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

[Derek P]

Please restrict your answers to criticisms of derivations of the Born Rule that are generally accepted by proponents of MWI. Please provide a verbal description of the issue where possible so that people like myself who are certainly graduate-plus* but rusty as hell have a chance of seeing what the point is.

*plus nearly half a century in my case

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

 

[PeterDonis]

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

In the sense that nobody has produced a derivation of the Born Rule using just the postulates accepted by the MWI, that has been generally accepted as being valid.

A previous PF thread on the subject is here:

https://www.physicsforums.com/threads/the-born-rule-in-many-worlds.763139/

 

[akvadrako]

Like PeterDonis said, you have to assume something. Take the simplest case possible, a fully symmetric wavefunction with 2 equal-weighted branches which have no interaction terms. How can you say the probability of finding yourself (or a particle) on either branch is 50% ?

Using only the axioms of unitary QM it doesn't seem possible.

 

[Derek P]

In the sense that nobody has produced a derivation of the Born Rule using just the postulates accepted by the MWI, that has been generally accepted as being valid.
A previous PF thread on the subject is here:
https://www.physicsforums.com/threads/the-born-rule-in-many-worlds.763139/

Well I read the first page which included a rapturous endorsement of the maths by no less than @bhobba. Of course he may have changed his mind since then. In addition @stevendaryl, @atyy, @vanhees71 to name but a few had their say, but I didn't see anything at all suggesting that the usual derivation is flawed. Should I continue?

 

[PeterDonis]

the usual derivation

What do you think is "the usual derivation"? Can you give a reference?

 

[DarMM]

As far as I understand there is no totally clear derivation of the meaning of non-uniform probabilities using purely unitary quantum theory with no projection postulates. Zurek's proof relies on hard to justify assumptions about subspaces of states and assumes some notion of probability to begin with, as it starts with a Schmidt state as the post-measurement state.

The Oxford style decision theory proofs only prove that in a many worlds situation one can construct a utility function whereby an agent is permitted to act as if the Born rule were true. However this doesn't show the utility function to be unique, nor is it obvious that it is the most sensible one. Even more strongly, just because I might act in a way that satisfies the axioms of decision theory, doesn't mean anything for the real observed frequencies of measurement results recorded by lab equipment. Also the Oxford proofs do too much handwaving at certain points about how the branching structure works in order to force the proof.

 

[Derek P]

What do you think is "the usual derivation"? Can you give a reference?

<shrug> My bad. "that there was no unflawed derivation" then.

 

[PeterDonis]

Should I continue?

I'm not sure if there's much point if we don't have any specific derivation to discuss.

 

[Stephen Tashi]

In what sense does any approach to QM derive the Born Rule? - as opposed to taking it as an assumption.

Perhaps that's a good topic for another thread.

 

[stevendaryl]

In what sense does any approach to QM derive the Born Rule? - as opposed to taking it as an assumption.

Perhaps that's a good topic for another thread.

Well, @bhobba likes to mention a theorem (Gleason's Theorem) that shows that if you've already decided that the quantum state gives probabilities for measurement outcomes, then under some pretty minimal assumptions, it turns out that the square of the amplitude is pretty much the only natural choice.

 

[stevendaryl]

Well, @bhobba likes to mention a theorem (Gleason's Theorem) that shows that if you've already decided that the quantum state gives probabilities for measurement outcomes, then under some pretty minimal assumptions, it turns out that the square of the amplitude is pretty much the only natural choice.

I'm not sure what's the best place to look for the proof, but here's one:

https://arxiv.org/pdf/quant-ph/9909073.pdf

I think that, in a certain sense, the interpretation of the square of the wave function as a probability seems baked into the very definition of a Hilbert space. The fact that $\int |\psi(x,t)|^2 dx = 1$ is certainly very suggestive that $|\psi|^2$ can be interpreted as a probability density.

 

[Mentz114]

Please restrict your answers to criticisms of derivations of the Born Rule that are generally accepted by proponents of MWI. Please provide a verbal description of the issue where possible so that people like myself who are certainly graduate-plus* but rusty as hell have a chance of seeing what the point is.

*plus nearly half a century in my case

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

In my understanding the Born rule has two parts
1. The square of the amplitude is probability
2. Doing a suitable measurement on a system (applying $\hat{A}_m$) results in the $m$th eigenstate of $\hat{A}_m$ with probability $|\alpha_m|^2$

I think MWI runs into trouble with the second using the original 'naive' splitting protocol.

The problem is that if we follow a particular line of splits for a lot of splits and count, frequencies don't all match the expected ones. For instance for 100 coin tosses.

This can be fixed easily by tinkering with the splitting protocol and introducing a (non-physical) memory $M$ for each instance of our observer.
P is the probability of getting a 1 on each trial of a binary process with outcomes '0' and '1'.
Let the splitting be controlled by the quantity $M/S$ where $M$ is the count of '1' splits and $S$ is the number of splits in the history.
The rule is
if M/S < P split into 1,1
if M/S = P split into 1,0
if M/S > P split into 0,0.

Now the limiting probability estimate M/S → P as required.

But the sequences are not random !
If P=1/2 the sequence becomes alternating 0's and '1's. This non-randomness happens for any value of P.
So, this is not a good solution ( no coconut). I think the reason is that MWI robs QM of doing anything that can give random results because that requires quantum indeterminacy, which is gone.
My feeling is that (even after one attempt to fix this) that one must have indeterminacy to make correct predictions.

[ I may be conflating indeterminacy and non-unitarity. It is the aim of MWI to remain unitary, but it seems that this inevitably loses randomness in some fundamental way]

 

[Nugatory]

In what sense does any approach to QM derive the Born Rule? - as opposed to taking it as an assumption.

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.

 

[akvadrako]

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.

MWI proponents reject reduction but most don't reject using an additional assumption from which they can derive the Born rule; that's more often something critics claim. It's clear the unitary axioms are not enough to even talk about probability and as a result MWI has no advantage over other interpretations in this regard. This says it better than I can:

In defence of the self-location uncertainty account of probability in the many-worlds interpretation
Kelvin J. McQueen and Lev Vaidman. February 13, 2018

Do we provide unconditional proof of the probability postulate in the MWI from the formalism of quantum mechanics as Everett and others hoped possible? Deustch [1999] and Wallace [2012] relied on some decision-theoretic principles of rationality to deduce their Born rule analogue. We used a particular metaphysical approach that relied on physical principles.8 Similar principles allowed us to “prove” the Born rule in the framework of standard quantum mechanics. The status of our proofs therefore greatly depends on the status of these physical principles. We welcome further research into these principles. Either way, we have shown that probability can be consistently and naturally introduced into the framework of the MWI and that there is no special advantage that collapse theories have over the MWI regarding derivation of the Born rule.

 

[Stephen Tashi]

In defence of the self-location uncertainty account of probability in the many-worlds interpretation
Kelvin J. McQueen and Lev Vaidman. February 13, 2018

In that paper we read:

MWI probability postulate: The probability of self-location in a world with a given set of outcomes is the absolute square of that world's amplitude,

...

In contrast to collapse theories, the postulate is not about some fundamental physical process,
it is about the experience of the observer. Each observer with records of some quantum exper-
iments understands that there are many other observers living in parallel worlds who obtained
dierent outcomes. The postulate concerns the subjective probability
for an observer to live in a particular world.

If one successfully appends the Born Rule to the Many Worlds Interpretation by taking the viewpoint of a single observer in the MWI, doesn't that say that a physicist may as well take the collapse viewpoint ? - seeing as the physicist is a single observer.

The verbal interpretations of MWI seem to take for granted that observers that are different (due to being on different branches) cannot be aware of each other. Intuitively that seems self evident, but how is that shown mathematically? How is "A is not aware of B " translated into a precisely defined physical process?

 

[akvadrako]

If one successfully appends the Born Rule to the Many Worlds Interpretation by taking the viewpoint of a single observer in the MWI, doesn't that say that a physicist may as well take the collapse viewpoint ? - seeing as the physicist is a single observer.

That does seem valid and as another way of saying it, Copenhagen can be derived from the MWI by considering just a single viewpoint, at least practically because of decoherence. But I guess not exactly due to Wigner's friend-type experiments. If I experience collapse, but I'm part of a larger isolated experiment, then it's still possible an outside experimenter will undo that collapse by causing me and other versions of me to interfere, causing the probability of some futures to become zero.

Even not regarding that very difficult experiment (at least with people), with many worlds you can analyse the viewpoint of multiple physicists together. If you try to do that with Copenhagen in a single world, you'll get inconsistencies.

The verbal interpretations of MWI seem to take for granted that observers that are different (due to being on different branches) cannot be aware of each other. Intuitively that seems self evident, but how is that shown mathematically? How is "A is not aware of B " translated into a precisely defined physical process?

Do you mean after decoherence, how it "not aware of" defined? I think about it as no longer interfering and can be written as separate terms that evolve independently.

 

[Derek P]

WMI

Do you mean MWI or is this some other species of interpretation that I don't recognise?

 

[akvadrako]

Do you mean MWI or is this some other species of interpretation that I don't recognise?

Thanks, fixed.

 

[Derek P]

How is that shown mathematically? How is "A is not aware of B " translated into a precisely defined physical process?

Right at the beginning. Note. I very strongly disagree that worlds separate at measurement, let alone before. They separate at decoherence. Before that, the two pre-world states are orthogonal. They absolutely cannot affect each other or be aware of each other. Separation of worlds is something completely different.

 

[Stephen Tashi]

Copenhagen can be derived from the MWI by considering just a single viewpoint, at least practically because of decoherence. But I guess not exactly due to Wigner's friend-type experiments. If I experience collapse, but I'm part of a larger isolated experiment, then it's still possible an outside experimenter will undo that collapse by causing me and other versions of me to interfere, causing the probability of some futures to become zero.

I can imagine such an un-collapse happening from the viewpoint of an "outside" observer C who can observe how observer B undoes the collapse of observer A. But isn't observer A really two (or more) different observers from the viewpoint of MWI and observer C ?

Do you mean after decoherence, how it "not aware of" defined? I think about it as no longer interfering and can be written as separate terms that evolve independently.

I have difficulty digesting arguments that treat complex systems (such as human observers) as simple mathematical structures - such as superimposed waves or Hilbert Space rays in QM ( or points on a world line in the theories of relativity). Perhaps this is conceptual limitation on my part, but it would please me to hear details of how decoherence can implement the common language notion of "A is not aware of B"

A realistic model of a human observer portrays him as a aggregation of matter, some of which "comes and goes" as far as what we consider being "part of" the observer. The matter has more properties than position and momentum. If we take the view that matter and everything else is really waves in various types of fields then how do you define which parts of the fields are observer A and which parts are observer B -especially in MWI where the notion of "observer A" seems only to make sense at a single instant in time. (An instant later, observer A splits up.)

A simple mathematical question about relations is: If E is aware of S and A is aware of S then is E necessarily aware of A? In the common language notion of "is aware of", I'd say No. So if earthling E is aware of star S and alien A is aware of star S then earthling E may not be aware of alien A.

If "is aware of" is replaced by "is entangled with" , then shouldn't the answer be Yes?

In another thread, I asked whether using "rational agents" to introduce probability into MWI assumes these are actual agents or are they merely conceptual agents - analogous to a conceptual "unit test mass" that we imagine in order to define a force field. I think this is a meaningful question due to the complexity of implementing actual agents.

 

[akvadrako]

I can imagine such an un-collapse happening from the viewpoint of an "outside" observer C who can observe how observer B undoes the collapse of observer A. But isn't observer A really two (or more) different observers from the viewpoint of MWI and observer C ?

In this case, A is like the particle in a delayed-choice experiment, right? So you normally don't say there are 2 particles, just that it's in a superposition of two states.

I have difficulty digesting arguments that treat complex systems (such as human observers) as simple mathematical structures.

One thing that's probably in between is considering the whole experiment running in a quantum computer, and the observers as simple preprogrammed agents with a few bits of storage.

If "is aware of" is replaced by "is entangled with" , then shouldn't the answer be Yes?

Maybe sticking with entanglement instead of awareness is just easier. Does anything else matter?

 

[Derek P]

I have difficulty digesting arguments that treat complex systems (such as human observers) as simple mathematical structures - such as superimposed waves or Hilbert Space rays in QM ( or points on a world line in the theories of relativity). Perhaps this is conceptual limitation on my part, but it would please me to hear details of how decoherence can implement the common language notion of "A is not aware of B"

So would I as they would be unaware of each other even if there were no decoherence (assuming that A and B are pre-worlds (my term) - states that will evolve into separate worlds through decoherence in due course)..

A realistic model of a human observer portrays him as a aggregation of matter, some of which "comes and goes" as far as what we consider being "part of" the observer. The matter has more properties than position and momentum. If we take the view that matter and everything else is really waves in various types of fields then how do you define which parts of the fields are observer A and which parts are observer B -especially in MWI where the notion of "observer A" seems only to make sense at a single instant in time. (An instant later, observer A splits up.)

You don't define what an observer is. You define what an observer observes. Hence the common statement "an observer can be anything that interacts with the system". The system has properties.

A simple mathematical question about relations is: If E is aware of S and A is aware of S then is E necessarily aware of A? In the common language notion of "is aware of", I'd say No. So if earthling E is aware of star S and alien A is aware of star S then earthling E may not be aware of alien A.

If "is aware of" is replaced by "is entangled with" , then shouldn't the answer be Yes?

No. The idea of awareness presumably includes events happening to A which then alters B. This is absolutely impossible with entanglement alone.

In another thread, I asked whether using "rational agents" to introduce probability into MWI assumes these are actual agents or are they merely conceptual agents - analogous to a conceptual "unit test mass" that we imagine in order to define a force field. I think this is a meaningful question due to the complexity of implementing actual agents.

You might well ask. I think it is a very awkward attempt to reinstate the idea of probability as a real property of the system. I don't know why anyone would want to do this as MWI seems to predict observation frequencies perfectly well without defining instrinsic probability or whatever you want to call it. But does it? This of course is why I started this thread - someone here had cast doubts on whether MWI actually does derive the Born Rule. I don't see the possible inability of MWI to define something it has no need of as a defect or even surprising. I am a very simple person. Vague too or so I'm told.

On the subject of which may I respectfully ask that people here stick to the topic.It's already getting diluted beyond any usefulness. Thanks!

 

[DarMM]

I don't know why anyone would want to do this as MWI seems to predict observation frequencies perfectly well without defining instrinsic probability or whatever you want to call it.

There's no problem with intrinsic probabilities nor the feeling that Many Worlds needs to include them. Rather the problem is precisely the first part of your sentence, MWI doesn't seem to predict observed frequencies very well.

Attempts to derive the Born Rule, like Zurek's, are attempts to show that expected values for frequencies within a single world match those given by textbook Quantum Mechanics.

That is, for a state:
$$\sqrt{\frac{1}{3}}| a \rangle + \sqrt{\frac{2}{3}}| b \rangle$$

That your odds of being in a world where "b" is observed are $2/3$. However there is no clear proof of this from MWI as of 2018.

 

[Derek P]

There's no problem with intrinsic probabilities nor the feeling that Many Worlds needs to include them. Rather the problem is precisely the first part of your sentence, MWI doesn't seem to predict observed frequencies very well.

Attempts to derive the Born Rule, like Zurek's, are attempts to show that expected values for frequencies within a single world match those given by textbook Quantum Mechanics.

That is, for a state:
$$\sqrt{\frac{1}{3}}| a \rangle + \sqrt{\frac{2}{3}}| b \rangle$$

That your odds of being in a world where "b" is observed are $2/3$. However there is no clear proof of this from MWI as of 2018.

First of all thank you very much for such a definitive statement. If the alleged problems with deriving the Born Rule apply strictly to frequencies then the question becomes very straightforward. There are a number of versions of a very simple proof such as this one from Michael Price. I would not like to get too tied into one person's semi-popular presentation but those who say MWI can't produce the Born Rule should be able to point to where Price, and many others with similar arguments, go wrong.

 

[PeterDonis]

There are a number of versions of a very simple proof such as this one from Michael Price.

I don't see how this is a proof of anything.

First, his argument relies on the B state being degenerate; but it is easy to write down states with unequal amplitudes for A and B terms, neither of which is degenerate. (The easiest is to do it with qubit states.)

Second, his argument relies on there being a simple integer ratio between the two amplitudes. But of course the vast majority of states do not obey that restriction. (In fact, it's simple to show that the pairs of amplitudes that do are a set of measure zero in the space of all possible pairs of amplitudes, and therefore, by this person's own argument, do not exist.)

Third, to address the obvious objection to the above, I don't buy his hand-waving argument for how his "proof" extends to the case of non-degenerate states or non-rational amplitude ratios.

But finally, even leaving all that aside: counting "numbers of worlds" and squaring their relative amplitudes and saying that those are the probabilities of orthonormal "worlds" is not proving the Born rule: it's assuming it. His "proof" is just arguing in a circle.

And finally finally, he misstates what DeWitt proved in his 1970 paper. DeWitt did not prove that "the norm of the worlds in which the Born rule is violated vanishes". He only showed that, in the limit of an infinite number of observations on an ensemble of identically prepared systems, the difference between the relative frequency of a particular outcome and the square of the amplitude of that outcome vanishes. That is not at all the same thing.

 

[PeterDonis]

I would not like to get too tied into one person's semi-popular presentation

Then you should be able to produce something which is not a semi-popular presentation, but an actual textbook or peer-reviewed paper. In other words, a valid source for PF discussion.

 

[akvadrako]

Then you should be able to produce something which is not a semi-popular presentation, but an actual textbook or peer-reviewed paper. In other words, a valid source for PF discussion.

For example, the paper I referenced from Vaidman - yes I know it’s from the philosophy department, so not really allowed - but he is an expert on born rule derivations and it’s one of the most modern views available. He builds on Deutsch, Wallace, Carroll and Zurek, in an aim to make it clear what must be assumed for it to work. I find it an easier read than that quora post.

 

[PeterDonis]

For example, the paper I referenced from Vaidman

Ah, I missed that. I'll take a look.

 

[DarMM]

Well, unless I'm missing something I don't really see how they demonstrate it.
Section 4.2 seems to be the main section proving it, but to me it only contains a clear argument for the uniform probability case.

So they say, imagine you have a state:

$$\sqrt{\frac{1}{3}} \left( | a \rangle + | b \rangle + | c \rangle \right) \tag 1$$

which represents a particle in a superposition of traveling toward either observer A, B or C. This will split into three worlds, depending on who observed the particle. Since it is in only one world that A will see the particle, it is rational for A to say the chance of detecting the particle is $1/3$, that's fine I think.

However, they then say that because of locality, A will not be aware of anything that might happen outside his local region $\mathcal{O}_{A}$, so even if the state was altered to:

$$\sqrt{\frac{1}{3}} | a \rangle + \sqrt{\frac{2}{3}} | d \rangle \tag 2$$

where D is basically an observer at another location to whom the particle is redirected from locations B and C, the observer at A should still apply probability $1/3$ to their situation, hence we can explain non-uniform cases like (2).

Again, maybe I'm being dumb, but I don't really see how this works. It's basically saying that non-uniform probabilities occur, because they are transformations of uniform ones, but the transformations occur in a region I can't see, so I keep the old uniform probability.

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

However ignoring all this, it still doesn't answer the objection I always have to these derivations. What is the model? What happens at a measurement in the case of non-equal coefficients that causes non-uniform probabilities. Taking about what's rational for agents is very weak in my opinion, it just means they have a way of assessing measurement set ups that obeys the axioms of decision theory, but that method could be completely detached from what's actually being recorded by the devices.

I have never heard an account of what is really going on, aside from branch counting, i.e. when we have a state like:
$$\sqrt{\frac{1}{3}} | \uparrow \rangle + \sqrt{\frac{2}{3}} | \downarrow \rangle \tag 3$$

Then after measurement there is one world with spin-up and two with spin-down.

However branch counting has currently untackled issues with conditional probabilities, in fact there are several papers by Adrian Kent and others that seem to pretty much show branch counting is inconsistent. (See his chapter in Many Worlds? Everett, Quantum Theory, & Reality, Oxford University Press, 2012)

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?

 

[bhobba]

Here is Wallace's proof:
https://arxiv.org/abs/0906.2718

He discusses objections against in in his book the Emergent Multiverse:
https://www.amazon.com/dp/0198707541/?tag=pfamazon01-20

He also has a proof - the so called non-contextuality theorem - that shows that MW is inherently non-contextual. Being non-contextual Gleason applies.

However as people have noted its controversial if the proofs are valid:
https://arxiv.org/abs/1803.08762

If you want to discuss an issue be specific ie give the assumption that is being attacked and exactly why. Also explain why such would invalidate the non-contextuality theorem because then Gleason comes into play and their is no out - it must be true.

Thanks
Bill

 

[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.