Why MWI cannot explain the Born rule

[yossell]

Hi Fredrik,

thanks for the response - I don't mean to be pedantic, but...

I wasn't arguing that the collapse view was any more legitimate - rather, I took you to be arguing that, without extra assumptions, the formalism supported MWI - and I was arguing against *that*. In this previous post, you seem to be saying that the formalism is neutral - in which case, we're on agreement on this point. I took you to be arguing for the stronger claim because you wrote (as I quoted before)

The argument I've been using to support the claim that we need an additional axiom to get rid of the other worlds goes like this:

and also "If we reject the idea that this describes what actually happens," - my point is given a reasonable interpretation of the density matrix, I see nothing in the formalism that the collapse theorist has to *reject* - but I've not been arguing in favour of one position over the other.

I think, though, given your last post, I agree with most of what you say - if I've read it right and it's that the theory is actually neutral. I'm not sure what's supposed to follow from the fact that all terms are described in the same way - a probabilistic theory that only tells you the likelihood of what will happen and stops short of telling you what will happen will treat
the possibilities in the same way - I can't see why that should make me think a many worlds interpretation is in any sense preferred.

 

[Fredrik]

I wasn't arguing that the collapse view was any more legitimate - rather, I took you to be arguing that, without extra assumptions, the formalism supported MWI - and I was arguing against *that*. In this previous post, you seem to be saying that the formalism is neutral -

Yes, in #28 and #30 I was saying that the axioms might be neutral on this, but I still need to think that through (unless someone does it for me).

I think, though, given your last post, I agree with most of what you say - if I've read it right and it's that the theory is actually neutral. I'm not sure what's supposed to follow from the fact that all terms are described in the same way - a probabilistic theory that only tells you the likelihood of what will happen and stops short of telling you what will happen will treat
the possibilities in the same way - I can't see why that should make me think a many worlds interpretation is in any sense preferred.

Those options are not the ones that are competing here. The assumption that QM doesn't tell us what actually happens is the "ensemble interpretation". That's an anti-realist interpretation. This is not a debate about realism vs. anti-realism. We're just trying to determine if the MWI is the minimal realist interpretation or not.

So our starting point is the assumption that the Dirac-von Neumann axioms are literally telling us what actually happens in any kind of interaction, including measurements. The part I'm not 100% clear on is if the axioms are telling us how to interpret probabilistic statements as statements about what actually happens, or if they're consistent with several such interpretations. (Maybe it would be immediately obvious to me if I just read the axioms again, but I have to go to bed now).

The above should make it clear why it's significant that all the terms are the same. To say that \rho\rightarrow \sum_i P_i\rho P_i is a description of what actually happens is definitely 100% saying that all the worlds are equally real. The only detail that requires further investigation is the question of whether the axioms imply this mathematical description or if this is just one of several possibilities that are consistent with the axioms.

 

[dmtr]

Here I propose a VERY SIMPLE and intuitive argument that MWI, with its MINIMAL set of assumptions, cannot explain the Born rule.

Any suggestions? Comments? Objections?

I think that this is correct, MWI cannot explain the Born rule, but the reasoning should be different. The Born rule have a hidden reference to the idealized observer in it. This idealized observer is not the part of the universe and is simply unphysical. MWI does not contain any unphysical entities and thus can not explain the Born rule.

Real observers can probably be defined as bit (or qbit) strings and I think it is very likely that you can derive a very good analog of the Born rule for these real observers using only MWI, symmetries and identity.

Consider a perfectly symmetric system {single photons source, beam splitter, detectors, counter}, if that counter is the classical observer and if the evolution is unitary, due to the symmetry of the system you'll probably be able to define the probabilities for that observer (counter) to find itself (BBP, identity) having some specific value.

 

[Dmitry67]

Consider a perfectly symmetric system {single photons source, beam splitter, detectors, counter}, if that counter is the classical observer and if the evolution is unitary, due to the symmetry of the system you'll probably be able to define the probabilities for that observer (counter) to find itself (BBP, identity) having some specific value.

So what's about the unfortunate branch where observer had detected all the 10000000 photons from the same side? Or in general, what's about low-probability observers?

Again, what is probability? If it is some correlations in the experience of any observer, then WHAT branch of an observer?

 

[dmtr]

So what's about the unfortunate branch where observer had detected all the 10000000 photons from the same side? Or in general, what's about low-probability observers?

Again, what is probability? If it is some correlations in the experience of any observer, then WHAT branch of an observer?

When you define probability in statistical mechanics you pretty much start from the same symmetry/unitarity assumptions. You have some symmetric system and you postulate that due to the symmetry probabilities to find the system in the symmetric states must be equal. You also postulate that the sum of probabilities must be equal to one.

After that you have the notion of probability. And you can apply the probability theory to find which states of the system are going to be probable or improbable.

Now in MWI we assume unitarity (sum of probabilities is 1). And again we can start with some symmetric system. The only troubling part is the observer. I think that the key here is the BPP (boundary of a boundary is zero) postulate - the identity - because the observer is calculating the probability to find himself in some state.

 

[yossell]

Those options are not the ones that are competing here. The assumption that QM doesn't tell us what actually happens is the "ensemble interpretation". That's an anti-realist interpretation. This is not a debate about realism vs. anti-realism. We're just trying to determine if the MWI is the minimal realist interpretation or not.

If by 'actually happens' you mean 'uniquely predicts a single outcome' then (a) it's pretty standard to say that QM doesn't predict a unique standard outcome; forget any funny business with cats half dead and alive - just is just what happens in the laws are probabilistic and, again, it's pretty standard to say that QM is indeterministic. This is not anti-realism. That the laws are irreducibly chancy has nothing to do with anti-realism. (b) There's a natural - and as far as I can tell - entirely standard way of understanding \rho\rightarrow \sum_i P_i\rho P_i: it's a description of how the probabiilties will actually evolve in certain conditions. One can be an objectivist about probabilities and think that there are objective facts about probability - so it's entirely neutral on the anti-realism debate.

Everything hinges on how you physically interpret these mathematical states, how you interpret probabilities, how you interpret the terms that will necessarily appear in probabilistic theories. There mere appearance of these terms in the mathematical description of the evolution of the probabilities doesn't get you anything about other the physical reality of worlds corresponding to these terms.

Actually - I'm still not sure I know which axioms you have in mind when you say Dirac-von Neumann. I tried to follow a previous link, but my Dutch is...ahh... a little rusty. Plus the pages didn't show. Do you have a link where the axioms are given explicitly - sorry if you already provided it somewhere.

 

[Dmitry67]

When you define probability in statistical mechanics you pretty much start from the same symmetry/unitarity assumptions. You have some symmetric system and you postulate that due to the symmetry probabilities to find the system in the symmetric states must be equal. You also postulate that the sum of probabilities must be equal to one.

After that you have the notion of probability. And you can apply the probability theory to find which states of the system are going to be probable or improbable.

Now in MWI we assume unitarity (sum of probabilities is 1). And again we can start with some symmetric system. The only troubling part is the observer. I think that the key here is the BPP (boundary of a boundary is zero) postulate - the identity - because the observer is calculating the probability to find himself in some state.

1 statistical mechanics describe single-history, in MWI we have multiple histories.

2 I don't see any problems with "the observer is calculating the probability to find himself in some state."

Please explain. I have random generator with 2 outcomes: Frequent (probability 99.9999%) and Rare (0.0001%). After a trial I have 2 branches: F and R (with the different 'measure of existence').

The question is, why I almost always end in F branch? That claim is naive from the MWI point of view. But denying it is denying any predictive power of QM.

MWI predicts 2 branches after the decoherence: F and R with surprised R-observer. Why, why F branch is 'more important'?

 

[pellman]

Demystifier, is the argument about the Born rule similar to what I am saying here https://www.physicsforums.com/showthread.php?t=230032&highlight=many+worlds ?

 

[Fredrik]

If by 'actually happens' you mean 'uniquely predicts a single outcome'

I mean that what the theory says is happening is actually happening. There's no way to define that concept in other terms.

The fact that you're talking about "outcomes" suggests that you're too focused on the results of experiments, or more generally, on the final state after an interaction. A description of "what actually happens" must also tell us how the state changes during any kind of interaction, including measurements. If it doesn't, it's not a realistic model, it's just a theory.

(I can't believe I just used the phrase "just a theory" ).

then (a) it's pretty standard to say that QM doesn't predict a unique standard outcome;

The standard is to talk about the results of experiments and not about the time evolution during a measurement, because all that's required by a theory is that it's able to make accurate predictions about probabilities of possible results.

forget any funny business with cats half dead and alive - just is just what happens in the laws are probabilistic and, again, it's pretty standard to say that QM is indeterministic. This is not anti-realism.

No, but it isn't realism either. A realistic model tells us what happens during a measurement as well as the final state after the measurement.

(b) There's a natural - and as far as I can tell - entirely standard way of understanding \rho\rightarrow \sum_i P_i\rho P_i: it's a description of how the probabiilties will actually evolve in certain conditions.

A realistic model tells us how the system evolves, not just how the probabilities of final results evolves.

Everything hinges on how you physically interpret these mathematical states, how you interpret probabilities, how you interpret the terms that will necessarily appear in probabilistic theories.

I don't think "interpretation of probability" is an idea that even makes sense, as I explained in the other thread. The interpretation of state operators are fixed by the requirement of realism: A state operator is a mathematical representation of the system.

There mere appearance of these terms in the mathematical description of the evolution of the probabilities doesn't get you anything about other the physical reality of worlds corresponding to these terms.

No, but the appearence of those terms in the evolution of the system does.

Actually - I'm still not sure I know which axioms you have in mind when you say Dirac-von Neumann. I tried to follow a previous link, but my Dutch is...ahh... a little rusty. Plus the pages didn't show. Do you have a link where the axioms are given explicitly - sorry if you already provided it somewhere.

That's weird. I did a Google search for "dirac-von neumann axioms" (without the quotes) and clicked the first link in the search results. It took me directly to the right page in the book. Then I copied and pasted the URL that appeared in the address field of the browser. I got a third URL for the page by using the "link" link in the upper right corner, and it worked, but only a couple of times. By the time I had included it here and previewed, it had stopped working. So I suggest you do the Google search yourself, and don't expect the link you find to work more than 2-3 times.

These axioms are the ones that appear in just about any book on QM, but sometimes in a slightly diffferent form. I don't know if many books call them "Dirac-von Neumann axioms" though.

 

[Fredrik]

MWI predicts 2 branches after the decoherence: F and R with surprised R-observer. Why, why F branch is 'more important'?

The MWI doesn't answer that. It just says that it is, by including the Born rule in its definition.

 

[Dmitry67]

Fredrik, I believe we are mixing 2 different things:

1. "Reptrospective" probability (frequentist view) - when I remember the history or look at the log of the experiment with many tries, I see frequent events more often.

2. "Predictive" probability (Bayesian view) - when I am observing Uranium atom, I would be quite surprised if it decays. For some reason "my consicusness" almost always end at one of the probable branches. If definitely ends at ALL branches, but it is stupid to deny that... hm... it does nto affect my life.

So both approaches can be applied to MWI. Discussing Born rule, what exactly are we discussing?

P.S. I always have an impressing that many people are somehow applying the frequentist approach to #2: like, if my consciousness jumps into random branch, it more probably ends at more probable one?

 

[Fredrik]

The definition of science (at least the definition I'm using) requires that we test theories by comparing the probabilities (numbers between 0 and 1) that a theory associates with possible results of experiments, to the relative frequencies of actual results in a large but finite number of actual experiments. This doesn't really have anything to do with interpretations of probability, which I consider meaningless nonsense, but it has a lot to do with the definition of science.

The Born rule assigns numbers between 0 and 1 to results that an observer can experience that he has obtained. It's necessary to say it this way, because in the MWI, an observer who hasn't performed the experiment yet will get all the results, and they will be correlated with different states of "his" memory. Because of that last thing, it's conventional to describe these memory states as properties of different observers.

 

[Dmitry67]

The Born rule assigns numbers between 0 and 1 to results that an observer can experience that he has obtained. It's necessary to say it this way, because in the MWI, an observer who hasn't performed the experiment yet will get all the results

The statement above is a denial of any predictive power of QM. Why are we trying to make devices more robust? Now matter how poor they are constructed, there are always branches where they will work!

We use Born rule for the past and for the future

 

[Fredrik]

Huh? This rule is what turns QM into an actual theory. Without it, it would be a model with nothing that connects it to results of experiments. This rule doesn't remove the predictive power of QM. It is the predictive power of QM.

 

[Dmitry67]

Huh? This rule is what turns QM into an actual theory. Without it, it would be a model with nothing that connects it to results of experiments. This rule doesn't remove the predictive power of QM. It is the predictive power of QM.

Ok, but is it a part of a PHYSICAL theory?

Imagine that our universe in virtual (purely mathematical). We emulate it on super powerful computer. I enter initial conditions, and I get the Universe wavefunction for any given moment. So I get Omnium(t).

Obviously, I don't need Born rule to solve the TOE equations and to calculate Omnium(t). Now, as for me this Universe is virtual I look at it from Gods/Birds view. I say: "ok, it is solved, next one please"

Frog: wait, wait, but what's about the probability?
Bird: what probability?
Frog: what observers see and what that mostly anticipate in the future
Bird: there is no future. These is a static solution. Let me take a project to some basis... yes, in that Universe frogs are talking about some "born rule"
Frog: But are they prepare to the most "probable" events, not to the almost impossible ones?
Bird: Yes, this is because of the natural selection. This is the correlation of their history and their consciousness. It is an illusion of their consciousness, like "flat space", "time flow", "past you can't change and fuzzy future". So it is purely psycological thing. Not physical.

 

[Fredrik]

Ok, but is it a part of a PHYSICAL theory?

I assume that what you mean by "physical theory" is what I've been calling a "realistic model". QM without the Born rule can be interpreted as a realistic model of the universe/omnium, but I'm reluctant to call this an interpretation of QM, because QM without the Born rule isn't QM. It isn't even a theory. It also doesn't tell us which things in the model that we should think of as "worlds" or "experiences".

 

[Dmitry67]

Why? What is wrong with a solution of Omnium(t)?

But I agree, there is some kind of anti-realism, you can not say "what had actually happened" until you specify the basis (decomposition into systems)

 

[Muppetmaster]

I really don't see where the parsimony of Occam's razor comes in here, that's a philosophical axiom at best, certainly it is a generalised trend rule not a rule of science?

Perhaps I'm missing something.

Plus Copenhagen Interpretation (CI) or MWI are indistinguishable from each other except one is deterministic the other probabilistic. So that basically means that MWI loses out as it has nothing to distinguish it from CI?

 

[Fredrik]

Why? What is wrong with a solution of Omnium(t)?

I didn't say that there's something wrong with it. It might very well be the correct objective description of what actually happens, but it still doesn't qualify as a theory if it doesn't tell us how to interpret the mathematics as predictions about the results of experiments. If it isn't a theory, then its description of what actually happens can't be considered an interpretation of a theory.

A different but still important concern is that "QM minus the Born rule" also doesn't tell us how to identify something in the mathematical model (the Hilbert space of the omnium) that we can think of as "worlds" or "experiences".

 

[Dmitry67]

So, what you are saying is "Mathematics is not enough. There must be also a way to translate Birds view into Frogs view". Is this correct?

 

[Fredrik]

Yes. Without that, we have a model that might possibly be correct, but doesn't qualify as a theory and shouldn't be called "many-worlds interpretation" since it doesn't tell us what the worlds are.

 

[Dmitry67]

Ok.
There is only one Bird, but infinitely many Frogs.
So it is required to translate Bird->Frog, then what is a choice of a Frog (preferred basis)?
What kind of basis should be used? All possible? Some subset? Based on what criteria?

 

[dmtr]

Please explain. I have random generator with 2 outcomes: Frequent (probability 99.9999%) and Rare (0.0001%). After a trial I have 2 branches: F and R (with the different 'measure of existence').

The question is, why I almost always end in F branch? That claim is naive from the MWI point of view. But denying it is denying any predictive power of QM.

Because of the symmetry and unitarity. Because of the claim that if the outcomes are symmetric there is an equal probability to see each outcome. And the claim that the sum of probabilities should add to 1.

 

[Dmitry67]

Because of the symmetry and unitarity. Because of the claim that if the outcomes are symmetric there is an equal probability to see each outcome. And the claim that the sum of probabilities should add to 1.

I don't follow your logic.

Or probably you don't realize that the verb "to see" needs additional clarification in the muti-history theory where basic can be define anyway you want.

F and R are symmetric from the 'number of observers', but not symmetric from 'measure of existence' approach.

 

[Fredrik]

Ok.
There is only one Bird, but infinitely many Frogs.
So it is required to translate Bird->Frog, then what is a choice of a Frog (preferred basis)?
What kind of basis should be used? All possible? Some subset? Based on what criteria?

It's not sufficient to just pick a basis. The first thing we need to do is to decompose the omnium/universe into subsystems. Mathematically this corresponds to expressing the Hilbert space as a tensor product of two (or more) Hilbert spaces: \matcal H=\mathcal H_1\otimes\mathcal H_2. Then we can consider bases for those two spaces. Let's say that \{|\psi_\mu\rangle\} is a basis for \mathcal H_1 and that \{|\phi_\alpha\rangle\} is a basis for \mathcal H_2. Then we can define |\psi_\mu,\phi_\alpha\rangle=|\psi_\mu\rangle\otimes|\phi_\alpha\rangle. This definition ensures that \{|\psi_\mu,\phi_\alpha\rangle\} is a basis for \mathcal H.

The question is now, which bases for \mathcal H_1 and \mathcal H_2 should we be using? We can use any pair of bases, but the ones that are of particular interest are the ones that are such that stable records of the system's state (e.g. the memory of having measured spin "up" in a physicist's brain) will exist for some time in the system's environment. I think that decoherence theory singles out a unique pair of bases (or perhaps a class of pairs of bases) that have that property.

The state operator (=density matrix) is almost diagonal in such a basis for \mathcal H. We interpret the terms of the state operator expressed in such a basis as representing "worlds". We can interpret the terms of the state operator expressed in some other basis as worlds too, but we prefer not to call them that, because they don't describe the environment as containing a stable record of the system's state, which means that they can't describe conscious observers with well-defined memory states.

It's possible that I'm way off about something here, because I don't know decoherence very well. Chances are pretty good that if you ask me something about the details of what decoherence theory says, I won't be able to answer.

 

[dmtr]

Or probably you don't realize that the verb "to see" needs additional clarification in the muti-history theory where basic can be define anyway you want.

I do realize that. It is likely that you'll need an additional postulate. Boundary of a boundary principle.

F and R are symmetric from the 'number of observers', but not symmetric from 'measure of existence' approach.

The relevant symmetry is the symmetry of a physical system (like a 50/50 beam splitter), not some imaginary symmetry of some not very well defined quantities.

 

[Count Iblis]

Muppetmaster:

Plus Copenhagen Interpretation (CI) or MWI are indistinguishable from each other except one is deterministic the other probabilistic. So that basically means that MWI loses out as it has nothing to distinguish it from CI?

That's only true in practice, not in principle. And we all know that thought experiments that you can only perform in principle have been very important in theoretical physics.

According to the MWI, it is possible to measure a system (which would irreversibly collapse the wavefunction in the CI), keep a record of the fact that the system has been measured but then erase the result of the measurement by applying the inverse of the unitary transform that describes the measurement process. This will then undo the collapse of the wavefunction. The observer does have the memory of having measured the system, but his memory of the measurement result has been erased. That information has been dumped back on the system itself, allowing the original state of the system to be restored.

The CI also makes a definite prediction for this thought experiment: After the exeriment the system, which was initially in a pure state, will be found in a mixed state. The attempt to restore the original state will fail because only one branch of the wavefunction after measurement really exists. So, the unitary transform describing the "inverse" measurement process only acts on one branch, not on all the branches.

 

[RUTA]

I do realize that. It is likely that you'll need an additional postulate. Boundary of a boundary principle.

Why BBP?

 

[dmtr]

Why BBP?

It's a guess, based on the criteria that the postulate should:
* have something to do with mathematical identity;
* be 'fundamental enough'.

Also BBP is a very safe guess. :)

 

[RUTA]

It's a guess, based on the criteria that the postulate should:
* have something to do with mathematical identity;
* be 'fundamental enough'.

Also BBP is a very safe guess. :)

Are you working on anything relating quantum and BBP? Do you have any papers on this topic? Read 0908.4348 to see why I ask.