Why MWI cannot explain the Born rule

[jensa]

That is odd. I can view it just fine. What happens if you use this link:

http://books.google.com/books?id=6eTHcxeNxdUC&lpg=PP1&pg=PA175#v=onepage&q=&f=false

 

[Demystifier]

That is odd. I can view it just fine. What happens if you use this link:

http://books.google.com/books?id=6eTHcxeNxdUC&lpg=PP1&pg=PA175#v=onepage&q=&f=false

It gives me no more than 101 pages plus the back cover. How many pages can you see?

 

[jensa]

It gives me no more than 101 pages plus the back cover. How many pages can you see?

Oddly I can see pages 134-180 + the first few pages with table of contents etc + backcover. I am a little confused as to how the preview feature works...perhaps its either 101 first pages or part of a chapter of your choosing.

You could try amazon as well, at least I am able to look at that particular section (searching for page 175).

https://www.amazon.com/dp/3540003908/?tag=pfamazon01-20

 

[Demystifier]

Oddly I can see pages 134-180 + the first few pages with table of contents etc + backcover. I am a little confused as to how the preview feature works...perhaps its either 101 first pages or part of a chapter of your choosing.

Perhaps different pages are allowed in different countries?
It is the fact that some web pages have different appearance in different countries. For example, on the top of page of this Forum I often see an advertisement for an IQ test in CROATIAN language (I live in Croatia).

 

[Fredrik]

Google books uses cookies for something. (I'm not sure what). So I would at least try clearing out the cookies from the browser, and also make sure that it allows cookies from that site.

 

[Demystifier]

*) See the book "Decoherence and the appearance of a classical world in quantum theory" by Joos, Zeh, Kiefer, Giulini, Kupsch and Stamatescu in the section entitled "True, False and Fake decoherence". I personally prefer this book over Schlosshauer's.

Now I have noticed that Schlosshauer also explains the notions of true and fake decoherence, but does not mention false decoherence. Could you be so kind to explain to me (by your own words) what false decoherence is and what is the difference between false decoherence and fake decoherence?

 

[jensa]

Now I have noticed that Schlosshauer also explains the notions of true and fake decoherence, but does not mention false decoherence. Could you be so kind to explain to me (by your own words) what false decoherence is and what is the difference between false decoherence and fake decoherence?

Well, I hope I didn't give the impression that I am some kind of expert in decoherence, I'm certainly not. But I can try to explain what I understand from the the book. Joos describes false decoherence as when "Coherence is trivially lost if one of the required components [of the wavefunction] disappears". I'm not sure really how to interpret this sentence, but he gives two examples of such cases 1) Relaxation 2) When we are considering only a subset of all available states.

Let's start with what I think is the easiest one no. 2). It is sometimes convenient to use a truncated model to describe a system. For example, many qubit implementations are in fact multilevel systems although we choose only to consider the two lowest energy levels as an operational subspace. It is clear that there is a certain probability of "leakage" out of this subspace, this process also produces suppression of the off-diagonal elements of our density matrix, however it also does not conserve probability (within the subspace). Coherence is not really lost here, it's just a consequence of our truncated model.

As for Relaxation, it should originate from interaction with an environment (since energy is not conserved). So I suppose the point he is trying to make is that interaction with environment can produce "true decoherence" which is a pure quantum effect, as well as an exchange of energy which also occurs for classical systems. Relaxation, although it also suppresses the off-diagonal elements of the density matrix, should be distinguished from "true decoherence".



If you would like, I could scan the pages and send them to you. Would be nice to see what somebody else makes of it.

 

[Demystifier]

Thanks jensa, it was a quite clear explanation.

 

[Dmitry67]

Here is a radical view. What is Born rule does not need to be derived from MWI? What if Born rule is not objective but subjective?

Usually we look it from the inside (frogs perspective): I am looking at the world and I observe more frequent event than events with lower probability. How is it explained by MWI, while all possible outcomes exist in omnium? I think that this question is incorrect. We should ask an inversed question instead: We know that all branches exist, why there is “more” conscious observers with the higher “intensity of existence?”

I can give you another example. Say, we have a very powerful computer and we know TOE. So we were able to calculate the “wavefunction of the Universe” from moment 0 (initial conditions at BB) to, say, 1 second. Not just one “branch”, but omnium at whole, the global solution. There are no conscious observers, however. But do we need a Born rule as soon as we know the global solution?

In fact, once you look at the Universe from ‘bird’s view’, having the solution of the evolution of the omnium, there is no Born rule. We have a solution, that’s it. Born rule appear when we try to jump inside the Universe, converting from a single birds view to some arbitrary frog’s view. At the moment of the choice you already implicitly use the Born rule, choosing “more intensive” branch. In such case the Born rule – which is so obvious when we look around – is nothing more then an illusion, created by our consciousness, similar to so obvious flow of time

 

[Demystifier]

Dmitry, I don't see that this view explains the Born rule in any quantitative sense.

 

[Dmitry67]

Well, my idea was that it Born rule is just an illusion created by our consiousness.
It is a part of theory of consciousness, not a part of physics.

 

[Dmitry67]

Suppose, there are 2 outcomes of some experiment: (F)requent (90%) and (R)are 10%
We repeat this experiment 3 times.
Obviously, there are 8 branches, begiining from F-F-F to R-R-R
Why do we see F events more often then R events?

Check the picture.
There is no difference between F-F-F and R-R-R at the birds view.
In the frog's view (asking "what observer will see?") you must define the basis (a point on my diagram). So we chose (for some mysterious reason, I agree) the point showed by red cross and ask: why in the history of that observer there are more Fs then Rs?

So Born rule is not about physics, it is about how consicous observers are chosen.

 

[pellman]

So we chose (for some mysterious reason, I agree) the point showed by red cross and ask: why in the history of that observer there are more Fs then Rs?

The "mysterious reason" is the crux of the whole question.

Although the question "why in the history of that observer there are more Fs then Rs?" is somewhat troublesome. There are 8 potential observers. One of them sees F-F-F. And we chose to label that one with the X.

But what is special about that observer?

 

[Count Iblis]

In the bird's view there is one timeless multiverse. What we experience as time evolution is just a unitary mapping from one sector of the multiverse to another. At the level of the whole multiverse, time does not exist.

In this case you have some sector containing the initial observer |i>, which maps under the time evolution operator to a |F> + b |R>. It is this time evolution operator which is subjective to the observer. In the bird's view time does not exist and you could have considered any other unitary operator.

For the observer things are different. An observer is essentially an algorithm that processes information. To define the algorithm as it is in any particular computational state, you have to list the outputs for each possible input. This defines an operator which is essentially a sort of coarse grained Hamiltonian which generates the time evolution experienced by the observer.

You cannot just define an observer by looking at a sequence of the states it runs through, because you can then map that to the states of a clock which does not execute a nontrivial algorithm. You have include the information about counterfactual inputs and the corresponding counterfactual outputs, which will not be mapped correctly from the observer to the clock.

Note that the information about the correct Hamiltonian is not present in the bird's view, because if H is the "correct" Hamiltonian, the whole multiverse satisfies an equation like:

H|psi> = 0

There is then no way you can construct H back from|psi>. You can replace H by any other Hamiltonian that except for |psi> has different eigenvectors.

 

[Dmitry67]

The "mysterious reason" is the crux of the whole question.

Although the question "why in the history of that observer there are more Fs then Rs?" is somewhat troublesome. There are 8 potential observers. One of them sees F-F-F. And we chose to label that one with the X.

But what is special about that observer?

I can provide a part of an answer by giving another example:

the moment of "NOW"

This is also a big red cross, a moment very special to consciousness. But as we believe in block time, physically there is absolutely nothing special about NOW!

So if our consicouness can break symmetry on T axis, creating an illusion of a special moment 'NOW', then why it can't create another illusion on B-axis (Branch axis. Yes, I am aware that it is not an axis at all)

 

[Dmitry67]

You cannot just define an observer by looking at a sequence of the states it runs through, because you can then map that to the states of a clock which does not execute a nontrivial algorithm. You have include the information about counterfactual inputs and the corresponding counterfactual outputs, which will not be mapped correctly from the observer to the clock.

This sounds very interesting but I am lost at the part I quoted
Could you elaborate?

 

[Demystifier]

Well, my idea was that it Born rule is just an illusion created by our consiousness.
It is a part of theory of consciousness, not a part of physics.

Does this mean that such a theory does not need to have a quantitative form?

 

[Dmitry67]

Yes, but we can use Born rule for practical purposes. As a rule of thumb

For example, NOW we are 13.7 billion years from the BB, or 8.2*10^60 plank times from the Big Bang

Can you derive somehow this integer dimensionless number (8.2*10^60) from BM?

 

[Demystifier]

You can use the Born rule as a rule of thumb.
However, if it is the best that MWI can do about the Born rule, then BM is much more successfull than MWI because in BM the Born rule is much more than a rule of thumb.

 

[Demystifier]

Can you derive somehow this integer dimensionless number (8.2*10^60) from BM?

Of course I can't. If you give me a theory that can, I will accept that theory immediately. Moreover, if that theory turns out to be incompatible with BM, I will reject BM as well. But in the meantime, I will keep BM as the most attractive possibility currently known.

 

[Dmitry67]

You can use the Born rule as a rule of thumb.
However, if it is the best that MWI can do about the Born rule, then BM is much more successfull than MWI because in BM the Born rule is much more than a rule of thumb.

Yes, there is some advantage, but it is too weak.
Observer is point is some space, and history is a curve.
While BM limits the number of curves to 1, it does not limit the curve to a single point. So I think the claim 'MWI does not explain what is observed now' is to the full extent applicable to BM. It just slightly limits the number of degrees of freedom, where we can put that red cross.

 

[Dmitry67]

Of course I can't. If you give me a theory that can, I will accept that theory immediately. Moreover, if that theory turns out to be incompatible with BM, I will reject BM as well. But in the meantime, I will keep BM as the most attractive possibility currently known.

Hm. Do you think that symmetry breaking (NOW vs not-NOW) must be explained by physical theiry? It sounds very Smolin-like.

 

[Dmitry67]

Note that the information about the correct Hamiltonian is not present in the bird's view, because if H is the "correct" Hamiltonian, the whole multiverse satisfies an equation like:

So are you saying that some properties visible in the frog's view can't be derived mathematically, even in principle, from birds view? So even when we have an ultimate TOE equation, we can't explain everyhting we observe?

 

[Demystifier]

Hm. Do you think that symmetry breaking (NOW vs not-NOW) must be explained by physical theiry?

Maybe yes, maybe not.

 

[toho]

Regarding deriving the Born rule from the MWI, here is a conjecture:

The Born rule is the only probability measure, Q, consistent with the criteria that
i) P(A)=0 iff L2-norm of psi over A = 0, for every A in the limit as time -> infinity
ii) Q is a function of psi, for any psi consistent with QM

I.e. if you want a probabilistic interpretation of the MWI interpretation of QM, it has got to be the Born rule. (It think it would be possible to relax the t->inf critera, btw)

 

[Dmitry67]

I.e. if you want a probabilistic interpretation of the MWI interpretation of QM, it has got to be the Born rule. (It think it would be possible to relax the t->inf critera, btw)

Yes, if you want to use MWI practically, you need to add Born rule as new axiom
That axiom is not purely mathematical (as MWI is deterministic and does not know the word "probability")
Also, sometimes Born rule is violated (Anthropic principle)

On the deeper level Born rule can not and should not be explained by MWI, but rather by a future theory of conciousness.

 

[Count Iblis]

So are you saying that some properties visible in the frog's view can't be derived mathematically, even in principle, from birds view? So even when we have an ultimate TOE equation, we can't explain everyhting we observe?

I believe you can explain in principle everything, however the predictions may be dependent on the specification of the frog.

 

[toho]

Yes, if you want to use MWI practically, you need to add Born rule as new axiom

On the deeper level Born rule can not and should not be explained by MWI, but rather by a future theory of conciousness.

No, my point is that you don't need to add the Born rule as a new axiom. (Provided that the conjecture is correct, which I am pretty sure it is. You do have to accept the axioms of probability theory, though.) The Born rule is the only sensible probabilistic interpretation of the wave function.

I don't think the Born rule has anything to do with conciousness at all. If you accept MWI, you will also have to accept that there are multiple versions of your conciousness in orthogonal branches of the universe.

A probabilistic interpretation is indeed practical in many situations, analogous to how a probablistic interpretation of a determistic but unpredictable outcome, such as a coin flip, can be practical.

(I edited the conjecture after your posting, by the way. It wasn't very clear as originally stated.)

 

[Dmitry67]

Ok, so you have an explanation, so I will challenge you.

What is Born rule in MWI? MWI is deterministic. So Born rule in MWI is not about what we see, but it is about how we chose the preferred basis for the consciousness In another words, it is about brid->frog transition, or about how particular frog is chosen

As "measure of existences" never fades to 0, there are all sorts of weird branches where all sorts of weird things happen (and one of such things is life). We should see all of them, then why FAPP we expect to see frequent events?

So I ask clarification on your waords about how we can use a probability interpretation in MWI. I think the root of the problem is there.

 

[Dmitry67]

P.S.
I can't find it right now but I am sure I have seen it somewhere, some form of "weak" Born rule: so if we assume that probability is some function of wavefunction, then we can derive the Born rule.

 

[toho]

Ok, so you have an explanation, so I will challenge you.

What is Born rule in MWI? MWI is deterministic. So Born rule in MWI is not about what we see, but it is about how we chose the preferred basis for the consciousness In another words, it is about brid->frog transition, or about how particular frog is chosen

Well, the MWI is only deterministic to an outside observer who is able to observe the wave function. An observer - a brain - in the universe is described by the wave function just as everything else. The state of that brain can be entangled with the outcome of an event, just as two spins can be entangled.

As "measure of existences" never fades to 0, there are all sorts of weird branches where all sorts of weird things happen (and one of such things is life). We should see all of them, then why FAPP we expect to see frequent events?

So I ask clarification on your waords about how we can use a probability interpretation in MWI. I think the root of the problem is there.

My point is, according to the MWI, our perception of the outcome of a future event is uncertain. If we want to use probabilities to describe the likelihood of (our perception of) the outcome, there is only one sensible probability measure (i.e. there is only one way to assign the probabilities). That is the Born rule. If I understand your interpretation of MWI, you believe that it is possible to assign probabilities in a way so that events that are exceedingly rare according to the Born rule will have much larger probabilities. I don't think that is possible to do in a systematic way without violating the axioms of probability.

P.S.
I can't find it right now but I am sure I have seen it somewhere, some form of "weak" Born rule: so if we assume that probability is some function of wavefunction, then we can derive the Born rule.

That is essentially what I am saying (you would need regularity conditions, I guess). But I don't see it as a weak form of the Born rule. If the universe is uniquely determined by the wave function, then any probability measure must surely be derived from the wave function as well. Otherwise there is additional information outside of the wave function.

 

[Dmitry67]

1
Well, the MWI is only deterministic to an outside observer who is able to observe the wave function. An observer - a brain - in the universe is described by the wave function just as everything else. The state of that brain can be entangled with the outcome of an event, just as two spins can be entangled.

2
My point is, according to the MWI, our perception of the outcome of a future event is uncertain. ... That is the Born rule.

3
If I understand your interpretation of MWI, you believe that it is possible to assign probabilities in a way so that events that are exceedingly rare according to the Born rule will have much larger probabilities. I don't think that is possible to do in a systematic way without violating the axioms of probability.

4
That is essentially what I am saying (you would need regularity conditions, I guess). But I don't see it as a weak form of the Born rule. If the universe is uniquely determined by the wave function, then any probability measure must surely be derived from the wave function as well. Otherwise there is additional information outside of the wave function.

1 Yes, so called "birds view" (c) Max Tegmark

2 Well, let's forget about our expectations of the future. How do you explain Born rule as statistics of the past events? Say, I have a substance with half decay time of 1 minute. I take 1'000 atoms and wait 1 minute. 500 (or 498 or 501 atoms) decay. But it is very unlikely the ALL of them decayed or NONE of them.

But in MWI where are 2**1000 branches (if we assign number to every atom and register everything) and all of them equally valid.

So how do you explain that?
So let's talk about frequentist probability, not ignorance (bayesian) probability

3 In general, no. With some exceptions (I can give clarifications)

4 Yes, I believe it is called officially "measure of existence". I agree on that, so letas return to the main issue - issue #2

BTW I found the article I menationed:
http://plato.stanford.edu/entries/qm-manyworlds/#6.3

What is true instead is that one can derive the Probability Postulate from a weaker postulate according to which the probability is a function of the measure of existence. The derivation can be based on Gleason's 1957 theorem about the uniqueness of the probability measure. Similar results can be achieved by the analysis of the frequency operator originated by Hartle 1968 and from more general arguments by Deutsch 1999

 

[toho]

2 Well, let's forget about our expectations of the future. How do you explain Born rule as statistics of the past events? Say, I have a substance with half decay time of 1 minute. I take 1'000 atoms and wait 1 minute. 500 (or 498 or 501 atoms) decay. But it is very unlikely the ALL of them decayed or NONE of them.

But in MWI where are 2**1000 branches (if we assign number to every atom and register everything) and all of them equally valid.

So how do you explain that?
So let's talk about frequentist probability, not ignorance (bayesian) probability

Explain what exactly? You have 2^1000 branches. In any probabilistic interpretation of physics where you would assign an equal probability to each sequence, then the sequence that you actually observed had a probability of 1/2^1000. The same very low probability as that of observing 1000 decays in a row. Any outcome observed will have been exceedingly improbable before the fact.

As soon as you start computing statistics you are in probability land, and then there is only one way to assign probabilities in MWI that makes sense - Born. Any other rule for assigning probabilities is unphysical in the sense that it would place you - your conscious, your brain - in a favoured position in the universe. If the same rule for assigning probabilities shall apply to the entire universe, then Born it is.

Of course, you can drop the probabilistic interpretation, but then the problem goes away. You just accept that you are in a branch that has 501 decays, conclude that it is consistent with QM and MWI and go to the beach.

 

[toho]

BTW I found the article I menationed:
http://plato.stanford.edu/entries/qm-manyworlds/#6.3

Thanks for the link. I will read as soon as I find time.

 

[Dmitry67]

Explain what exactly? You have 2^1000 branches. In any probabilistic interpretation of physics where you would assign an equal probability to each sequence, then the sequence that you actually observed had a probability of 1/2^1000. The same very low probability as that of observing 1000 decays in a row. Any outcome observed will have been exceedingly improbable before the fact.

As soon as you start computing statistics you are in probability land, and then there is only one way to assign probabilities in MWI that makes sense - Born. Any other rule for assigning probabilities is unphysical in the sense that it would place you - your conscious, your brain - in a favoured position in the universe. If the same rule for assigning probabilities shall apply to the entire universe, then Born it is.

So we agreed that we can't use frequentists approach over the set of branches, because it gives incorrect results (like having 50% change to win in any lottery because there are 2 branches - where I win and where I dont)

But the part marked Bold looks as axiom. I agree with that sentence (so we have almost agreed), but you are just introducing a Born rule as axiom. FAPP it works, but I just wanted to dig deeper, so let me play devils advocate.

Returning to the toy example I provided many pages ago. There are 2 outcomes Frequent (90%) and Rare (10%). I do 3 tries, I get 8 branches from FFF (72.7 %) to RRR (0.1%).

Now we are both Gods, using the Bird's view, looking at the wavefunction from the outside. As God, I also have a magic monitor: I doubleclick on the desired branch and monitor shows how that branch looks like. I click on RRR and I hear the words of experimenter "What the hell? Is it mulfunctioning?" I click on FRF and I hear "As expected... Boooring..."

Now I ask - what does the probability means from that point of view? Can "Gods" talk about the Born rule?

 

[toho]

But the part marked Bold looks as axiom. I agree with that sentence (so we have almost agreed), but you are just introducing a Born rule as axiom.

No, I am not introducing it as an axiom. I mean that sentence in the context of the conjecture before (and you provided a link that says there is indeed a theorem saying more or less the same thing). There is only one set of probabilities that are sensible (in a mathematically precise way). That happens to be the Born rule. Thus, there is a way to derive the Born rule from first principles. You may (probably will) need more axioms than the two posted at the beginning of this thread, or at least you would need to make them more precise, but you don't need to postulate the Born rule. It can be derived.

You can use probability theory whenever there is incomplete information, but the result of your calculations depend on the probability measure you are using. Usually, there is no favoured probability measure, such as when calculating the probability of future stock market moves. Sometimes you can use statistics to fix probabilities. Sometimes you can use symmetry arguments to fix probabilities, and that allows you to make much more precise calculations. But in MWI there is no freedom with regards to the probability measure.

Returning to the toy example I provided many pages ago. There are 2 outcomes Frequent (90%) and Rare (10%). I do 3 tries, I get 8 branches from FFF (72.7 %) to RRR (0.1%).

Now we are both Gods, using the Bird's view, looking at the wavefunction from the outside. As God, I also have a magic monitor: I doubleclick on the desired branch and monitor shows how that branch looks like. I click on RRR and I hear the words of experimenter "What the hell? Is it mulfunctioning?" I click on FRF and I hear "As expected... Boooring..."

Now I ask - what does the probability means from that point of view? Can "Gods" talk about the Born rule?

Using probability is a choice. It is a way to mathematically treat a system where you don't have complete information. So, sure, we as Gods can use probability theory, if only to calculate probabilities of outcomes that the actors in the world we are observing should reasonably expect with the information available to them.

Because of Bell's theorem, I guess many physicists tend to look at probability in QM as something fundamentally different from probability in deterministic games of chance with incomplete information, such as coin flipping. But I think that is really just a mystics position. Probability is probability. It is a way to calculate, and it follows from a few simple axioms. Nothing more, nothing less.

 

[Dmitry67]

Well, I tried to explain but apparently my explanation was confusing.

So let me continue playing the devil's advocate, to take my logic to extreme and to deny the Born rule. In the toy example above, I claim that F and R have the same probability. Here is a logic:

There are 8 branches: RRR, RRF, RFR, RFF, FRR, FRF, FFR, FFF
I incarnate myself in all of them and count the number of times I see F and R
I get 12 R and 12 F. Hence I have 50% chance to observe R and F.
Prove that I am wrong :)

Now you say: but wait, more often we appear in the more probable branches, like FFFFRFFFFRFFFFF..., not like RRRRRRRRRRRRRR... So this is what you do: you get all branches, then you prepare some artificial subset of them based on the Born rule (thinking: I have more chances to appear in the ..FFFF.. branch), then you say: look, the number of Fs and Rs obey the Borns rule! So it is cyclical.

Born rule is encoded not when you count the number of F's and R's in some branch, but when you select that branch.

 

[toho]

So let me continue playing the devil's advocate, to take my logic to extreme and to deny the Born rule. In the toy example above, I claim that F and R have the same probability. Here is a logic:

There are 8 branches: RRR, RRF, RFR, RFF, FRR, FRF, FFR, FFF
I incarnate myself in all of them and count the number of times I see F and R
I get 12 R and 12 F. Hence I have 50% chance to observe R and F.
Prove that I am wrong :)

You have not created a probability measure over the whole state space of the universe. If you accept my conjecture, then you can not do that and get equal chances for F And R.

Now you say: but wait, more often we appear in the more probable branches, like FFFFRFFFFRFFFFF..., not like RRRRRRRRRRRRRR...

No, that is not quite what I am saying. All I am saying is that you will compute a higher probability value for some branches when you apply mathematics correctly. I am not saying anything about more often, whatever that is supposed to mean.

So this is what you do: you get all branches, then you prepare some artificial subset of them based on the Born rule (thinking: I have more chances to appear in the ..FFFF.. branch), then you say: look, the number of Fs and Rs obey the Borns rule! So it is cyclical.

I have now read throgh the document that you kindly provided a link to before. I think I maybe understand what you are saying a bit better now. But I think that the problems that are described in section 4 of that document, are really misunderstandings rooted in semantics. The same or at least a very similar problem plagued the mathematics of probability for a long time before the current system of axioms. The author of that document argues that there are different kinds of probability, ignorance probability and (I guess) some kind of real or objective probability. But those are semantic concepts. Mathematically, a probability is just a value you get when you perform a certain type of calculation. If you would argue that probability is really something else or something more, then you would have to provide a new set of axioms to describe that something, otherwise it is just metaphysics.

If you like, you can choose to view MWI in a probabilistic manner. IF you do that THEN you can mechanically compute probabilities. Those probabilities are unique, and those are the only set of probabilities that make sense when trying to predict the future in the probabilistic framework.

Alternatively, you can choose not to use probability. Then QM and MWI will only tell you what is possible. As far as I am concerned, that is a completely valid interpretation. You and I both exist in branches where (QM based) statistics seems to have made sense, based on our experience. But there are also weird branches where statistics does not seem to have made sense, based on the experience in those branches. According to this non-probabilistic interpretation of MWI, those weird branches are no less real or likely than our own.

But what does not make sense IMHO is to first choose the non-probabilistic interpretation, and then try to use probability theory to try to prove or disprove things.

 

[Dmitry67]

1
No, that is not quite what I am saying. All I am saying is that you will compute a higher probability value for some branches when you apply mathematics correctly. I am not saying anything about more often, whatever that is supposed to mean.

2
The author of that document argues that there are different kinds of probability, ignorance probability and (I guess) some kind of real or objective probability. But those are semantic concepts. Mathematically, a probability is just a value you get when you perform a certain type of calculation. If you would argue that probability is really something else or something more, then you would have to provide a new set of axioms to describe that something, otherwise it is just metaphysics.

1 I agree that for FFFF some value is higher. But why do you call this value "probability"? So I see tha gap: you calculate some numbers, for F it is higher then for R, cool. hen you say: "So the probability of F..." wait, wait... What probability? (check #2)

2 No. There are 2 fundamentally different approaches to the probablility: on the phylosophical level bayesian vs frequentist, with 2 different formalisms and sets of axioms: Cox and Kolmogorov (Cox = Bayesian and Kolmogorov = Frequentist)

So when you say "probability" in MWI, whay approach do you use?

 

[Dmitry67]

... wanted to add:

The problem as I see is:
1. If you use the Kolmogorov probability, then you get incorrect result (counting Fs and Rs you get the same number)
2. You can't use Bayesian probability because MWI is deterministic.

But

1. You can use Kolmogorov theory if you make an adjustment: values must be normalized (multiplied to) "measure of existence". So instead of count of occurances of Fs and Rs you get a weighted sum. But it is a new axiom.
2. You can use Bayesian view (Cox axioms) adding an axiom that "Cox plausibility" = "measure of existence". But again, it is a new axiom.

 

[toho]

... wanted to add:

The problem as I see is:
1. If you use the Kolmogorov probability, then you get incorrect result (counting Fs and Rs you get the same number)
2. You can't use Bayesian probability because MWI is deterministic.

But

1. You can use Kolmogorov theory if you make an adjustment: values must be normalized (multiplied to) "measure of existence". So instead of count of occurances of Fs and Rs you get a weighted sum. But it is a new axiom.
2. You can use Bayesian view (Cox axioms) adding an axiom that "Cox plausibility" = "measure of existence". But again, it is a new axiom.

I am not really familiar with the Cox axioms. In the Kolmogorov framework, calculated probabilities are explicitly dependent on a probability measure. A Kolmogorov probability is not the same as just counting frequencies.

In the document you linked to, one such probability measure is called "measure of existence" (if I understand it correctly). However, in general there may be more than one probability measure, and each new measure may give different probabilities. But I believe that in MWI, the probability measure is unique, given some reasonable assumptions. That is the gist of my conjecture. There seems to be some support for that argument in the document you linked to.

1 I agree that for FFFF some value is higher. But why do you call this value "probability"?

Because it is a probability. It is equal to P(FFFF) for some probability measure P. In addition, P happens to be unique in some sense.

Again, everything I say is dependent on my conjecture being true - which of course you are free to doubt, since I can't prove it (but at the very least a similar theorem is true as described in section 6.3). However, if you just assign an arbitrary probability 0.5 to an experiment (such as having observed an atom decay after time T) where the probability according to the Born rule is, say 0.9, then at least you will get some rather strange phenomena at times before T, such as having the probability of having observed a decay go down with time (or alternatively, having the atom "undecay" with positive probability). But I can't make a stringent argument to prove that a probability measure with those properties does not exist, in the sense that it can not be systematically created from the wave function.

 

[Dmitry67]

However, if you just assign an arbitrary probability 0.5 to an experiment (such as having observed an atom decay after time T) where the probability according to the Born rule is, say 0.9, then at least you will get some rather strange phenomena at times before T, such as having the probability of having observed a decay go down with time (or alternatively, having the atom "undecay" with positive probability). But I can't make a stringent argument to prove that a probability measure with those properties does not exist, in the sense that it can not be systematically created from the wave function.

Yes, and I do observe such weird behavior - in many "weird" branches.

Of course, normal branches are more, say, self-consisent. In some sense, they have minimum "magic".

But why do we care about how Born rule is respected in "normal" branches, but don't care about "waird" ones?

Imagine an example: a branch where mean lifetime of Uranium is not 4by but 2by. For all atoms. By pure accident. So scientists there routinely measure higher decay rate, but they fail to explain it using QM.

Or what if life is extremely rare event, so there is a huge gap between primitive molecules and DNA, which can be created only by pure chance, so all we are on such weird branch?

 

[toho]

So let me continue playing the devil's advocate, to take my logic to extreme and to deny the Born rule. In the toy example above, I claim that F and R have the same probability. Here is a logic:

There are 8 branches: RRR, RRF, RFR, RFF, FRR, FRF, FFR, FFF
I incarnate myself in all of them and count the number of times I see F and R
I get 12 R and 12 F. Hence I have 50% chance to observe R and F.
Prove that I am wrong :)

I will try this again. I will try to be more precise, because we seem to be talking past each other. Let me give you the outline of a proof:
Assume that F and R have equal probabilities according to a probability measure P, different from the Born rule probability measure Q.
Then in the limit when the number of experiments -> infinity all branches will have equal number of F and R with probability 1 (#F / #R -> 1 P almost surely). However, the wave function will -> 0 if you sum over all branches with (almost) equal F and R.
There will also be other branches with different proportion of Fs and Rs for which the sum of the wave function -> 0 in the limit. The union of those branches have P-probability 0 in the limit.
Thus, assigning equal probabilities to F and R (or indeed any other probability than the Born rule probability Q) is not consistent with the probability measure P being derived from (the value of) the wave function in the limit as time -> infinity.

 

[Demystifier]

Regarding deriving the Born rule from the MWI, here is a conjecture:

The Born rule is the only probability measure, Q, consistent with the criteria that
i) P(A)=0 iff L2-norm of psi over A = 0, for every A in the limit as time -> infinity
ii) Q is a function of psi, for any psi consistent with QM

I.e. if you want a probabilistic interpretation of the MWI interpretation of QM, it has got to be the Born rule. (It think it would be possible to relax the t->inf critera, btw)

Let me try to challenge this conjecture by proposing a probabilistic interpretation that seemingly satisfies i) and ii), but is not the Born rule:

Due to decoherence, the wave function psi develops branches that (at t->inf) do not overlap in the configuration space. Let us postulate that each branch has the same probability. Obviously, it satisfies ii) because the notion of branches is defined by psi. It also satisfies i) because a branch with zero norm has a vanishing wave function, so it is not a branch at all. Still, the Born rule is not satisfied because different branches may have different (larger than zero) norms, but they all have the same probability.

Why nature could not take this probabilistic rule?

 

[Dmitry67]

I will try this again. I will try to be more precise, because we seem to be talking past each other. Let me give you the outline of a proof:
Assume that F and R have equal probabilities according to a probability measure P, different from the Born rule probability measure Q.
Then in the limit when the number of experiments -> infinity all branches will have equal number of F and R with probability 1 (#F / #R -> 1 P almost surely). However, the wave function will -> 0 if you sum over all branches with (almost) equal F and R.
There will also be other branches with different proportion of Fs and Rs for which the sum of the wave function -> 0 in the limit. The union of those branches have P-probability 0 in the limit.
Thus, assigning equal probabilities to F and R (or indeed any other probability than the Born rule probability Q) is not consistent with the probability measure P being derived from (the value of) the wave function in the limit as time -> infinity.

Got it.
Let me think about your idea...

 

[toho]

Let me try to challenge this conjecture by proposing a probabilistic interpretation that seemingly satisfies i) and ii), but is not the Born rule:

Due to decoherence, the wave function psi develops branches that (at t->inf) do not overlap in the configuration space. Let us postulate that each branch has the same probability. Obviously, it satisfies ii) because the notion of branches is defined by psi. It also satisfies i) because a branch with zero norm has a vanishing wave function, so it is not a branch at all. Still, the Born rule is not satisfied because different branches may have different (larger then zero) norms, but they all have the same probability.

Why nature could not take this probabilistic rule?

I am not sure I follow your argument. However, there is an infinite number of branches, and any individual branch will have probability 0 in the limit. You will have to think in (unions of) countable sequences of branches, and show that i) and ii) is true for each such countable sequence.

 

[Demystifier]

I am not sure I follow your argument. However, there is an infinite number of branches, and any individual branch will have probability 0 in the limit. You will have to think in (unions of) countable sequences of branches, and show that i) and ii) is true for each such countable sequence.

You are making good points. But let me improve my proposal. After a finite time, a finite number of branches develops the overlap between which is smaller then some fixed positive but small number epsilon. (This definition of a branch is somewhat vague, but there is a way to define it in a more precise, even if artificial, way.) Then I propose that each branch has the same probability. I think it is obvious that i) and ii) are satisfied, but that the Born rule is not. What do you think?

By the way, in your axiom ii), do you require that the probability is a function of psi ONLY? Or do you allow that it may depend also on something else?

 

[Demystifier]

Regarding deriving the Born rule from the MWI, here is a conjecture:

The Born rule is the only probability measure, Q, consistent with the criteria that
i) P(A)=0 iff L2-norm of psi over A = 0, for every A in the limit as time -> infinity
ii) Q is a function of psi, for any psi consistent with QM

I.e. if you want a probabilistic interpretation of the MWI interpretation of QM, it has got to be the Born rule. (It think it would be possible to relax the t->inf critera, btw)

Another challenge. Let P(x) be some probability density that satisfies the axioms above. For definiteness, let us take it to be the Born rule. Then consider a DIFFERENT probability density
P'(x)=\frac{P^2(x)}{\int dx' P^2(x')}
Clearly, P' is not the the Born rule. Yet, P' also satisfies i) and ii).

Or, if P' does not really satisfy i) and ii), then it can only be due to the fact that i) insists on L2-norm, and not on some other norm such as L4. But then the natural questions is: Why one should insist on L2-norm and not some other norm? Isn't the requirement of L2-norm rather than some other norm actually a hidden ASSUMPTION of the Born rule? If so, can such a "derivation" of the Born rule really be considered an explanation of the Born rule?

 

[toho]

You are making good points. But let me improve my proposal. After a finite time, a finite number of branches develops the overlap between which is smaller then some fixed positive but small number epsilon. (This definition of a branch is somewhat vague, but there is a way to define it in a more precise, even if artificial, way.) Then I propose that each branch has the same probability. I think it is obvious that i) and ii) are satisfied, but that the Born rule is not. What do you think?

By the way, in your axiom ii), do you require that the probability is a function of psi ONLY? Or do you allow that it may depend also on something else?

i) is still not satisfied in the limit as t->inf for any A (e.g. any countable sequence of branches), as far as I can see. If the probability measure is a function of psi (and not time) then your method of assigning probabilities must hold for any point in time.

Secondly, it is really not enough to assign a probability to each branch. We have to assign a probability to each testable outcome at each point in time. This makes a difference, at least if we have a finite number of branches.

The rationale of ii) is that every feature of the universe should be described by psi, so I guess no. I am however, not very precise, and that is probably because I don't really know exactly what I mean.

 

[toho]

Another challenge. Let P(x) be some probability density that satisfies the axioms above. For definiteness, let us take it to be the Born rule. Then consider a DIFFERENT probability density
P'(x)=\frac{P^2(x)}{\int dx' P^2(x')}
Clearly, P' is not the the Born rule. Yet, P' also satisfies i) and ii).

It will not necessarily satisfy i) in the limit, i.e. you can find an A for which P(A) -> p, p>0, while P'(A) -> 0.