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

[bhobba]

Well I read the first page which included a rapturous endorsement of the maths by no less than @bhobba.

I often say it's mathematically very beautiful.

So?

It does not mean its not controversial.

Thanksa
Bill

 

[Derek P]

I often say it's mathematically very beautiful.
So? It does not mean its not controversial.
Thanks
Bill

Which is why I asked why.
Thanks for the links. The Mandolesi paper ,which I'm slowly skimming through, looks very interesting but I still haven't got my head round why it is the responsibility of MWI to define probability other than as frequencies in a history, let alone in terms of how best to place bets. It seems a flaky idea given that the reward is subject to one's personal whims - which might include a penchant for quantum suicide. In which case all bets are off. So I find myself regularly echoing the words of Lethbridge-Stewart "One imagines some of those words were attached to actual meanings of some sort".
Thanks again for the links.

 

[stevendaryl]

It seems to me that arguments about proving Born's rule (using decision theory or some other logic) are sort of beside the point. Maybe there is a kind of "anthropopic principle" for the existence of viable theories, like there is one for the existence of intelligent life.

Suppose you have a nondeterministic theory of physics. This theory gives rise to a set of possible histories. Among those histories, only some of them will be "typical", where relative frequencies for repeated trials of random events are calculable from the theory. So even if the theory is "correct", only in the typical worlds will intelligent beings bother to develop that theory.

In the typical worlds, people will use the theory, even if they don't have a rigorous justification for it. In the atypical worlds, people will not use the theory, even if they do have a rigorous justification for it. For those in the typical worlds, there might additional satisfaction if they can prove that the set of atypical worlds has measure zero, but such a proof is neither necessary nor sufficient for them to use the theory.

 

[Derek P]

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

Frequencies in a history? One history, one branch. No branch counting. But apparently frequencies are not enough, or else they don't emerge. I thought they were and they did but apparently they aren't or they don't, so we have to use decision theory in order to explain the projection rule.

 

[stevendaryl]

I would be more interested in the interpretation that they come up with in the atypical worlds if they do manage to prove that their worlds have measure zero! I bet God comes into it.

As I was saying (in this thread?) to A Neumaier, I don't think that QM is particular unusual in having such problems. The "theory" that a coin toss gives 50/50 results predicts possible histories with every conceivable relative frequency for heads and tails. 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. In this history, scientists would just be baffled. They would examine a coin, and see that it seems completely symmetric between heads and tails. They might come up with some "hidden variable" theory for why there are more heads than tails---even though they can't detect a difference, they would believe that there is a difference.

So there is another special thing about the typical histories, which is that they obey some kind of symmetry principle---that if there is no reason to favor outcome $A$ over outcome $B$, then they will have equal relative frequencies. That kind of aesthetic beauty only exists in some possible worlds. In the other ones, physics might not even develop --- but engineering and, as you say, religion probably would.

 

[DarMM]

Frequencies in a history? One history, one branch. No branch counting. But apparently frequencies are not enough, or else they don't emerge. I thought they were and they did but apparently they aren't or they don't, so we have to use decision theory in order to explain the projection rule.

I'm not aware of a solid proof that they emerge from Frequencies in a given history. It would be essentially equivalent to branch counting and run into the same problems. Regardless I haven't seen such a proof, or seen one mentioned where a frequency within a history approach is used that is different from branch counting.

 

[Derek P]

So there is another special thing about the typical histories, which is that they obey some kind of symmetry principle---that if there is no reason to favor outcome $A$ over outcome $B$, then they will have equal relative frequencies. That kind of aesthetic beauty only exists in some possible worlds. In the other ones, physics might not even develop --- but engineering and, as you say, religion probably would.

Well one would have to assume that the symmetry-breaking observations only and always took place in a laboratory but everywhere else things were typical, otherwise common sense would have evolved to expect symmetry-breaking and to regard the 50/50 case as strange!

 

[DarMM]

It seems to me that arguments about proving Born's rule (using decision theory or some other logic) are sort of beside the point. Maybe there is a kind of "anthropopic principle" for the existence of viable theories, like there is one for the existence of intelligent life.

Suppose you have a nondeterministic theory of physics. This theory gives rise to a set of possible histories. Among those histories, only some of them will be "typical", where relative frequencies for repeated trials of random events are calculable from the theory. So even if the theory is "correct", only in the typical worlds will intelligent beings bother to develop that theory.

The problem with Many-Worlds is more that you can't even prove there are any typical histories, aside from uniform ones.

There has to be someway of deriving the association between the amplitudes and probabilities, regardless of if your world is similar to the expected values or not.

EDIT: Removed last line as it was written poorly

 

[stevendaryl]

The problem with Many-Worlds is more that you can't even prove there are any typical histories, aside from uniform ones.

I'm not sure what you mean. Let's pick an experiment: Say, I pick two directions in space, $\vec{a}$ and $\vec{b}$, and I repeatedly perform the experiment:

1. Put an electron into the state of having spin-up along the $\vec{a}$ axis.
2. Later, measure its spin along the $\vec{b}$ axis, and write down either "U" or "D" in my notebook.
3. Beside each entry, I also calculate the relative frequencies of $U$ versus $D$, so far.

In the many-worlds interpretation, there will be many versions of my notebook. Some of them will have relative frequencies close to that predicted by quantum mechanics ($cos^2(\frac{\theta}{2})$ where $\theta$ is the angle between $\vec{a}$ and $\vec{b}$), and some will not.

 

[DarMM]

ome of them will have relative frequencies close to that predicted by quantum mechanics ($cos^2(\frac{\theta}{2})$ where $\theta$ is the angle between $\vec{a}$ and $\vec{b}$), and some will not.

I've no problem that there will be worlds where the ratio matches the Born Rule.

Let me take a simpler case, the state of the particle is:

$$\sqrt{\frac{1}{3}} | \uparrow \rangle + \sqrt{\frac{2}{3}} | \downarrow \rangle \tag 1$$

and I repeatedly perform a set of measurements on the spin.

What shows that distribution of the observations across branches "peaks" around worlds where the frequency of observing spin-down is twice that of observing spin up?

That is to say that there is a higher weight of worlds "near" the 2:1 ratio. Or "more" worlds with the 2:1 ratio.

I don't see how the world structure is any different from the one resulting from repeated experiments on:

$$\sqrt{\frac{1}{2}} | \uparrow \rangle + \sqrt{\frac{1}{2}} | \downarrow \rangle \tag 2$$

 

[Derek P]

Currently it would seem that every world should see uniform frequencies.

I don't see why. 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. At the same time the probabilities that each microstate contributes when the mess is finally observed add linearly. Where's the catch?

 

[DarMM]

That was poorly worded, #40 to stevendaryl is more my point.

 

[stevendaryl]

Let me take a simpler case, the state of the particle is:

$$\sqrt{\frac{1}{3}} | \uparrow \rangle + \sqrt{\frac{2}{3}} | \downarrow \rangle \tag 1$$

and I repeatedly perform a set of measurements on the spin.

What shows that distribution of the observations across branches "peaks" around worlds where the frequency of observing spin-down is twice that of observing spin up?

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.

That is to say that there is a higher weight of worlds "near" the 2:1 ratio. Or "more" worlds with the 2:1 ratio.

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?

I don't see how the world structure is any different from the one resulting from repeated experiments on:

$$\sqrt{\frac{1}{2}} | \uparrow \rangle + \sqrt{\frac{1}{2}} | \downarrow \rangle \tag 2$$

That's a puzzling philosophical question, but what I would say is that there is nothing particularly quantum-mechanical about the puzzle. You can do the same thing with classical probabilities:

You flip a coin many times and convince yourself that it has a 50/50 chance of heads versus tails. Now, completely unknown to you, God (or some computer scientist, if you happen to be a simulation inside a supercomputer) does the following: Every time you flip a coin, God makes an exact copy of the world, and makes sure that in this copy, the opposite result occurs. He does the same for every new world: Every time someone in any of the worlds flips a coin, there are two copies made, one where the result is "heads" and the other where the result is "tails".

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.

But now, suppose purely on a whim, God changes his mind, and changes his rule so that there are 2 copies made where the result is "heads" and only one copy where the result is "tails"? On the one hand, from the branch counting point of view, that makes heads have a probability of 66% while tails has a probability of 33%. That seems like a drastic change to the laws of physics. But surely, the extra copies have no effect on the existing copies? The existence or nonexistence of an alternate world can't possibly affect my empirically derived notion of relative frequencies. Regardless of how many copies are produced with each flip, some of the worlds will observe 50/50 relative frequencies, and will be happy because it agrees with their notions of symmetry. Others will observe other ratios and will be puzzled by the lack of symmetry in relative frequencies.

 

[akvadrako]

1. How do I know which "fictional" uniform case my state is a modification of to say my probability is $1/N$?

Any choice should work, as long as the magnitude of each branch is equal.

2. How does this apply to something where a non-uniform probability state is measured by a single observer. Like electrons coming from a silver oven toward one detector.

This is all about what a single observer will experience, so I don't think I understand your point here.

However ignoring all this, it still doesn't answer the objection I always have to these derivations. What is the model?

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.

Also it is in essence an extra axiom, as unitary QM only gives you the state above (3), which under a naive MWI reading is two-worlds. You have to add the assumption that the value of the amplitude also tells you how many copies there are of each world, e.g. in (3) there are two "down worlds".

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

There are extra axioms, indeed. In Vaidman's approach, they are locality and symmetry principles, which say that if you decompose your state into equal-weighted branches, then branch counting agrees with the Born rule.

 

[akvadrako]

That is to say that there is a higher weight of worlds "near" the 2:1 ratio. Or "more" worlds with the 2:1 ratio.

I don't see how the world structure is any different from the one resulting from repeated experiments on:

$$\sqrt{\frac{1}{2}} | \uparrow \rangle + \sqrt{\frac{1}{2}} | \downarrow \rangle \tag 2$$

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.

Akin to how the one real world in Bohmian mechanics can be represented as a point-particle guided by the wavefunction, instead of considering splitting it works to consider every point on the wavefunction as a possible world. And when they diverge, there will be a higher density of worlds/points following the higher-magnitude branches. At least for me, this is one approach I've found illustrative.

 

[stevendaryl]

If a river branches 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.

Akin to how the one real world in Bohmian mechanics can be represented as a point-particle guided by the wavefunction, instead of considering splitting it works to consider every point on the wavefunction as a possible world. And when they diverge, there will be a higher density of worlds/points following the higher-magnitude branches. At least for me, this is one approach I've found illustrative.

In Many-Worlds, there are two different notions of measure/probability that come into play: The probability of a world, and the relative frequencies within one world. They are related, in that if you use the Born rule to compute probabilities of possible worlds, then you will find that "most" worlds have relative frequencies that are given by the Born rule, as well.

 

[ftr]

Many Worlds, the Born Rule, and Self-Locating Uncertainty
https://arxiv.org/abs/1405.7907

but from my point of view the derivation of Schrödinger equation should come first and since there is no such a thing yet, I see the whole exercise as futile.

 

[bhobba]

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. Although not generally discussed there are a number of interpretations of probability - decision theory is just one of them. Actuaries use it a lot in deterministic systems (probably chaotic like financial markets but no assumption of such is made) so it's mathematically valid - as always meaning is the issue.

Is that a valid issue with the interpretation?

Blowed if I know - like all interpretive stuff it often just degenerates into counter-productive heated arguments. They are of zero value. No interpretation is right or wrong - they are just interesting and educational ways to look at the formalism. You can decide what you like or do not like - no need for long threads about it.

What we do here interpretation wise is clarify what they say - not argue about it.

Fact - the Born Rule in MW is often justified using decision theory. You can decide if its a valid approach - arguing about it will simply result in people like me as a mentor keeping it on track. Then we have the non-contextuality theorem and Gleason. You have to get a hold of the book and study it to see if it can be attacked and post a specific question about it. It looks tight to me. But of course the whole issue of probability in a deterministic theory is an issue. 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.

Here is an example from the paper I linked to criticizing it. It says. 'Deutsch has proposed using decision theory to show that, under Everettian conditions, it would be rational to decide on bets about the results of quantum measurements as if they were probabilistic and followed the Born rule.' Well BM is deterministic yet all you can't predict things because of the uncertainty principle - only probabilities. This is similar to decision theory as a discipline - in some situations things are deterministic - we just, for some reason or another, do not know the outcome. So IMHO this argument against it is invalid. But these types of things are more philosophical on what probability is and generally just generate a lot of verbose discussion that in the end just says I view probability this way - scientifically IMHO dubious value.

If you want to go down that path, discussion that inherently go no-where will, correctly, be stomped on by mentors like me. They are counterproductive to the aims of the forum. By all means point out statements like the above from the paper are arguable - but don't argue it because it won't really resolve anything.

Thanks
Bill

 

[Stephen Tashi]

Let's pick an experiment:

That again raises (in my mind) the question of whether defining probability in terms of an observer refers to an actual observer or an conceptual observer that we only imagine. For example, for a given type of experiment, can there be a history (or "world") where it is never repeated?

Are we using the term "history" in the same sense as the "consistent histories" formulation of QM?

A complicated experiment like "Try to build a transatlantic tunnel" might never be attempted. Is it implicit in any formulation of QM that Nature is composed of "elementary" phenomena that may be always be regarded forming independent repeated experiments? This is different that the question of whether such independent repeated experiments in a world have the "correct" limiting frequencies of outcomes. (Mathematically, a sequence might not have any limiting outcome at all.)

 

[stevendaryl]

Are we using the term "history" in the same sense as the "consistent histories" formulation of QM?

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.

A complicated experiment like "Try to build a transatlantic tunnel" might never be attempted. Is it implicit in any formulation of QM that Nature is composed of "elementary" phenomena that may be always be regarded forming independent repeated experiments? This is different that the question of whether such independent repeated experiments in a world have the "correct" limiting frequencies of outcomes. (Mathematically, a sequence might not have any limiting outcome at all.)

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?

 

[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