Why MWI cannot explain the Born rule

[Demystifier]

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

The argument goes:

The minimal set of assumptions defining MWI is:
1. Psi is a solution of a linear deterministic equation.
2. Psi represents an objectively real entity.

Assume that the Born rule can be derived from the assumptions above. Then ANY system satisfying these assumptions must necessarily have the probabilistic interpretation defined by the Born rule. Therefore, in order to show that the Born rule cannot be derived from the assumptions above, it is sufficient to find one counterexample of a system that satisfies these assumptions but does not have the probabilistic interpretation. And it is very easy to find such an example; just take some appropriate wave equation from CLASSICAL mechanics of fluids. Q.E.D.

Of course, this is just a rough idea for the argument. I'm sure it can be further refined, e.g. by replacing 1. above with something more specific and yet sufficiently general. Since the idea is so simple, I'm sure that many readers of this can contribute to further developments of the idea.

Any suggestions? Comments? Objections?

 

[Count Iblis]

This is also my opinion. You can also consider simulating a virtual quantum world on a classical computer. The classical computer simulates the world according to the Schrödinger equation, and then you can consider the internal virtual observers. Do they indeed have observations consistent with the Born rule? It is easy to see that the probabilities for their observations are uniformly distributed over the set of all possible experimental outcomes and therefore not given by the Born rule.

 

[Demystifier]

Count Iblis, your idea to simulate the Schrodinger equation on a classical computer is even better than mine. Excellent!

 

[Fredrik]

The minimal set of assumptions defining MWI is:
1. Psi is a solution of a linear deterministic equation.
2. Psi represents an objectively real entity.

I don't think those are sufficient to define the MWI. The "worlds" in the MWI are just correlations between states of subsystems, so we can't expect to see any worlds unless we assume that the universe has subsystems.

How would we define the "internal virtual observers" in the suggested computer simulation? Aren't they supposed to be subsystems?

I still think that assuming that the Hilbert space of the universe is the tensor product of the Hilbert spaces of the subsystems is essentially equivalent to assuming the Born rule. (And I still don't have a complete proof of that).

 

[Demystifier]

I don't think those are sufficient to define the MWI. The "worlds" in the MWI are just correlations between states of subsystems, so we can't expect to see any worlds unless we assume that the universe has subsystems.

I don't think that it is fair to define "worlds" as correlations, because a correlation is, by definition, something that has to do with statistics and probability, whereas we want to DERIVE statistics and probabilities.

Of course, you are allowed to postulate any axioms you want that will help you to incorporate the Born rule, but then the main virtue of MWI - the minimal set of axioms - is lost.

 

[Demystifier]

Fredrik, consider a CLASSICAL wave psi(x1,x2,x3,t). You can think of 3 coordinates x1, x2, x3 as 3 space dimensions x, y, z. But you can also think of these 3 coordinates as 3 particles moving in 1 dimension. The mathematics is the same. Yet, in this second case you have a natural and intuitive definition of subsystems (each particle is another subsystem). So, how the words which say that 3 coordinates represent 3 particles, without changing the mathematics, imply that the classical wave above has a probabilistic interpretation?

 

[Dmitry67]

Lets begin from the very beginning.
How you define the 'probability' in the determenistic theory?
You can not apply Born rule 'as is' in MWI

 

[Demystifier]

How you define the 'probability' in the determenistic theory?

Through ignorance of some deterministic property.

For example, in BM (which is the most classical interpretation of QM), the particle lives in one many-world branch only, but you just don't know which one. Therefore, your knowledge about the position of the particle can be described in terms of probability. Unfortunately, there is no analog of a Bohmian-particle-in-a-single-branch within pure MWI. That's why it seems reasonable to supplement MWI with an additional axiom of particles, which leads to BM. BM is nothing but MWI with one additional axiom that allows to explain the Born rule in a simple and intuitive way.

 

[kote]

Just because the probabilities always happen to be 1 doesn't mean they aren't probabilities, does it ?

I'm actually serious here. I think it's easier for us to consider that deterministic theories are realistic while probabilistic theories are not. But do we have any more evidence for the reality of deterministic theories than probabilistic theories? EPR's criterion for reality requires determinism, but do we accept determinism as the ultimate test of reality? Logically it's fallacious to draw any such conclusions about the nature of reality.

There is a tradition of associating determinism with reality, from both EPR and the classical paradigm. Deterministic theories really are just special cases of probabilistic theories though. The fact (I think - might need more research) that we can always replace a probabilistic function with a deterministic function regarding uncertain variables further deprives such differences of much meaning.

I guess I'm not sure how this applies to MWI in particular. I think you make a good point by comparing current theories vying for "reality" to similar theories that make no such claim. Can we come up with a criterion for reality that doesn't lead to contradictions when applied to disparate theories? If not, and I suspect that we can't, then what justification do we have for claiming the reality of any theory?

 

[Fredrik]

I don't think that it is fair to define "worlds" as correlations, because a correlation is, by definition, something that has to do with statistics and probability, whereas we want to DERIVE statistics and probabilities.

I think this is like starting with a manifold without a metric and trying to derive the physical significance of special relativistic inertial frames (which would be defined using the isometries of the Minkowski metric). It's impossible by definition, no argument necessary.

Of course, you are allowed to postulate any axioms you want that will help you to incorporate the Born rule, but then the main virtue of MWI - the minimal set of axioms - is lost.

I agree that it loses something significant, but I think Everett, Tegmark and others were just plain wrong when they said that the MWI can be defined this way.

So, how the words which say that 3 coordinates represent 3 particles, without changing the mathematics, imply that the classical wave above has a probabilistic interpretation?

They can't. I'll go even further and say that no mathematical model can ever imply anything about the real world. We need something more than just mathematics. A theory of physics is defined by a set of axioms that tells us how to interpret the mathematics as predictions about probabilities of possible results of experiments. The Born rule is an essential part of the axiomatic structure of QM. Without it, the theory is crippled. It's not even a theory anymore.

This model of 3 particles hasn't been endowed with axioms that turn it into a theory. It's not clear what mathematical objects represent observables, or how they can be measured.

 

[Demystifier]

I agree that it loses something significant, but I think Everett, Tegmark and others were just plain wrong when they said that the MWI can be defined this way.

So we basically agree on that.

A theory of physics is defined by a set of axioms that tells us how to interpret the mathematics as predictions about probabilities of possible results of experiments.

I guess you allow also the deterministic theories as a special case, in which all probabilities are either 1 or 0.

 

[hamster143]

Suppose you start with a wavefunction that describes a repeatable experiment (say, an asymmetric coin toss) and an observer. The observer counts the number of heads-up outcomes. There's a decoherence factor involved, which has the effect of adding a uniformly distributed random phase to all observers after each toss.

You evolve the system for 100 coin tosses, and you end up with a superposition of 101 observers who observed 0, 1, ..., 100 heads-up outcomes.

I claim that you'll find that, however you define the measure on the Hilbert space, you'll observe that the outcome distribution will conform to the Born rule, i.e. the mean number of observed heads-ups is 100*|c1|^2/(|c1|^2+|c2|^2), where c1 and c2 are complex amplitudes for heads-up and tails-up transitions in each coin toss.

 

[Demystifier]

I claim that you'll find that, however you define the measure on the Hilbert space, you'll observe that the outcome distribution will conform to the Born rule, i.e. the mean number of observed heads-ups is 100*|c1|^2/(|c1|^2+|c2|^2), where c1 and c2 are complex amplitudes for heads-up and tails-up transitions in each coin toss.

Of course you will, this is the experimental fact. The issue is to find a theoretical EXPLANATION of this fact.

 

[hamster143]

Of course you will, this is the experimental fact. The issue is to find a theoretical EXPLANATION of this fact.

This is a purely theoretical explanation. Write some code to simulate this using transitions described, and you'll get the expected outcome.

My point is, Born rule naturally arises through a series of repeated experiments in presence of random decohering influence.

 

[Dmitry67]

I was thinking too about the 'measure of existence' vs the 'number of observers'

Say, I go to work.
In 1 branch I had successfully came to office. But there are 1000000 low probability branches, where I made something stupid – robbed the gas station, killed the cop, etc etc. So there is 1 high-intensity branch versus many low intensity branch. So there are more weird observers than normal ones.

But then I had realized that reality is even more complicated. Single ‘normal’ branch is an illusion. I can not be aware of all macroscopic degrees of freedom of my body and even of my brain, because my consciousness have less degrees of freedom like computer chip can stored less information than is encoded in all matter it consists of.

So and observer is not only a branch, but a huge ensemble of branches. And you actually don’t have a SINGLE basis, but rather a sum over ensemble of basis. How it is relevant to the Born rule? Not sure. But I think that before we talk about the ‘probability’ we must check if ‘the number of distinct observers’ approach is equivalent or not to ‘the measure of existence’ approach.

 

[Dmitry67]

My point is, Born rule naturally arises through a series of repeated experiments in presence of random decohering influence.

Noturalu arises... to what observer?
Say, I flip a coin 1000000 times.
There are 2^10000000 observers:

Observer 1: 0000...000000 (1000000 times)
Observer X: 0101010101...0101011010101
Observer Y: 101011010...1010101010011
Observer (last): 1111111...111111 (1000000 times)

Why observers X and Y are more 'natural' then 1 and last?

 

[Demystifier]

My point is, Born rule naturally arises through a series of repeated experiments in presence of random decohering influence.

Is there a paper where more details can be found?

 

[hamster143]

How?
Say, I flip a coin 1000000 times.
There are 2^10000000 observers:

Observer 1: 0000...000000 (1000000 times)
Observer X: 0101010101...0101011010101
Observer Y: 101011010...1010101010011
Observer (last): 1111111...111111 (1000000 times)

Why observers X and Y are more 'natural' then 1 and last?

They are not. But there's only one observer who saw 0 heads and only one observer who saw 1000000 heads, and there's an enormous number of observers who saw 500000 heads.

Furthermore, if you assign a factor of 0.6 * exp(i*random_phase) to all heads flips and a factor of 0.8 * exp(i*random_phase) to all tails flips, and you sum up things to construct meta-observers called "0 heads", "1 heads", ..., "1000000 heads", the meta-observer with the largest factor will be very close to the value predicted by the Born rule, 1000000 * (0.6*0.6) / (0.6*0.6+0.8*0.8) = 360000 heads.

 

[hamster143]

Is there a paper where more details can be found?

There probably is, and it's probably dated 1930 or so, and therefore I have no idea how to find it. If there isn't, feel free to write one and include me as a reference.

 

[Demystifier]

There probably is, and it's probably dated 1930 or so, and therefore I have no idea how to find it. If there isn't, feel free to write one and include me as a reference.

I am both disappointed and satisfied with this answer. Disappointed for obvious reasons. Satisfied because my arguments still seem cogent to me.

 

[Dmitry67]

They are not. But there's only one observer who saw 0 heads and only one observer who saw 1000000 heads, and there's an enormous number of observers who saw 500000 heads.

Furthermore, if you assign a factor of 0.6 * exp(i*random_phase) to all heads flips and a factor of 0.8 * exp(i*random_phase) to all tails flips, and you sum up things to construct meta-observers called "0 heads", "1 heads", ..., "1000000 heads", the meta-observer with the largest factor will be very close to the value predicted by the Born rule, 1000000 * (0.6*0.6) / (0.6*0.6+0.8*0.8) = 360000 heads.

Yes. But this is circular.

As there are more observers who don't observe weird things (like all 1s or zeros) then if I impersonate randomly with some observer, I have more chances to see see the most expected value of 1s and 0s.

I like MWI, but I don't buy the explanation above.

 

[Fredrik]

I guess you allow also the deterministic theories as a special case, in which all probabilities are either 1 or 0.

Yes.

In classical physics, the possible states of a physical system are represented by the points in a set called "phase space". (Each point represents a value of position and momentum). Observables are represented by functions from the phase space to the real numbers. For example, "energy" is represented by a function f_E that takes a state s to the energy f_E(s) that the system has when it's in state s. Now consider sets of the form $f_E^{-1}(A)$, where A is a subset of the real numbers. Such a set consists of all the states in which the system has an energy that's a member of A. Because of this, each such set is considered a representation of a "property" of the system, or equivalently, an "experimentally verfiable statement". We can now define a probability measure $\mu_s$ for each state s, on the set of all such sets (constructed from all observables of course, not just energy), by $\mu_s(Z)=1$ if $s\in Z$ and $\mu_s(Z)=0$ if $s\notin Z$.

 

[Fredrik]

My point is, Born rule naturally arises through a series of repeated experiments in presence of random decohering influence.

There probably is, and it's probably dated 1930 or so, and therefore I have no idea how to find it. If there isn't, feel free to write one and include me as a reference.

Decoherence theory is much more recent than that, and it relies on the assumption that the Hilbert space of a system is the tensor product of the Hilbert spaces of its subsystems, which is usually justified by the Born rule.

 

[yossell]

I find Demystifier's original argument utterly compelling *if* his assumptions define MWI. But these assumptions are so minimal I don't see how we get MWI - I don't see how we get any physics at all. Assumption (1) has purely mathematical content - no physical content at all that I can see. (2) is extremely weak and still has arguably no physical content. A mathematical platonist could agree with (1) and (2) - but he's no many worlds theorist.

Surely, at some level, some view about of realism about Many Worlds must appear in the MWI, something that links the maths up with their world view, something about how the different terms that appear in the superposition correspond to different branches. Otherwise, well, it's just not MWI - it's not even physics. As you've stated the argument, it looks as though you're asking the MWI to get the Born rule out of pure mathematics - that obviously isn't going to happen.

 

[helenk]

You might want to read;

http://bjps.oxfordjournals.org/cgi/content/abstract/57/4/655"

Abstract

I consider exactly what is involved in a solution to the probability problem of the Everett interpretation, in the light of recent work on applying considerations from decision theory to that problem. I suggest an overall framework for understanding probability in a physical theory, and conclude that this framework, when applied to the Everett interpretation, yields the result that that interpretation satisfactorily solves the measurement problem.

Introduction

What is probability?
2.1 Objective probability and the Principal Principle
2.2 Three ways of satisfying the functional definition
2.3 Cautious functionalism
2.4 Is the functional definition complete?

The Everett interpretation and subjective uncertainty
3.1 Interpreting quantum mechanics
3.2 The need for subjective uncertainty
3.3 Saunders' argument for subjective uncertainty
3.4 Objections to Saunders' argument
3.5 Subjective uncertainty again: arguments from interpretative charity
3.6 Quantum weights and the functional definition of probability

Rejecting subjective uncertainty
4.1 The fission program
4.2 Against the fission program

Justifying the axioms of decision theory
5.1 The primitive status of the decision-theoretic axioms
5.2 Holistic scepticism
5.3 The role of an explanation of decision theory

Conclusion

and/or

http://bjps.oxfordjournals.org/cgi/pdf_extract/47/2/233?ssource=mfc&rss=1"


:)

 

[Dmitry67]

Whats wrong with mathematical platonism?
MWI is an ideal ingidient for MUH (or vice versa?)

 

[yossell]

Whats wrong with mathematical platonism?
MWI is an ideal ingidient for MUH (or vice versa?)

Nothing's wrong with it - that wasn't the point of the post. The point was that a mathematical platonist, who had no interest in physics, would happily sign up to 1 and 2 - 1 and 2 cannot define MWI which is a substantive view about the physical world for they are compatible with a completely minimal view about the physical world

edit: like to second helenk's comment - Wallace is building on some ideas of Deutch - the rough idea is that the weights that quantum theory assigns to branches deserves to be understood as *probability* because they track exactly how a rational agent ought to bet in quantum situations. It's the link with an agent's rational betting behaviour that's does the important work, rather than an attempt to derive probabilities from frequencies. It's very interesting

 

[Fredrik]

Is it really true that the MWI is the minimal realistic interpretation, and that it takes an additional axiom to get rid of the many worlds?

I've been saying this myself, after some time of saying the opposite. Now I'm back to being confused. I define the MWI as the assumption that QM as defined by the Dirac-von Neumann axioms (which include the Born rule) can be taken as a description of what actually happens. It's supposed to be the assumption of realism, and nothing else.

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:

What the formalism says is that a measurement of an observable A changes a pure state $\rho=|\psi\rangle\langle\psi|$ into a mixed state:

$$\rho\rightarrow \sum_i P_i\rho P_i$$

where the $P_i=|a_i\rangle\langle a_i|$ are the projection operators of the one-dimensional subspaces that represent the possible states after the measurement. If we reject the idea that this describes what actually happens, then we can certainly say that only one of the terms represent the state of the system after the measurement. But if we're going to claim that the above is a description of what actually happens, we're going to have to deal with the fact that there's nothing that even suggests that one of the terms is more real than the others.

This argument is only valid if the Born rule plus realism is equivalent to the time evolution described above. But is it really? The form of the Born rule that I'm the most used to seeing is the one that appears in the introduction of this article. It doesn't even mention mixed states. So it seems that at the very least we need to start with a formulation of QM that uses state operators instead of unit rays as a mathematical representation of "states" (operationally defined equivalence classes of preparation procedures).

Suppose that we modify the Dirac-von Neumann axioms to use state operators instead of unit rays, and add the assumption of realism. Does my argument work now? Unfortunatly I don't have time to think of an answer right now, because I have to go and do something else for a while. (I suspect that the argument doesn't work unless the axioms are very specific about how to interpret a state operator).

By the way, the article I referenced contains a nice statement about the sort of things discussed in the OP:

The necessity for an assumption like (3) in the derivation
of the Born rule can be traced back to a fundamental
statement about any probabilistic theory: We cannot derive
probabilities from a theory that does not already contain
some probabilistic concept; at some stage, we need
to “put probabilities into get probabilities out”.

 

[yossell]

I define the MWI as the assumption that QM as defined by the Dirac-von Neumann axioms (which include the Born rule) can be taken as a description of what actually happens. It's supposed to be the assumption of realism, and nothing else.

Not a point of substance - but I think this claim is misleading. At least in all my experience (which, I know, is limited) the von Neumann axioms include the collapse postulate - that after measurement, the state has collapsed to an eigenspace. MWI is notorious for trying to do away with this postulate. Von Neumann did not and is not known for giving a probability interpretation to the wavefunction. That was Born.

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:

I was just in the process of trying to compose a response to this very move. My worries about it are not obviously the same as yours though - but I'd be interested in your opinion.

Your argument seems to be that the assumption that only one possibility is realized is in conflict with the fact that we end up with a particular matrix that represents the various possible states after measurement, as if the view that only one possibility is realized is in some sense in tension with the correctness of this matrix.

But the issue is what this matrix represents. If it just represents the various *possible* results, with the chances that each obtains, then I see no conflict with the theory, no rejection of anything, to believe that only one of these *possibilities* is realized. Indeed, that's exactly how we normally think of possibilities. Indeed, there are those who don't see how, if all possibilities are realized it even makes sense to talk of different probabilities being attached to them. There's nothing in the formalism which says that everything which is possible is also actual.

It's true that QM doesn't tell us which state is realized - it only gives us the probabilities of various states being realized - but to think that one and only one of the states is realized is not to conflict with QM - it's just a natural way of understanding what these arrays of numbers represent - the chances of one of the possibilities being realized.

The Many-worlds vs the single world view are interpretational issues - depending upon how we interpret the matrices that QM produces. But insofar as the formalism itself is concerned, it's not a matter of contradicting the formalism to say that one possibility is realized. The production of the matrix itself, along with the minimal interpretation that these are possible results of measurement, is at best silent on the issue - and so I don't think it can be used to argue that MWI is minimal in this respect.

 

[Fredrik]

Not a point of substance - but I think this claim is misleading. At least in all my experience (which, I know, is limited) the von Neumann axioms include the collapse postulate - that after measurement, the state has collapsed to an eigenspace. MWI is notorious for trying to do away with this postulate. Von Neumann did not and is not known for giving a probability interpretation to the wavefunction. That was Born.

If we use the version of the Born rule that I linked to, we have to add the axiom that if we measure the same thing again (immediately), we'll get the same result with probability 1. From the observer's point of view, this is indistinguishable from a collapse, but note that this form of the axioms allows for the possibility that this "collapse" is subjective rather than objective. The term "collapse" is usually only used when we have an objective physical process in mind, so I would say that these axioms do not include a collapse.

I got the term "Dirac-von Neumann axioms" from "Introduction to the mathematical structure of quantum mechanics", by Franco Strocchi...and hey, another look at it (link) reveals that they do start with state operators and not unit rays. I remembered that part wrong.

Your argument seems to be that the assumption that only one possibility is realized is in conflict with the fact that we end up with a particular matrix that represents the various possible states after measurement, as if the view that only one possibility is realized is in some sense in tension with the correctness of this matrix.

I'm just saying that there's nothing in the right-hand side that gives us a reason to interpret one of the terms in a different way than the others.

But the issue is what this matrix represents.

Agreed, that's what my previous post was about.

There's nothing in the formalism which says that everything which is possible is also actual.

Is there something that says the opposite? The formalism describes all possibilities exactly the same, and we know from experience that at least one of them is realized.

...to think that one and only one of the states is realized is not to conflict with QM - it's just a natural way of understanding what these arrays of numbers represent - the chances of one of the possibilities being realized.

I have never said it's in conflict with QM. I've been saying that it's in conflict with realism, the idea that QM literally tells us what actually happens. (For the record, I don't think QM tells us what actually happens).

The Many-worlds vs the single world view are interpretational issues - depending upon how we interpret the matrices that QM produces. But insofar as the formalism itself is concerned, it's not a matter of contradicting the formalism to say that one possibility is realized. The production of the matrix itself, along with the minimal interpretation that these are possible results of measurement, is at best silent on the issue - and so I don't think it can be used to argue that MWI is minimal in this respect.

I still haven't really thought this through, so I don't have a good answer yet. Apparently you don't either, since the above (the last quote) is a set of statements rather than a set of arguments. We need to analyze the axioms a bit deeper. You also seem to be consistently overlooking the fact that all the terms are described the same way.

 

[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.