frequencies perfectly well without defining instrinsic probability or whatever you want to call it. But does it? This of course is why I started this thread

Without assuming the concept of probability, how would you define being successful at predicting observational frequencies?

If we use the concept of probability, there are familiar ways to define what it means to be successful at predicting observational frequencies - namely that the prediction method predicts a frequency that has a high probability of being the actual probability. However, what definition can we make without the concept of probability?

There have been attempts to found probability theory on actual frequencies, such as the "collectives" of Richard von Mises. However, I'm not aware of any that meet the modern standards of mathematical rigor.

There is the fundamental problem that "frequency" is a concept (initially) defined for finite numbers of events. For infinite collections of events, frequency must be defined as a limit. To define that limit, some way of considering only a finite number of events from the infinite collection must be specified.

Taking pains to speak only of frequencies, how do we decide if MWI predicts the Born Rule? It has to be something like "On the the most frequent branches ( i.e. the most frequent "wolds") , the frequency of events observed in a repeated experiment is approximately the frequency given by the Born Rule." The delicate part of that argument is how to define what finite sets of branches are used in computing the frequency of branches.

There is a non-circular and non-trivial aspect to the above argument. It is not self-evident that there is a single set of weights than can be used in defining how we pick finite sets of branches to use in defining their frequency that also works to produce the frequencies of events observed in experiments within the frequent branches. If such a set of weights exists, how do we know it is unique? If the weights exist and are unique, we still have to show they correspond to the numbers (for probabilities) given by the Born Rule.

The above type of argument using only frequencies can be disparaged as "branch counting". However, without taking the notion of probability as fundamental, I don't see any alternative approach.

 

[A. Neumaier]

It seems odd to me that in a theory that is about the deterministic evolution of the universe, with probability only arising from local subjective viewpoints, the deterministic evolution would have the mathematical form of a probability theory.

Isn't this just the result of classifying $C^*$ algebra factors? Or do you ask specifically why MWI assumes $I_\infty$?

 

[stevendaryl]

Yes, this is exactly what I was paraphrasing..

Well, your "paraphrase" substituted a completely different meaning. It isn't what I meant. It isn't what I said. I didn't say that because that's not what I meant.

 

[A. Neumaier]

Well, your "paraphrase" substituted a completely different meaning. It isn't what I meant. It isn't what I said. I didn't say that because that's not what I meant.

But that's how your arguments sound to me. The words you are using convey nothing significant, since you simply postulate a lot of unseen worlds and a probabilistic law for them to justify with a weak argument only what you know already about the world you see.

 

[stevendaryl]

But that's how your arguments sound to me. The words you are using convey nothing significant, since you simply postulate a lot of unseen worlds and a probabilistic law for them to justify with a weak argument only what you know already about the world you see.

No, that wasn't what I was doing, at all. When you're confused about what someone means, ask questions. There is no point in responding to you if you just ignore what I say and pretend I said something else.

 

[stevendaryl]

Taking pains to speak only of frequencies, how do we decide if MWI predicts the Born Rule? It has to be something like "On the the most frequent branches ( i.e. the most frequent "wolds") , the frequency of events observed in a repeated experiment is approximately the frequency given by the Born Rule."

Probability in quantum mechanics is definitely not a matter of "counting branches". You can have just two branches, and that doesn't mean that they have equal weight.

If you prepare an electron in a state that is spin-up in the z-direction, and then measure its spin along an axis that is 5^o away from the z-axis, you'll get spin up with probability .998 and spin-down with probability .002. In a Many-Worlds type interpretation, there are now two branches, but they're not equally weighted.

 

[DarMM]

Isn't this just the result of classifying $C^*$ algebra factors? Or do you ask specifically why MWI assumes $I_\infty$?

I think the question I'm asking might be more on the philosophical side after some thought. It was basically that the mathematical structure of any quantum theory is that of a C*-algebra, i.e. a generalised probability theory. In MWI, probability is not a fundamental component of the theory and yet the theory has the mathematical structure of a probability theory. I was wondering if there was any discussion of this odd mathematical structure, given the non-probabilistic nature of the theory.

 

[Derek P]

The real question is: In what sense does MWI predict Born's rule? I cannot see any coherent argument.

Really? I would understand if you had said "valid argument" but to deny the existence of a coherent argument is patently ridiculous.

 

[Stephen Tashi]

Probability in quantum mechanics is definitely not a matter of "counting branches". You can have just two branches, and that doesn't mean that they have equal weight.

Yes, if you take the usual notion of "probability" as fundamental, but my remarks are directed at attempts that begin with the MWI sans probabilities and attempt to substitute actual frequencies for probabilities. If you take the MWI, you can have unequal weights, but these have no direct interpretation as probabilities - at least not without a lengthy proof. If you want to deal only with actual frequencies, you have to figure out a way to define your frequencies in such a manner that branches that have higher weights are "frequent". For example, you could assume there are more copies of them.

 

[Derek P]

Just another question, doesn't decoherence already require the Born rule, to permit tracing over the environment? Hence without the Born Rule, how do you show the state vector is of essentially Schmidt form to permit the clear branching structure without the Born Rule?

No. Decoherence does not require the Born Rule! It's a physical process that does not resolve a state as a proper mixture anywhere.
I may have failed to understand your question about the Schmidt form but I would assume that the answer lies in the fact that we define the subsystems - the original system, the detector, the environment, the observer - and each of these has its own state space. So the state space must be factorizable and the state is then a sum of product states.

 

[DarMM]

No. Decoherence does not require the Born Rule!

How is Tracing justified without the Born Rule?

 

[Derek P]

How is Tracing justified without the Born Rule?

No idea. Why would you want to?

 

[Derek P]

Yes, if you take the usual notion of "probability" as fundamental, but my remarks are directed at attempts that begin with the MWI sans probabilities and attempt to substitute actual frequencies for probabilities. If you take the MWI, you can have unequal weights, but these have no direct interpretation as probabilities - at least not without a lengthy proof. If you want to deal only with actual frequencies, you have to figure out a way to define your frequencies in such a manner that branches that have higher weights are "frequent". For example, you could assume there are more copies of them.

Well there are. What you are calling a branch is a branch of the original superposition - the state of the system under observation. It's not a world.

 

[stevendaryl]

No idea. Why would you want to?

Well, it's the most general form of a quantum prediction. If you have a state that is described by a certain density operator $\rho$, and you have an operator $A$, then the expected value of $A$# is $tr(\rho A)$.

 

[DarMM]

No idea. Why would you want to?

Tracing is required to obtain decoherence and tracing requires the Born Rule. Hence decoherence does require the Born rule.

I'm reading Wallace's book right now, so I'll see what he says.

 

[Derek P]

Tracing is required to obtain decoherence and tracing requires the Born Rule. Hence decoherence does require the Born rule.
I'm reading Wallace's book right now, so I'll see what he says.

Well, it's the most general form of a quantum prediction. If you have a state that is described by a certain density operator $\rho$, and you have an operator $A$, then the expected value of $A$# is $tr(\rho A)$.

You're both missing something. The Born Rule and tracing are not assumed in MWI, they are derived from the unitary model. Indeed it is hard to imagine how a treatment that concludes with quantum predictions could fail to reproduce the Projection Postulate - except that it would no longer be a postulate but a theorem. But let's not be too ambitious - this thread is about the Born Rule, not the entirety of MWI's mathematical underpinnings. It can be tackled by explicitly expanding the global state as a sum of many ket products. It is a result, not an assumption. And of course once the Born Rule is established, you can hit the density matrix with it if that's what floats your boat.

 

[stevendaryl]

You're both missing something. The Born Rule and tracing are not assumed in MWI, they are derived from the unitary model. Indeed it is hard to imagine how a treatment that concludes with quantum predictions could fail to reproduce the Projection Postulate - except that it would no longer be a postulate but a theorem.

He wasn't saying that tracing was assumed. He was asking how it was justified. A theorem is a good justification, if there is a theorem.

 

[A. Neumaier]

tracing requires the Born Rule

Actually, tracing and decoherence do not involve measurement, hence tracing is strictly speaking an additional assumption independent of the Born rule in its conventional formulation. The latter is a statement about measurement results, but things not yet measured have no results, hence the conventional form of Born's rule is inapplicable.

 

[Derek P]

He wasn't saying that tracing was assumed. He was asking how it was justified. A theorem is a good justification, if there is a theorem.

Okay. So let's pretend there isn't a theorem and tracing is not justified. You can (I believe) derive the Born rule by expanding the state as a sum of components. Of course you will actually be taking a trace but you won't be relying on a theorem to give you permission.

 

[Derek P]

Look into Gleason.

Thanks
Bill

You need to change your signature to "Every serious student of QM needs a copy of Ballentine and to Look into Gleason" !

 

[Derek P]

However, if it's not branch-counting, what is it?

No, it's state counting.

 

[bhobba]

I know the proof of Gleason's theorem, but it has never genuinely helped me comprehend the MWI arguments as it comes from a very different direction, Wallace argues that his proof is a separate line of argumentation to Gleason. I think I need to read Wallace's book in full perhaps.

Look into the non-contextuality theorem in the appendix of Wallace. It may have an error - but I couldn't find it. That means Gleason applies.

IMHO the real issue is, yes there are physical reasons one can ague about regarding the proof but the real bug bear is - how to introduce probabilities into a deterministic theory. I won't say what I think, you probably have guessed it, arguing positions is not something I enjoy that much. A little bit is OK. I think its much better to understand the pro's and con's of different views - to that end form and elucidate your view but arguing just seems to go on and on not really getting anywhere. Just my view of course - mentors as a group will ensure it all remains under control.

Interesting to hear what you think after reading Wallace. I actuay found it very illuminating of decoherent histories as much as MW - interesting. I like its theorem/proof approach due to my math background but I know its not every-ones cup of tea.

Thanks
Bill

 

[stevendaryl]

Okay. So let's pretend there isn't a theorem and tracing is not justified.

Then it doesn't agree with experiment, because the tracing prediction is what we observe.

 

[bhobba]

Tracing is required to obtain decoherence and tracing requires the Born Rule. Hence decoherence does require the Born rule. I'm reading Wallace's book right now, so I'll see what he says.

Tracing does require the Born rule - and other things - but the mixed state after decoherence may or may not be what we call an observation. One out, one I actually like, is simply define it that way - but it just semantics.

Decoherent histories replaces measurement with consistency - but the details require to study a text on it.

I personally like Schlosshauer's book - it looks at a lot of interpretations and decorehrence implications:
https://www.amazon.com/dp/3540357734/?tag=pfamazon01-20

Thanks
Bill

 

[stevendaryl]

No, it's state counting.

What do you mean by "state counting"? In the MWI, there is only one state for the entire universe, so counting states always gives you the answer 1.

Do you mean counting the number of terms in the superposition? If so, then how is that different from branch counting?

 

[bhobba]

Okay. So let's pretend there isn't a theorem

But there is a theorem - you can't waive away Gleason. The only out with Gleason is non-contextuality - non-contextual interpretations usually have there own way of handling the measurement issue.

The measurement issue has problems - but we know things about it the early pioneers did not - there is no getting around it. Progress will not be made by - let's suppose something we know is true is not.

As I have said please please try to understand interpretations pro's and cons. I think your question has been answered. But if you or anyone wants to keep it going - go ahead. It will be shut down if it becomes simply argumentative.

Thanks
Bill

 

[DarMM]

Okay, just to say, I'm going to stay away from this thread until I finish Wallace and Schlosshauer's books. I'm reading Wallace's book using Mandolesi's papers as a guide, both the 2015 and 2018 ones.

 

[Derek P]

What do you mean by "state counting"? In the MWI, there is only one state for the entire universe, so counting states always gives you the answer 1.

Do you mean counting the number of terms in the superposition? If so, then how is that different from branch counting?

There are googols of terms in "the superposition" but only a small number of worlds. It's up to you which one you call "branches" but they are certainly very different.

 

[bhobba]

Okay, just to say, I'm going to stay away from this thread until I finish Wallace and Schlosshauer's books. I'm reading Wallace's book using Mandolesi's papers as a guide, both the 2015 and 2018 ones.

Its a good approach, but you chose a tough one. I don't know if I could do it. I would post my latest thinking as I learn more. You have my very humble admiration.

Thanks
Bill

 

[A. Neumaier]

You can (I believe) derive the Born rule by expanding the state as a sum of components.

This only gives a decomposition but assigns no meaning or frequencies/probabilities to the terms. You have to postulate that each term has a given frequency/probability. This is a postulate far worse than Born's rule since Born's rule has at least an empirical support, but assigning probabilities to worlds of which only one is observed is completely arbitrary.

 

[A. Neumaier]

Okay, just to say, I'm going to stay away from this thread until I finish Wallace and Schlosshauer's books. I'm reading Wallace's book using Mandolesi's papers as a guide, both the 2015 and 2018 ones.

Schlosshauer's book is probably the best modern discussion of the measurement problem.