Question regarding the Many-Worlds interpretation

[tom.stoer]

stevendaryl, The_Duck,

In the meantime I am totally confused whether it makes sense to talk about a well-defined may-worlds-interpretation at all, or whether there is only a collection of guesses.

I copied the following text from Wikipedia, but I could use other refences (Zurek, Zeh, Wallace, ...) as well. This is what I understood and this is what I am talking about here. And this is what makes sense to me and what has the potential to turn an interpretation into a well-defined theory:

A consequence of removing wavefunction collapse from the quantum formalism is that the Born rule requires derivation, since many-worlds claims to derive its interpretation from the formalism. Attempts have been made, by many-world advocates and others, over the years to derive the Born rule, rather than just conventionally assume it, so as to reproduce all the required statistical behaviour associated with quantum mechanics. There is no consensus on whether this has been successful.[24][25][26]

Everett (1957) briefly derived the Born rule by showing that the Born rule was the only possible rule, and that its derivation was as justified as the procedure for defining probability in classical mechanics. ... Andrew Gleason (1957) and James Hartle (1965) independently reproduced Everett's work, known as Gleason's theorem[27][28] which was later extended.[29][30]

Bryce De Witt and his doctoral student R. Neill Graham later provided alternative (and longer) derivations to Everett's derivation of the Born rule. They demonstrated that the norm of the worlds where the usual statistical rules of quantum theory broke down vanished, in the limit where the number of measurements went to infinity.

MWI removes the observer-dependent role in the quantum measurement process by replacing wavefunction collapse with quantum decoherence. Since the role of the observer lies at the heart of most if not all "quantum paradoxes," this automatically resolves a number of problems ... MWI, being a decoherent formulation, is axiomatically more streamlined than the Copenhagen and other collapse interpretations; and thus favoured under certain interpretations of Occam's razor.

So what we are talking about here is a physical branching on the level of the state vector. It becomes a superposition of "nearly uncoupled" or "dynamically disconnected" superselection sectors (= branches) which are stable w.r.t. time evolution.

This means that branching, number of branches and especially the measure, factorization in orthogonal subspaces and their stability etc. must follow from the theory, i.e. Hilbert space + Schrödinger equation + decoherence (or some other physical process). It means that postulating Born's rule again doesn't help since a) then we exactly replace the unphysical collapse by unphysical branching (which is no progress but choosing between the devil and the deep blue sea) and since b) it does not resolve the problem of the bottom-up perspective (which I tried to explain a couple of times). And I would say that this is mainstream; many agree that Born's rule has to follow as a result, and many have worked on a derivation.

 

[bhobba]

It means that postulating Born's rule again doesn't help since a) then we exactly replace the unphysical collapse by unphysical branching (which is no progress but choosing between the devil and the deep blue sea) and since b) it does not resolve the problem of the bottom-up perspective (which I tried to explain a couple of times). And I would say that this is mainstream; many agree that Born's rule has to follow as a result, and many have worked on a derivation.

It still follows from Gleason's and non contextuality is very reasonable in the MWI.

The issue though is not that the Born Rule is not derivable within the MWI, it's why you get probabilities at all from a deterministic theory. That's what the other proofs like Wallace's are trying to do. They have a rational definition of probability based on decision theory and derive it via that method. Whether it accomplishes what they want is open to debate. The theorem is valid but exactly what's its saying, or even if it not circular (here meaning decision theory itself contains an implicit appeal to a notion of probability making the whole thing circular like probabilities based entirely on the frequentest interpretation is) is arguable.

These are deep questions and all I can suggest is get a hold of a good book on it and go through it for yourself eg the one I am studying right now:
https://www.amazon.com/dp/0199546967/?tag=pfamazon01-20

Thanks
Bill

 

[tom.stoer]

It still follows from Gleason's and non contextuality is very reasonable in the MWI.

The issue though is not that the Born Rule is not derivable within the MWI, ...

That's one step, but by no means sufficient.

What we need in addition is

... a physical branching on the level of the state vector, a superposition of "dynamically disconnected" superselection sectors (= branches) which are stable w.r.t. time evolution.

This means that branching, number of branches and especially the measure, factorization in orthogonal subspaces and their stability etc. must follow from the theory, i.e. Hilbert space + Schrödinger equation + decoherence (or some other physical process).

It seems that the idea behind MWI is compelling philosophically, but by no means complete mathematically.

 

[bhobba]

That's one step, but by no means sufficient.

Can you elaborate. Gleason's plus non-contextuality implies Born.

If you mean it also requires the assumption the outcomes are described by a probability measure then yes - that is an assumption at odds with the foundations of MWI.

Thanks
Bill

 

[tom.stoer]

Here's a "toy" universe that has some of the properties of MWI:

The universe is deterministic, except for a mysterious, one-of-a-kind perfect coin. When you flip it, it's completely impossible to predict ahead of time whether it will end up "heads" or "tails".

Whenever someone flips the coin, God ... stops everything for a moment, and makes two copies of the world that are identical in every respect, except that ...

So you can't deduce from the many-worlds theory (I'm talking about the theory of the many worlds created by God or the programmer, not Everett's Many Worlds) what the relative frequency must be, because it's different in different possible worlds. You can assume that you live in a world with a particular relative frequency, but that's an additional assumption; it doesn't follow from the theory.

This toy model is irrelevant for MWI as I understand it.

I am talking about a theory with a Hilbert space, a Hamiltonian H and a time evolution operator U = exp(-iHt), and a singe state (ray) to start with.

Nobody copies states or Hilbert spaces.
Everything follows from H w/o additional assumption.
The problem "top-down" vs. "bottom-up" does not arise for 50% probability.

 

[tom.stoer]

Can you elaborate. Gleason's plus non-contextuality implies Born.

I think I did this already a couple of times.

What we need in addition is

... a physical branching on the level of the state vector, a superposition of "dynamically disconnected" superselection sectors (= branches) which are stable w.r.t. time evolution.

This means that branching, number of branches and especially the measure, factorization in orthogonal subspaces and their stability etc. must follow from the theory, i.e. Hilbert space + Schrödinger equation + decoherence (or some other physical process).

It seems that the idea behind MWI is compelling philosophically, but by no means complete mathematically.

 

[bhobba]

What we need in addition is

QM follows from the two axioms in Ballentine. The first axiom is simply the existence of observables. The second is the Born Rule. What MWI assumes contains observables. It simply needs the Born Rule to imply all of QM.

The logic is as follows. QM proceeds exactly as normal with decoherence occurring at an observation. The elements of mixed states from decoherence are interpreted as separate worlds. The issue is why are they experienced with a probability related to their weight in the mixed state. The Born rule implies that.

Thanks
Bill

 

[kith]

Honestly, I am a bit confused by a couple of recent posts here. mfb and stevendaryl seem to claim that the Born rule is not needed at all in MWI. I get that we can deduce the Born rule from our actual history, just like people who observed results not in accordance with the Born rule could deduce a different rule from their history. So a priori, the Born rule would be nothing special. Is this the basis of your argument?

 

[tom.stoer]

No we are running round in circles.

We do not introduce Born's rule as an axiom, even if Ballantine does, simply b/c we do not discuss Ballantine.

Even if we are able to explain how Born's rule can be derived, it's by no means clear why the bottom-up perspective within one branch should care about Born's rule (which applies to the top-down perspective).

 

[S.Daedalus]

Can you elaborate. Gleason's plus non-contextuality implies Born.

How does that work? As things look to me, non-contextuality doesn't hold in QM, and besides, the contextuality of QM can essentially be derived from Gleason's theorem; that's how Bell did it originally.

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.

 

[tom.stoer]

Honestly, I am a bit confused by the last few posts.

I am confused as well b/c I still think that there is not one well-defined MWI but only a collection of related ideas.

 

[mfb]

Even if we are able to explain how Born's rule can be derived, it's by no means clear why the bottom-up perspective within one branch should care about Born's rule (which applies to the top-down perspective).

For future experiments.

I guess my previous post was too long :(.

 

[tom.stoer]

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.

THANKS A LOT!

This is what I try to explain here!

 

[bhobba]

Honestly, I am a bit confused by the last few posts. mfb and stevendaryl seem to claim that the Born rule is not needed at all in MWI.

It is needed.

What the hope of MWI adherents is it can be deduced from the Hilbert space formalism alone. There are a number of proofs about that purport to do that.

Its a matter of opinion and debate if they do. I believe on their own terms they do just that - but the key caveat is - ON THEIR OWN TERMS. One issue for example, as I mentioned, is if the decision theory proof they use subtly contains what they are trying to prove in its assumptions. Other proofs based on the idea of envariance exist as well that have been criticised for circularity as well. The debate rages and the issues are complex and suptle.

There are also issues associated with decoherence such as the so called factoring problem, but they need further investigation to be resolved one way or another.

Thanks
Bill

 

[mfb]

but what the MWI needs is to make sense of the notion of 'probability of finding yourself in a certain branch'.

A simple question: Why?
What is wrong if we do not have this?
What does "probability" even mean in a deterministic theory?

 

[bhobba]

How does that work? As things look to me, non-contextuality doesn't hold in QM,

Come again. Its built into the basic axioms of QM.

A state can be expanded in any basis and it makes no difference to probabilities.

Thanks
Bill

 

[bhobba]

A simple question: Why? What is wrong if we do not have this?
What does "probability" even mean in a deterministic theory?

Exactly. That is the key issue with the MWI. Their adherents believe they are able to derive the subjective impression of probabilities from the formalism.

That's where the debate lies.

Thanks
Bill

 

[kith]

A simple question: Why?
What is wrong if we do not have this?
What does "probability" even mean in a deterministic theory?

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?

 

[S.Daedalus]

A simple question: Why?
What is wrong if we do not have this?

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?

Come again. Its built into the basic axioms of QM.

A state can be expanded in any basis and it makes no difference to probabilities.

Thanks
Bill

Ah, I think we're using separate concepts of contextuality. I use it in terms of the Kochen-Specker theorem, i.e. as meaning the possibility of assigning definite values to observables, regardless of the measurement context, i.e. of other, compatible observables measured simultaneously.

 

[bhobba]

What is the significance of the Born rule then?

That depends on the version you adhere to. In the decision theoretic view that Wallace espouses in the reference I gave its what's required for a rational agent to make sense of it. There is also an evarience proof and probably others I am unaware of.

Thanks
Bill

 

[stevendaryl]

This means that branching, number of branches and especially the measure, factorization in orthogonal subspaces and their stability etc. must follow from the theory, i.e. Hilbert space + Schrödinger equation + decoherence (or some other physical process). It means that postulating Born's rule again doesn't help since a) then we exactly replace the unphysical collapse by unphysical branching (which is no progress but choosing between the devil and the deep blue sea) and since b) it does not resolve the problem of the bottom-up perspective (which I tried to explain a couple of times). And I would say that this is mainstream; many agree that Born's rule has to follow as a result, and many have worked on a derivation.

I don't see how the collapse hypothesis helps anything, if you think there is a conceptual problem with the Born rule. They are distinct assumptions in orthodox quantum mechanics:

1. When you perform measurement corresponding to operator O, you will always get an outcome that is some eigenvalue o of O, with probabilities given by the square of the projection of the wavefunction onto the normalized eigenstate of O corresponding to eigenvalue o.

2. AFTER the measurement, the wave function will be in that eigenstate.

The second step is the collapse hypothesis, and it's not needed for the first step. And without the second step, you basically have MWI.

 

[tom.stoer]

I think I should try to explain again what the problem is.

Suppose we have a state of a single photon and x- and y-polarizatations

$|\gamma\rangle = a_x|x\rangle + a_y|y\rangle$

Suppose that this photon is part of larger system including a measurement device, a pointer, me as an observer and environment d.o.f. Suppose that (due to decoherence or something similar) the system evolves unitarily into a state

$|\psi\rangle = a_x |\psi_x\rangle + a_y |\psi_y\rangle$

The two states on the r.h.s. describe the full system including the photon, the measurement device, the pointer, the sheet of paper with the result string s="x" or s="y" etc.

1) Suppose I have some operator S_x which can be used to calculate the probability to find the whole system in a state with result string s="x". This can e.g. be a projector to the first term in the equation above, i.e. to the branch $|\psi_x\rangle$

So I find the probability

$p(x) = \langle \psi|S_x|\psi\rangle = |a_x|^2$

This is what I call the top-down perspective b/c it is used not from the perspective of the observer within that branch but from a god-like perspective observing the full Hilbert space. Everything is fine, if decoherence produces such a split and guarantuees its stability w.r.t. time evolution than many issues are solved. Especially Born's rule for probability to find the system in state x has been derived.

2) Now suppose I am an observer finding myself in front of a paper with s="x". This is what I call the bottom-up perspective b/c I do not know anything about the other branch. Perhaps I even do not know the MWI.

Now my statement is the following: if somebody explains to me that the world has just split in two branches, one with s="x" and one with s="y", and if he explains to me how to calculate the probability to find a specific result s (from the top-down perspective!) this does by no means explain that I will find myself in the "x"-branch with probability

$p(x) = \langle \psi|S_x|\psi\rangle = |a_x|^2$

There are two branches, two observers, two results. My claim is that the following statements cannot be shown to be contradictory:

A) I will find myself in a specific branch with pobability 50%
B) The probability to find x from a top-down perspective is 90%

 

[stevendaryl]

This toy model is irrelevant for MWI as I understand it.

The point is to illustrate that measures on a set of possible worlds doesn't need to be, and I don't think CAN be, an object fact about that set of possible worlds. Measure is something used by a resident of one of the possible worlds in order to do probabilistic reasoning within HIS world.

Nobody copies states or Hilbert spaces.

That's why I called it a "toy model". It's not an actual model of quantum mechanics. It's just illustrating an issue with nondeterminism. You seem to think that if the theory is deterministic, then that means that there is no role for an axiomatic probability measure, and I'm saying that's wrong. For someone living in one "branch" of a branching history, he MUST assume an axiomatic probability measure to apply probability in his branch. The probabilities for HIS branch aren't derivable from the structure of the branching model, because different branches have different subjective probabilities.

 

[bhobba]

Ah, I think we're using separate concepts of contextuality. I use it in terms of the Kochen-Specker theorem, i.e. as meaning the possibility of assigning definite values to observables, regardless of the measurement context, i.e. of other, compatible observables measured simultaneously.

Well actually Kochen-Specker is a weaker form of Gleason (in the sense its a simple corollary) that is considered easier to prove, although I personally think modern proofs are not that hard.

What Kochen-Specker says is definite values of variables and non-contextuality are not possible. Bohmian mechanics gets around it by being contextual. What Gleason shows is if every variable is non-contextual you get the Born Rule.

Thanks
Bill

 

[stevendaryl]

2) Now suppose I am an observer finding myself in front of a paper with s="x". This is what I call the bottom-up perspective b/c I do not know anything about the other branch. Perhaps I even do not know the MWI.

Now my statement is the following: if somebody explains to me that the world has just split in two branches, one with s="x" and one with s="y", and if he explains to me how to calculate the probability to find a specific result s (from the top-down perspective!) this does by no means explain that I will find myself in the "x"-branch with probability

$p(x) = \langle \psi|S_x|\psi\rangle = |a_x|^2$

There are two branches, two observers, two results. My claim is that the following statements cannot be shown to be contradictory:

A) I will find myself in a specific branch with pobability 50%
B) The probability to find x from a top-down perspective is 90%

What is the meaning of that first 50% probability in A)?

It seems to me that reconciling what you call the "top-down" perspective and the "bottom-up" perspective is not just a problem for MWI, it's a problem for ANY notion of nondeterministic evolution. If you flip a coin, you either get "heads" or you get "tails". What sense does it make, from your bottom-up perspective, to say that you get heads with "50%" probability, or 90% probability (with a biased coin)? It doesn't have any objective meaning in the "bottom-up" perspective.

You can say that the meaning of 90% probability is that if you flip a coin 100 times, you'll get about 90 results of heads, but that's not literally true. There is a possibility of getting all heads. There's a possibility of getting all tails, there's a possibility of getting anything in between. You can work out probabilities, and say that, if the probability of heads for a single flip is 90%, then if you flip a coin 100 times, you'll get between 85 and 95 heads with a probability of 99% (or whatever it is). But that's circular--you're giving the meaning of probability in terms of probability.

The only two ways out, it seems to me, are (1) take the "top-down" approach, where you start with a measure for the set of possible histories, and use that to derive a probability for events, or (2) you use SUBJECTIVE probabilities, and give up on there being a unique, "right" probability to use, in the bottom-up case.

I agree that there are philosophical difficulties with probability, but I don't see how MWI makes things any worse.

 

[tom.stoer]

I don't see how the collapse hypothesis helps anything, if you think there is a conceptual problem with the Born rule.

I never said it does. All what I am saying is that
A) dropping Born's rule as an axiom requires its derivation as a theorem
B) derivation of Born's rule along is not sufficient to make MWI a viable theory

The only two ways out, it seems to me, are (1) take the "top-down" approach, where you start with a measure for the set of possible histories, and use that to derive a probability for events, or (2) you use SUBJECTIVE probabilities, and give up on there being a unique, "right" probability to use, in the bottom-up case.

I agree that there are philosophical difficulties with probability, but I don't see how MWI makes things any worse.

MWI does not make things worse, but it requires more precise theorems than a collapse interpretation.

In the collapse interpretation there are postulates like "the collapse to a specific branch happens with probability p which is then equal to the probability calculated by Born's rule for a specific measurement". The problem is shifted to the philosophical level and solved by a postulate. In the MWI the idea is to solve it w/o additional postulates, so it does not become worse, but harder b/c proofs are required instead of postulates.

 

[S.Daedalus]

Well actually Kochen-Specker is a weaker form of Gleason (in the sense its a simple corollary) that is considered easier to prove, although I personally think modern proofs are not that hard.

What Kochen-Specker says is definite values of variables and non-contextuality are not possible. Bohmian mechanics gets around it by being contextual.

Yes to all of that.

What Gleason shows is if every variable is non-contextual you get the Born Rule.

This I don't think I get, especially since the Born rule holds in Bohmian mechanics (or can be made to hold assuming suitable initial conditions), which as you say is contextual. In general, at least the way of talking I'm familiar with is that since Kochen-Specker says non-contextuality and definite values are incompatible with QM, one says that QM is contextual; that's the same as saying in the case of Bell's theorem, since locality and definite values are incompatible with QM, it's non-local.

 

[stevendaryl]

There are two branches, two observers, two results. My claim is that the following statements cannot be shown to be contradictory:

A) I will find myself in a specific branch with pobability 50%
B) The probability to find x from a top-down perspective is 90%

You didn't like my coin-flipping branching universe analogy, but can I ask, do you think that probability makes sense in such a universe, from the "bottom-up" perspective? It seems to me, that it perfectly well does. But the "bottom-up" probability isn't objective, it's subjective.

To see that it is subjective, suppose that God starts off with the branching where there is one copy of the world for each outcome. Then he adds new branches so that there are 9 copies of the universe for each outcome of "heads", but still only one copy for each outcome of "tails". This changes the "top-down" perspective, but for anyone living in one of the branches, it makes no difference: how can the existence of other branches affect probabilities in THIS branch? So the people living in the branches can still use 50/50 probabilities, instead of 90/10 probabilities.

 

[tom.stoer]

In addition there is a way out, namely deriving the probabilities from branch counting.

Just replace

$|\psi\rangle = a_x |\psi_x\rangle + a_y |\psi_y\rangle$

by something like

$|\psi\rangle = \int_{\mathcal{A}(x)} d\mu(\alpha)\, |\psi_x,\alpha\rangle + \int_{\mathcal{A}(y)} d\mu(\alpha)\, |\psi_y,\alpha\rangle$

Here alpha means any other d.o.f. or whatever that is required to specify a state with a specific result string, so especially environment, small fluctuations leaving the result string invariant etc. 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?

 

[stevendaryl]

I never said it does. All what I am saying is that
A) dropping Born's rule as an axiom requires its derivation as a theorem

But you specifically were contrasting the collapse interpretation with MWI (a no-collapse interpretation). That's basically the only difference between MWI and Copenhagen, is that in MWI, we get rid of the collapse. So I don't understand what the collapse hypothesis has to do with probabilities. You can tease apart two different axioms in the standard interpretation:

1. Measuring an observable gives an eigenvalue, with a probability given by the square of the projection of the wavefunction onto the corresponding eigenstate.

2. After a measurement, the wave function is in the eigenstate corresponding to that eigenvalue.

What makes Copenhagen different from MWI, it seems to me, is axiom 2. MWI is trying to make sense of QM without axiom 2 (instead, you just have smooth unitary evolution). But the probabilities are in axiom 1. So I don't see how dropping axiom 2 requires you to come up with a derivation of Born probability. It would be nice if you could do that, but I don't see why you think it's required.

 

[S.Daedalus]

From my perspective, the problem is as follows, and it's indeed worse than in a collapse interpretation (which of course have their own problems---that I by and large consider even greater):

In a collapse interpretation, you have a well defined notion of the probability of an event, i.e. A happening to the exclusion of B. This probability has a connection to Gleason's theorem and thereby the Born rule, since an event occurring is essentially finding the state vector in a certain subspace of Hilbert space. The collapse postulate does the work of ensuring that one will always find the system to be in a state corresponding to an eigenstate of the observable one measures. That this occurs in accordance with the Born rule is ultimately what is postulated: one could, in principle, conceive of a theory in which Gleason's theorem holds, but collapse occurs according to different rules. However, the measure provided by Gleason's theorem is a natural one to use in order to define probabilities in this case, because of the connection between events and linear subspaces.

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.

 

[stevendaryl]

In addition there is a way out, namely deriving the probabilities from branch counting.

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.

There are special cases where the counting measure for possibilities is the most natural choice, because of some symmetry among the set of possibilities. For example, a six-sided die has 6-fold symmetry (well, if you disregard the effect of the dots), so it would be weird to give different probabilities to the different outcomes.

I guess there is a mathematical question at work, as to whether all quantum-mechanical probabilities can be understood in terms of those sorts of symmetries. Some of them certainly can. If you prepare a spin-1/2 particle in a state with spin-up in the z-direction, then later measure the spin in the x-direction, you can reason: There is a symmetry between the two axes, the +x direction and the -x direction. So the probability of measuring spin-up in the x-direction should be the same as the probability of measuring spin-up in the -x direction. But they can't both hold at once. So the probability must be 1/2.

 

[stevendaryl]

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.

I don't quite understand your reasoning. You're saying that (because of Gleason's theorem, which I don't quite remember the statement of), it's very natural (or provable, even?) to use the rule that a measurement of an observable results in an eigenvalue, with probability given by the square of the projection of the wave function. But you're saying that doesn't imply a measure on possible histories? It seems to me that it does.

Imagine that there is a super-Wikipedia that records everything that every happens, no matter how insignificant. This super-Wikipedia is a physical object, which can be in a superposition of states. But if you read its pages, it collapses into a state in which there is a definite number of articles, and each has a definite collection of words. So if Gleason's theorem can be used to justify the Born rule for observables, then when applied to an object recording the history of the world, it should give a probability for the alternative histories, right?

 

[S.Daedalus]

I don't quite understand your reasoning. You're saying that (because of Gleason's theorem, which I don't quite remember the statement of), it's very natural (or provable, even?) to use the rule that a measurement of an observable results in an eigenvalue, with probability given by the square of the projection of the wave function. But you're saying that doesn't imply a measure on possible histories? It seems to me that it does.

For present purposes, Gleason's theorem says that the only measure on a Hilbert space is the one furnished by the density matrices, in such a way that the measure of some linear subspace is given by $Tr(\Pi\rho)$, where $\Pi$ is the projection operator onto the subspace. Since the subspaces are essentially what determines the properties of some system, it makes sense to associate this measure with the probability of a certain system having a given property---a certain event, in my previous post. But this notion disappears in the MWI.

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'.

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.

 

[tom.stoer]

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.

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

And that's why I am saying that from the top-down perspective everything could work out correctly for calculating expectation values including all branches, but that the bottom-up perspective for exactly one branch does not work w/o additional assumptions.

Now one could say that Born's probability derived and as a multiplicative pre-factor for one branch is sufficient - but then another assumption is introduced.

Or one could try to derive all probabilities from branch counting and coarse graining.