Why MWI cannot explain the Born rule

[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.

 

[Demystifier]

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.

It must 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]

It must 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?

Well, the fact that P and P' have measure 0 for different sets (i.e. they are not what I believe is called equivalent) is not a consequence of the choice of L2-norm. However, the fact that P is preferrable over P' obviously is.

But, the L2-norm is natural as it is defined by the inner product, and we do have an inner product space.

 

[Demystifier]

But, the L2-norm is natural as it is defined by the inner product, and we do have an inner product space.

OK, but why this natural norm should have anything to do with probability? For example, the linear wave equation describing water waves also shares the same mathematical structure, including the inner product of solutions, yet this inner product of water wave solutions has nothing to do with probability.

 

[toho]

OK, but why this natural norm should have anything to do with probability? For example, the linear wave equation describing water waves also shares the same mathematical structure, including the inner product of solutions, yet this inner product of water wave solutions has nothing to do with probability.

The water wave is deterministic to an outside observer of the wave, just as the wave function of MWI would be deterministic to an outside observer. However, to an observer inside a universe according to MWI the future (as well as the past, btw) is indeed unpredictable. This is (as I understand MWI) because of entanglement of the brain of the observer with the state of the universe.

Now, the choice to use probability is indeed a choice (in my opinion), but if you do choose to use probability, I argue that there is only one probability measure that works. Why then would you expect or want the probability to be zero when the norm of the wave function is zero? Well, when the norm of the wave function is zero the wave function contains no information, i.e. it can't possibly describe the universe in those branches, or for those events (or rather, for projections of the wave function down to those sub-spaces). On the other hand, if the norm is not zero, (the projection of) the wave function will influence the future of the universe, so you don't want the probability to be zero.

(Just a further point with regards to the water wave analogy. Imagine that you are not an outside observer of the wave, but a perfect sine wave propagating in an idealised euclidean sea. Your perception of the sea is determined by your interaction with the rest of the waves in the sea. But since you are orthogonal to most of them you will not perceive much of the other waves. Despite the fact that you as a sine wave "are everywhere", you will not have information to predict the sea.)

 

[Demystifier]

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.

My intuition tells me that examples supporting this claim have a rather pathological form and do not correspond to cases of physical interest in practice. If I am right, then the axiom i) makes sense only if it is a fundamental axiom, not only a rule relevant in practical applications. But then MWI is FUNDAMENTALLY a probabilistic theory.

 

[toho]

My intuition tells me that examples supporting this claim have a rather pathological form and do not correspond to cases of physical interest in practice. If I am right, then the axiom i) makes sense only if it is a fundamental axiom, not only a rule relevant in practical applications. But then MWI is FUNDAMENTALLY a probabilistic theory.

I beg to differ. Let me give you an example:
Let the universe at any time have N+1 branches, where N is an unbounded, increasing function of time. Let one branch, called A, have a P-probability of 0.5. Furthermore, let each of the remaining branches (the union of which we call B) each have a P-probability of 0.5/N. Then P(B) = 0.5. Let's calculate:
P'(A) = 0.25/(0.25 + N*0.25/N^2) = N/(N+1)
P'(B) = 1/(N+1)
Thus, P'(B) quickly -> 0 with time. In general, the rate of branching will determine probability under P'.

 

[Demystifier]

I beg to differ. Let me give you an example:
Let the universe at any time have N+1 branches, where N is an unbounded, increasing function of time. Let one branch, called A, have a P-probability of 0.5. Furthermore, let each of the remaining branches (the union of which we call B) each have a P-probability of 0.5/N. Then P(B) = 0.5. Let's calculate:
P'(A) = 0.25/(0.25 + N*0.25/N^2) = N/(N+1)
P'(B) = 1/(N+1)
Thus, P'(B) quickly -> 0 with time. In general, the rate of branching will determine probability under P'.

I see. Now I see why axiom i) is formulated by referring to the limit t->inf. But what if I don't consider the limit t->inf? For example, if I only want to know what is the probability NOW, why is it relevant how the system will evolve LATER?

 

[Demystifier]

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.

I'm afraid I don't understand it. In my understanding of practical utility of quantum mechanics, each branch at a given time corresponds to one and only one possible measurement outcome at this time. Therefore, at each time it is enough to assign a probability to each branch at that time. How do you comment on this?

 

[toho]

I see. Now I see why axiom i) is formulated by referring to the limit t->inf. But what if I don't consider the limit t->inf? For example, if I only want to know what is the probability NOW, why is it relevant how the system will evolve LATER?

That's a good point. I want a theory to make sense not just now, but for any future time. One way to assure this is to include the infinity limit as well.

I am not sure that the limit t->inf is strictly necessary, though. As long as the Hilbert space is infinitely-dimensional, you can in principle construct infinite countable sequences by splitting up sub-spaces into ever smaller sub-spaces, and computing probabilities for projections onto such sub-spaces. Then you could construct a similar argument with regard to a non-Born probability measure without any reference to the t->inf limit (I think).

 

[Demystifier]

That's a good point. I want a theory to make sense not just now, but for any future time. One way to assure this is to include the infinity limit as well.

I would accept that argument if probability were a fundamental part of the theory. But if probability is EMERGENT (as MWI usually claims), then it is emergent at the time I am doing the experiment. In this case the probabilistic interpretation is not a part of the fundamental self-consistent theory, but is a rule of thumb applicable at the time the experiment is done.