Question regarding the Many-Worlds interpretation

[mfb]

Their adherents believe they are able to derive the subjective impression of probabilities from the formalism.

MWI can do hypothesis testing. If someone wants to interpret that as probability is a matter of taste.

You said we need the Born rule for future experiments. Yet you don't think it somehow gives the 'probability of finding yourself in a certain branch'. What is the significance of the Born rule then?

I said we can care about the rule. If you are looking for a rule (for whatever reason), the Born rule is the only reasonable one.

Well, what's the empirical content of your theory if you can't use it to predict relative frequencies of experimental outcomes? And how could any observations ever lead to the acceptance of such a theory---after all, every sequence of outcomes would be equally consistent with it, and thus, can't be used to increase confidence in it?

See my previous post. Is it invisible?

 

[tom.stoer]

I don't think that really makes sense, since there are infinitely many branches, corresponding to infinitely many different eigenvalues for operators such as position and momentum.

Also, I don't know why the measure should necessarily be derived from the counting measure. That's one possibility, but there is nothing forcing that to be the case. It's an assumption.

I never said that the expression I have written down is the correct one. I never said that I know the measure to be used. Please note all the "should", "could", "something like that", ...

All what I am saying is that dropping collapse and Born as postulates requires provable theorems to replace them. All what I am saying is that MWI must be able to provide a measure that works i.e. that is acceptable physically (top-down and bottom-up). And all what I have are "dynamically disconnected branches" emerging from decoherence, so I tend to believe in them and try to figure out how branch counting, coarse graining or something like that provides a way out.

I am not sure that this will work, but I am absolutely convinced that if this (or something like this) does not work, then MWI is not a viable approach. The status seems to be far from complete.

 

[S.Daedalus]

Daedalus is saying exactly what I want to explain for days. He got it precisely.

To be honest, I thought this was the usual way of thinking about the problem of probability in the MWI---it's what I got from the standard discussions by Hemmo/Pitowsky, Kent, Albert etc. But I've run into what I think can only be a failure to communicate a couple of times in discussion of this; to me, the problem seems crystal clear, but it seems other people just have different intuitions on this.

 

[Jazzdude]

Now we have something like a measure

\int_{\mathcal{A}(x)} d\mu(\alpha) = k \,|a_x|$\int_{\mathcal{A}(y)} d\mu(\alpha) = k \,|a_y|$

This is what should replace Born's rule to be physically acceptable and to solve the issues I tried to discuss.

Is this something MWI + decoherence try to achieve?

This has been attempted, even quite early in the development of MWI. Unfortunately it's relatively straight forward to see that the linearity of QT and the linearity of the measure make the resulting measure independent of the amplitudes, so the Born rule cannot follow.
The only way to avoid this conclusion is to introduce some form of non-linearity. Everett himself proposed to ignore branches that asymptotically decay to 0, which is effectively introducing a non-linear cutoff. Similar cutoff approaches followed, arguing that one could take the limit of the cutoff parameter approaching zero and "recovering" the linear theory result this way. Of course, this argument is flawed.

So, I'm afraid the measure approach failed.

Cheers,

Jazz

 

[tom.stoer]

To be honest, I thought this was the usual way of thinking about the problem of probability in the MWI---it's what I got from the standard discussions by Hemmo/Pitowsky, Kent, Albert etc. But I've run into what I think can only be a failure to communicate a couple of times in discussion of this; to me, the problem seems crystal clear, but it seems other people just have different intuitions on this.

I agree. I was quite confused by some statements here ...

 

[tom.stoer]

This has been attempted, even quite early in the development of MWI. Unfortunately it's relatively straight forward to see that the linearity of QT and the linearity of the measure make the resulting measure independent of the amplitudes, so the Born rule cannot follow

But the term "branch" and the "number of rays belonging to one branch" has not yet been defined.

 

[Jazzdude]

But the term "branch" and the "number of rays belonging to one branch" has not yet been defined.

Yes, but you have very strict constraints on what a branch can be, if you want to respect the linearity of the theory, which implies the linearity of the measure.

Cheers,

Jazz

 

[stevendaryl]

Consider a hat, containing different numbers of blue and red balls, say 70% red, 30% blue. This is the natural measure of the balls: if you draw balls from the hat, you'd expect the relative frequency of balls to approach these numbers. These are my 'events'.

To make the analogy with repeatable quantum measurements closer, let's assume that after selecting a ball, you replace it in the hat, so there's a possibility of drawing 100 red balls in a row.

So I have two comments about this analogy: First, the conclusion that the relative frequency of red balls should approach 70% isn't provable. It doesn't logically follow from the mere fact that 70% of the balls are red. You have to make some kind of "equally likely" assumption, which means that you're making some assumptions about probability. Furthermore, as I said, the conclusion doesn't follow, even then. It's possible to draw 100 blue balls in a row. So what is being claimed about the relative frequency of red balls? It seems to me, that it amounts to putting a measure on the set of possible histories, and showing something along the lines of "the set of histories whose relative frequencies for red don't approach 70% has a measure zero.

I don't see how things are any worse for a Many Worlds interpretation. Well, there is the "preferred basis" problem, but I don't think that has much to do with the meaningfulness of probabilities.

With a nondeterministic physics, you have two choices: You can model things using stochastic processes in "single world" model, or you can model things using deterministic evolution for a "many worlds" or "ensemble" model. To me, these seem exactly equivalent.

 

[bhobba]

MWI can do hypothesis testing. If someone wants to interpret that as probability is a matter of taste.

Hmmmm. Good point. That actually helped with understanding Wallice's derivation in the Emergent Universe.

Its good to see some of the later posts are actually getting to what I consider the critical issue with the MWI. Its not that its incomplete, inconsistent or anything like that. One can add assumptions to fix all those issues if they are present. Its whether it accomplishes what its adherents claim it does - have a completely deterministic theory that includes the Born rule.

Thanks
Bill

 

[bhobba]

because of Gleason's theorem, which I don't quite remember the statement of

Check out:
http://kof.physto.se/theses/helena-master.pdf

Its very simple to state. The only possible probability measure that can be defined on a Hilbert space is the Born rule if one requires non-contextuality ie the measure does not depend on the basis the elements are expressed in. Since basis independence is what vector spaces in general are about its almost a trivial requirement. But physically its another matter - different basis means different measuring contexts so what you are saying physically is its not dependent on the measuring context. It immediately implies the Kochen-Specker theorem because the Born rule does not admit an assignment of 0 and 1 only to all elements which means non-contextuality is incompatible with definite values assigned to variables.

Mathematically since a Hilbert space formalism is what QM, including the MWI, is about, non-contextuality is more or less what you have by using such a formalism - if you don't have it the question is - why use such a formalism in the first place - and theories like Bohmian mechanics answer that by saying it is not its true basis - the pilot wave is. It can be applied to MWI to justify the Born rule, but the question then is why do we have probabilities in a deterministic theory. It doesn't invalidate it or anything like that - it simply seems to be at odds with the aim of the theory.

There are other approaches such as the one by Wallace that makes use of decision theory and base it on how to rationally decide which branch you are in, but there is plenty of debate if it really does solve the issue. Again it doesn't invalidate the theory - merely if it does what the adherents claim.

Thanks
Bill

 

[tom.stoer]

I am still not covinced that the Born rule is sufficient. It misses what I called "bottom-up" perspective.

The Born rule says that
- results of a measurement of an observable A will always be one of its eigenvalues a
- the probability for the measurement of a in an arbitrary state psi is given by a projection to the eigenstate

$p(a) = \langle\psi|P_a|\psi\rangle$

This is a probability formulated on the full Hilbert space.

I still do not see how this is answers the question:

What is the probability p(B_a) that I find me as an observer in a certain branch B_a where a is realized as measurement result?

One could reformulate the problem as follows: The Born rule says that the probability to find a is p(a). What I am asking for is the probability to find a, provided that I am in a certain branch B_a where a is realized (the expectation is 100%, so we need some kind of Bayesian argument to extract the probability p(B_a) the branch)

I would like to see a mathematical expression based on the MWI assumptions which answers this question.

The Born rule as stated above is formulated on the full Hilbert space and therefore provides a top-down perspective, but I as an observer within one branch do have a bottom-up perspective. I still don't see why these two probabilities are identical and how this can be proven to be a result of the formalism. There is some additional (hidden) assumption.

 

[bhobba]

The Born rule says that
- results of a measurement of an observable A will always be one of its eigenvalues a
- the probability for the measurement of a in an arbitrary state psi is given by a projection to the eigenstate

That's the 'kiddy' version. The real version is given an observable O a positive operator of unit trace P exists such that the expected value of O is Tr(PO). P is defined as the state of the system. States of the form |u><u| are called pure states and is what you more or less are talking about above. However it implies more that what you said above.

This is actually quite important in MWI because a collapse actually never occurs - instead it interprets the pure states of an improper mixed state Ʃpi |ui><ui| after decoherence as separate worlds. Interpreting the pi as probabilities of an observer 'experiencing' a particular world is what the Born rule is used for in the MWI and that requires its full version.

Thanks
Bill

 

[tom.stoer]

That's the 'kiddy' version.

Nobody is able to answer, but it's kiddy; strange conclusion.

The real version is given an observable O a positive operator of unit trace P exists such that the expected value of O is Tr(PO). P is defined as the state of the system. States of the form |u><u| are called pure states and is what you more or less are talking about above. However it implies more that what you said above.

This is actually quite important in MWI because a collapse actually never occurs - instead it interprets the pure states of an improper mixed state Ʃpi |ui><ui| after decoherence as separate worlds.

I know this, but that's not my question.

Interpreting the pi as probabilities of an observer 'experiencing' a particular world is what the Born rule is used for in the MWI and that requires its full version.

So it's an interpretation.

Sorry to say that, but that's not an answer to my question. I think you do not even get my question.

How can you justify that it is allowed to use the probabilities pi contained in the full state (which is the top-down perspective not accessable to a single observer) as a probability observed (bottom-up) by an observer within one branch?

Top-down you derive the probability for a result of a measurement in full Hilbert space.
Bottom-up you find (as a single observer) that within your branch the same probabilities apply.
Why? What's the link?

 

[tom.stoer]

I found a good discussion including many refences of the question I asked in the last days

http://arxiv.org/abs/0712.0149
The Quantum Measurement Problem: State of Play
David Wallace
(Submitted on 3 Dec 2007)

In 4.6 Wallace discusses what he calls the "Quantitative Problem"

The Quantitative Problem of probability in the Everett interpretation is often posed as a paradox: the number of branches has nothing to do with the weight (i. e. modulus-squared of the amplitude) of each branch, and the only reason- able choice of probability is that each branch is equiprobable, so the probabilities in the Everett interpretation can have nothing to do with the Born rule.

...

As such, the ‘count-the-branches’ method for assigning probabilities is ill- defined.But if this dispels the paradox of objective probability, still a puzzle remains: why use the Born rule rather than any other probability rule?

...

But it has been recognised for almost as long that this account of probability courts circularity: the claim that a branch has very small weight cannot be equated with the claim that it is improbable, unless we assume that which we are trying to prove, namely that weight=probability.

...

The second strategy might be called primitivism: simply postulate that weight=probability. This strategy is explicitly defended by Saunders; it is implicit in Vaidman’s “Behaviour Principle”; It is open to the criticism of being unmotivated and even incoherent.

...

The third, and most recent, strategy has no real classical analogue ... This third strategy aims to derive the principle that weight=probability from considering the constraints upon rational actions of agents living in an Everettian universe. It remains a subject of controversy whether or not these ‘proofs’ indeed prove what they set out to prove.

I hope this reference explains (from a different perspective and in a more reliable and sound manner) that there is a problem regarding probabilities and weights in the Many Worlds Interpretation.

 

[S.Daedalus]

So I have two comments about this analogy: First, the conclusion that the relative frequency of red balls should approach 70% isn't provable. It doesn't logically follow from the mere fact that 70% of the balls are red. You have to make some kind of "equally likely" assumption, which means that you're making some assumptions about probability.

Yes, that is what I mean when I say that the distribution provides a natural measure on the events, i.e. the drawing of balls from the hat. This is an assumption, yes, but it's a natural one to make; assuming something different would need some additional justification. And from this, you can derive probabilities from sequences of draws, thus showing that 100 reds is very unlikely.

However, if you start out with sequences, there's simply no analogy to this reasoning. Again, the reason is that the notion of event, i.e. the drawing of a ball, doesn't make sense in the MWI. There, the natural measure to consider would be considering every history to be equally likely---as in the case of the balls in the hat.

Basically, the problem you raise is a problem in the philosophy of probability as a whole; but provided there's a solution, it still seems to me that the MWI has some additional problem to answer.

 

[bhobba]

How can you justify that it is allowed to use the probabilities pi contained in the full state (which is the top-down perspective not accessable to a single observer) as a probability observed (bottom-up) by an observer within one branch?

I have zero idea what you mean by bottom up and top down. The pi's are the pi's - that's it, that's all. Now by the definition of state as having unit trace the sum of the pi's is one and are positive suggesting they be interpreted as probabilities - not proving anything - but suggesting.

There are a number of arguments associating the pi with probabilities - I know of two:

1. A proof based on decision theory:
http://arxiv.org/abs/0906.2718

2. Envarience
http://arxiv.org/pdf/1301.7696v1.pdf

They however have been criticized as being circular. I personally don't think the decision theory one is - but rather relies on what one thinks of decision theory and rational behavior as a basis for probabilities being introduced in a deterministic theory. I suspect the envarience one is circular.

But Gleason's is possible if you assume non contextuality.

To introduce probabilities you simply imagine a suitably large number of repetitions of the same observation which will give some expected value and hence associate the pi with probabilities. But again why is it you get probabilities from a deterministic theory? Or to put it another way since the wavefunction is split into a number of worlds why do the observers in those worlds not experience each world as equally likely but instead as a probability determined by the pi in the mixed state? That is rather weird in terms of a deterministic theory.

I have David Wallace's book and he argues its not an issue based on the option that was posted previously: 'The third, and most recent, strategy has no real classical analogue ... This third strategy aims to derive the principle that weight=probability from considering the constraints upon rational actions of agents living in an Everettian universe. It remains a subject of controversy whether or not these ‘proofs’ indeed prove what they set out to prove.'

I am not personally convinced.

That's the real issue IMHO - why do you get probabilities in a deterministic theory?

I think that's what you may be getting at - or am I off the mark?

Thanks
Bill

 

[tom.stoer]

Bill, I think the problem has been explained several times.

This isn't available in the MWI, however. The reason is that the notion of an event doesn't make any sense anymore: A doesn't occur in exclusion to B, but rather, both occur. This makes the natural entities to associate probabilities with not events, but branches, or perhaps better histories, i.e. chains of observations; the sequence of values observed in elementary spin experiments, say. But there's no grounds on which one can argue that the likelihood of 'drawing' a history from all possible histories should be such that it is more likely to draw a history in which the relative frequencies are distributed according to the Born rule. If one were to associate a measure with histories at all, it seems that the only natural measure would be a uniform one---which would of course entail that you shouldn't expect to observe outcomes distributed according to the Born rule.

The proponent of many worlds is then, in my eyes, faced with justifying the use of a non-uniform measure on the set of histories, about which Gleason's theorem doesn't really say anything, it seems to me. Now of course, one can always stipulate that 'things just work out that way', but in my eyes, this would significantly lessen the attractivity of MW-type approaches, making it ultimately as arbitrary as the collapse, at least.

Also, I'm not at all sure I see how Gleason's theorem is relevant to probability in the MWI. What it gives is a measure on the closed subspaces of Hilbert space; but what the MWI needs is to make sense of the notion of 'probability of finding yourself in a certain branch'. It's not obvious to me how the two are related. I mean, sloppily one might say that Gleason tells you the probability of a certain observable having a certain value, but there seems to me a gap here in concluding that this is necessarily the same probability as finding yourself in the branch in which it determinately has that value. I could easily imagine a case in which Gleason's theorem, as a piece of mathematics, were true, but probability of being in a certain branch follows simple branch-counting statistics, which won't in general agree with Born probabilities.

But in the MWI, you don't draw a ball to the exclusion of another; rather, you always draw both a red and a blue ball. The distribution of the balls in the hat has no bearing on this; it's just not relevant. What you get is all possible strings of the form 'bbrbrr...', i.e. all possible 'histories' of drawing blue or red balls. In only a fraction of those do you observe the statistics given by the distribution of the balls; furthermore, the distribution of the balls has nothing at all to say about the distribution of the strings. You then need an argument that for some reason, those in which the correct statistics hold are more likely than those in which they don't. That the original distribution is of no help here can also be seen by considering that there isn't just one measure that does the trick: you could for instance attach 100% probability to a history in which the frequencies are correct, or 50% to either of two, or even some percentage to incorrect distributions; the setting leaves that question wholly open. And so does the MWI.

The third, and most recent, strategy has no real classical analogue ... This third strategy aims to derive the principle that weight=probability from considering the constraints upon rational actions of agents living in an Everettian universe. It remains a subject of controversy whether or not these ‘proofs’ indeed prove what they set out to prove.

Top-down: you derive the probability for a result of a measurement in the full Hilbert space.
Bottom-up I can ask: What is the probability that I find myself as a single observer in a certain branch where a is realized?
Why? What's the link?

I do not question here whether probabilities in agreement to Born's rule can be derived. I question whether these probabilities for a result of a measurement on full Hilbert space have anything to do with the probability to find myself in a certain branch.

I think you got it here

Or to put it another way since the wavefunction is split into a number of worlds why do the observers in those worlds not experience each world as equally likely but instead as a probability determined by the pi in the mixed state? That is rather weird in terms of a deterministic theory.

 

[bhobba]

I think you got it here

Great :thumbs::thumbs::thumbs::thumbs::thumbs:

That's exactly my concern - how does a deterministic theory accommodate probabilities.

I am not persuaded by Wallace's arguments in the book I am reading. It doesn't invalidate it but it means it doesn't do what its adherents would like - a totally deterministic theory.

Thanks
Bill

 

[tom.stoer]

That's exactly my concern - how does a deterministic theory accommodate probabilities.

This may be an even deeper concern.

Mine is that we get a probability for a result of a measurement which is implicitly assumed to be valid for an observer in a specific branch. Of course a derivation of Born's rule is required in MWI, but it has a different meaning than in a collapse interpretation.

Anyway - my original idea was to start ein branch counting, but I had to accept that this is impossible.

 

[tom.stoer]

Bill, it seems that we have identified the same two problems as Wallace in the aforementioned paper.

The Incoherence Problem: In a deterministic theory where we can have perfect knowledge of the details of the branching process, how can it even make sense to assign probabilities to outcomes?

The Quantitative Problem: Even if it does make sense to assign probabilities to outcomes, why should they be the probabilities given by the Born rule?

 

[audioloop]

even deeper concern.

maybe not so deep, block universe, there is not probabilities per se.
likewise mwi, anything happen, no probabilities..

 

[bhobba]

Bill, it seems that we have identified the same two problems as Wallace in the aforementioned paper.

I agree.

Interestingly when I started my sojourn into the MWI my concern with it was this exponentially increasing branching just seems unbelievably extravagant. But after looking into it it has now shifted.

Again it doesn't disprove it, or show it's inconsistent - but it doesn't do what its adherents (at least some anyway) would like.

Thanks
Bill

 

[Maui]

maybe not so deep, block universe, there is not probabilities per se.
likewise mwi, anything happen, no probabilities.


.

Even in that scenario the probabilities still exist in the calculations. Why would that be?

 

[kith]

I am still not covinced that the Born rule is sufficient. It misses what I called "bottom-up" perspective.

I don't think it makes sense to talk about probabilities from the top down perspective. The only reason to introduce probabilities is that in experiments, we observe that we end up in a single branch with a probability according to the Born rule. Imagine a godlike top down observer who just sees the evolution of the universal state (he isn't allowed to interct with it because this would lead to entanglement and thus make him a bottom up observer). Why should he assign probabilities to the coefficients? He simply sees that there are now multiple observers which can't interact with each other.

The Born rule says that
- results of a measurement of an observable A will always be one of its eigenvalues a

I think this is what decoherence explains. However, Jazzdude seemed to object.

- the probability for the measurement of a in an arbitrary state psi is given by a projection to the eigenstate

$p(a) = \langle\psi|P_a|\psi\rangle$

This is a probability formulated on the full Hilbert space.

This isn't correct if P_a simply projects the system to an eigenstate und does nothing else. Your expression only gives the Born probability if the full state is a product state |ψ_system>ꕕ|ψ_rest> which corresponds to a universe with only one branch.

In the general case, p(a) is a sum over all branches, so you have a sum of Born probabilities. You have to project on a specific branch to get the correct expression. Which branch? The single branch a specific observer perceives. So this is really the bottom up view.

 

[tom.stoer]

I agree to nearly everything, except for

The only reason to introduce probabilities is that in experiments, we observe that we end up in a single branch with a probability according to the Born rule.

Of course you are right; this is the MWI interpretation. But as I said a couple of times, it's unclear why the expectation value of an observable evaluated on full Hilbet space and the probability to be within one single branch has anything to do with each other.

If there is a state like a|x> + b|y> it's an interpretation that being in branch "x" has anything to do with |a| squared; you can't prove it.

 

[kith]

I said we can care about the rule. If you are looking for a rule (for whatever reason), the Born rule is the only reasonable one.

Why? In Copenhagen, it is a postulate about probability distributions. What is it in the MWI?

 

[kith]

But as I said a couple of times, it's unclear why the expectation value of an observable evaluated on full Hilbet space and the probability to be within one single branch has anything to do with each other.

I agree. My objection was against using the term "probability" wrt to the top down perspective.

It is also unclear to me how the connection could be made.

 

[Quantumental]

Just flat out postulating the Born rule also has implications for the hypothesis. In Copenhagen it leads you to a ugly collapse which needs a physical explanation of sorts.

What the **** is mwi's ontological explanation supposed to be ? God cuts the branches he doesn't like?

 

[mfb]

I said we can care about the rule. If you are looking for a rule (for whatever reason), the Born rule is the only reasonable one.

Why? In Copenhagen, it is a postulate about probability distributions. What is it in the MWI?

Gleason's theorem. Every other assignment would lead to results we would not call "probability".

 

[tom.stoer]

Gleason's theorem. Every other assignment would lead to results we would not call "probability".

Gleason's theorem states that the only possible probability measure assigned to a subspace with projector P (in a system with density operator ρ) is tr(Pρ).

But the theorem cannot explain why tr(Pρ) shall be a probability for an observer to find himself in that subspace. This is an interpretation, and our discussion (including Wallace's paper) shows that it's controversial and not convincing to everybody. It is unclear why - in a deterministic theory - a probability shall arise at all.

 

[Quantumental]

probability in deterministic theories are just talk of ignorance, i don't see how this is controversial by itself? look at the bohmian interpretation. We don't know which outcome is going to occur as we cannot measure the pilot wave itself, but I don't see how this is controversial.

*if* branch counting had worked for MWI then there would be no problem With explaining why probability arises, it would simply be branch location ignorance

 

[mfb]

tom.stoer: I don't see how your post is related to the specific question I answered.

It is unclear why - in a deterministic theory - a probability shall arise at all.

To me, it is unclear why you are looking for (wanting?) probabilities, indeed.

 

[bhobba]

Just flat out postulating the Born rule also has implications for the hypothesis. In Copenhagen it leads you to a ugly collapse which needs a physical explanation of sorts. What the **** is mwi's ontological explanation supposed to be ? God cuts the branches he doesn't like?

It's not flat out postulated in any interpretation where the Hilbert space formalism is fundamental because of Gleason - unless of course you think in vector spaces non-contextuality is not reasonable - most would consider contextuality quite ugly.

Collapse is not ugly in any interpretation that considers the quantum state is simply knowledge about a system, like probabilities are, any more than throwing a dice collapses anything 'real' when it goes from a state where the state vector has all entries 1/6 to one with an entry of 1. Interpretations like that include Copenhagen and the Ensemble interpretation. In those interpretations its simply an example of a generalized probability model with nothing more mysterious going on than modelling something by probability.

What the issue is (with such interpretations) is people push against the idea that the world may be fundamentally probabilistic and want an underlying explanation for it. The problem lies in them - not the theory. Or to put it another way - the interpretation is fine - they just don't like it.

Thanks
Bill

 

[bhobba]

probability in deterministic theories are just talk of ignorance, i don't see how this is controversial by itself? look at the bohmian interpretation. We don't know which outcome is going to occur as we cannot measure the pilot wave itself, but I don't see how this is controversial. *if* branch counting had worked for MWI then there would be no problem With explaining why probability arises, it would simply be branch location ignorance

In BM probabilities enter into it due to lack of knowledge about initial conditions. In MWI we have full knowledge of what it considers fundamental and real - the quantum state.

Thanks
Bill

 

[bhobba]

To me, it is unclear why you are looking for (wanting?) probabilities, indeed.

I am not quite following your point here.

The reason probabilities come into it is Born's Rule ie given an observable O its expected value is Tr (OP) where P is the state of the system.

How can probabilities not be involved?

I agree there is debate over if the experience of an observer requiring probabilities is an issue in MWI, and Wallace discusses it in his book, but I don't think there is anyway of circumventing that probabilities are involved.

Thanks
Bill