# Postulates of many worlds interpretation of QM

I am interested to see a clear enumeration of the postulates of this interpretation. There seem to be something fuzzy about how MWI describes what we call "reality". It might be a problem with the theory or, more probable, a misunderstanding of my part.

Here is a list I found on Google:

http://vergil.chemistry.gatech.edu/notes/quantrev/node20.html" [Broken]

Tell me if you agree with this list, and if not, what needs to be rejected/added/modified or give a link to a better one. Thanks.

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Check here: http://en.wikipedia.org/wiki/Many-worlds_interpretation#Axiomatics

"The existence of many worlds in superposition is not accomplished by introducing some new axiom to quantum mechanics, but on the contrary by removing the axiom of the probabilistic collapse of the wave pack"

That is why you cant add any.

I usually avoid interpretational questions but I am also trying to avoid work right now so here goes:

I never really understood the argument that Many worlds theory reduces the number of axioms by dropping the measurement postulate. I mean I understand that it assumes the observer becomes entangled with the system in a way that the combined system is in a superposition |observer_a>|eigenstate a> + |observer_b>|eigenstate_b> but I still feel like there needs to be some postulate that tells us how the observer experiences being in such a superposition.. I mean, so I do an experiment and I evolve into a superposition.. then why do I only experience one result? and what determines the probability of experiencing this result? I suppose the answer to these questions is the subject of the interpretation, but something must still be said (postulated) about how the probability of experiencing one or the other result is related to the superposition (i.e. amplitude of eigenstate a/b squared).
Or am I missing something?

I mean, so I do an experiment and I evolve into a superposition.. then why do I only experience one result?

You experience BOTH results
So YOU (observer_a) is asking "But why I observe eigenstate a, not eigenstate b?"
Observer_b is asking "But why I observe eigenstate b, not eigenstate a?"
The total picture is deterministic and symmetrical in terms of the result.

Regarding the probability check the same Wiki article

Hartle[35] showed that in Everett's relative-state theory, Born's probability law

The probability of an observable A to have the value a in a normalized state is the absolute square of the eigenvalue component of the state corresponding to the eigenvalue a:

no longer has to be considered an axiom or postulate. It can rather be derived from the other axioms of quantum mechanics

Thanks for the reply Dimitry, although I am not sure it addressed the point (or maybe it did).

You experience BOTH results
So YOU (observer_a) is asking "But why I observe eigenstate a, not eigenstate b?"
Observer_b is asking "But why I observe eigenstate b, not eigenstate a?"

This quote illustrates the question... So I exprience both results but at the same time only one???

I understand that from a birds view it can be interpreted that the observer experiences both results since he/she is in a superposition but since I am the observer it makes it suddenly very unclear what being in a superposition means... what can I expect to observe? It seems to become somewhat of a philosophical question. I don't see how anything about what I can expect to experience follows logically from the mathematical result of me being in a superposition without some extra postulate/interpretation. IMHO this is precisely what the "measurement postulate" is for... whether it is thought of as a collapse of the wave-function or if it is a recipe for the interpretation of being in a superposition... I still think one needs it.

EDIT: I should point out that when I say "measurement postulate" I don't refer to the collapse of the wave-function but simply the recipe for finding the probability of observing result a or b. I am pretty sure that it can be stated in a interpretation-unspecific way.

Regarding the probability check the same Wiki article

I suppose I should read the reference Hartle[35] someday when I have time. Maybe all the answers are there. Anyway, thanks for help.

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1 So I exprience both results but at the same time only one???
2 what can I expect to observe?

There are 2 YOU's. Each YOU observes only one result, not a superposition

There are 2 YOU's. Each YOU observes only one result, not a superposition

Well, I feel like this is getting nowhere, so I will drop my question. thanks anyway.

Fredrik
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This is a good article about axioms of many-worlds interpretations.

Frederick, BOTH articles were published in 1997.

Before major improvement in the understandig of the Quantum Decoherence
And before
"The many worlds interpretation has, controversially, been seen by some as offering the possibility of deriving the Born rule and the appearance of quantum probabilities from simpler assumptions. In fact, this was first attempted by Everett and DeWitt in the 1950s. In a September 2007 conference[11] David Wallace reported on what is claimed to be a proof by Deutsch and himself of the Born Rule starting from Everettian assumptions[12]. "

In fact, both articles are very pro-MWI because it appears that 1997 is the last year when there were any anti-MWI articles :)

---------------

But it appears that since Sept 2007, MWI was able to get rid of TWO, not ONE axiom: both apparent "collapse" AND Born rule can be derived from the pure QM.

Check here: http://en.wikipedia.org/wiki/Many-worlds_interpretation#Axiomatics

That is why you cant add any.

In the above link there is no clear statement of the MWI postulates. So, I will repeat my question:

Here is a list I found on Google:

http://vergil.chemistry.gatech.edu/n...ev/node20.html [Broken]

Tell me if you agree with this list, and if not, what needs to be rejected/added/modified or give a link to a better one.

I understand that you will not add anything but remove one postulate. It's OK. Can you tell me if you agree to the rest of them? Are they correctly formulated as far as MWI is concerned?

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It does not work.
At least now, I will check later

It does not work.
At least now, I will check later

Sorry, this is the good one:

http://vergil.chemistry.gatech.edu/notes/quantrev/node20.html" [Broken]

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Fredrik
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But it appears that since Sept 2007, MWI was able to get rid of TWO, not ONE axiom: both apparent "collapse" AND Born rule can be derived from the pure QM.
Other articles have pointed out that the reasoning used in Zurek's derivation of the Born rule was circular. He derived the Born rule using the technique of "tracing out" other degrees of freedom, but the justification for that technique is...the Born rule.

The "tracing out" technique is also used in derivations of the apparent collapse of the wave function. This means that you can't take the MWI axioms to be what you get by taking the Copenhagen axioms and removing the collapse axiom including the Born rule. It seems to me that what you can do is to take the Copenhagen axioms and just weaken the collapse axiom a bit. Instead of postulating that the collapse is exact, you postulate that it's approximate. But you absolutely have to keep the Born rule in the axioms of the MWI.

Now, if you do this, the existence of "many worlds" is not implied by the axioms. It's just one of several possible interpretations of the statement that a measurement entangles the eigenstates of the measured system with macroscopically distinguishable states of a system that can be described as classical to a very good approximation. A measurement of an observable B changes the density matrix $\rho=|\alpha\rangle\langle\alpha|$ according to

$$\rho\rightarrow\sum_b P_b\rho P_b$$

where $P_b$ is the projection operator onto the eigenspace of B corresponding to eigenvalue b. If the spectrum is non-degenerate, we have $P_b=|b\rangle\langle b|$.

I don't think decoherence has weakened Kent's arguments against the various MWIs in any way.

Fredrik
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I didn't notice this before, but Tegmark actually included a few comments about Kent's paper in his. (There's no link to the Tegmark paper in this thread, so I'm putting one here. His article is actually a very good). This one is interesting:

In Section II.A, the author states that “one needs to define
[...] the preferred basis [...] by an axiom.” According
to what preconceived notion is this necessary, since
decoherence can determine the preferred basis dynamically?
I don't think this is correct. The "preferred" basis is defined as the one in which the density matrix is approximately diagonal. You find this basis by "tracing out" the environment's degrees of freedom, but that's a move that's justified by the Born rule, which is an axiom that we had supposedly dropped.

I haven't been through the details of the arguments and counterarguments, so it's possible that I'm wrong. Let's just say that I'm unconvinced, and that the anti-MWI argument above seems plausible to me. (Note that it's not really an argument against many worlds. It's an argument against dropping the Born rule).

I want to make it clear that I'm not advocating an exact collapse of the wavefunction. I'll try to explain how I think about QM: We should weaken the Copenhagen axioms by assuming that the collapse (as seen from the perspective that Tegmark calls "the inside view") is only approximate. This is sufficient to guarantee that the Born rule axiom doesn't contradict the axiom that Tegmark takes as the definition of the MWI: Any isolated system evolves according to the Schrödinger equation. So we keep the Born rule as an axiom, along with an approximate collapse. The real significance of decoherence is that it tells us what a measurement is: It's an interaction that entangles the eigenstates of some observable, selected dynamically by the interactions between the system and the environmen, with macroscopically distinguishable states of a system that's approximately classical. This process (which only exists in the theory if the axioms include the Born rule) defines the preferred basis, which defines the worlds in the MWI.

All of this is perfectly consistent with the idea of many worlds, but it doesn't imply that many worlds exist. A measurement turns a pure state into a mixed state, represented by a density matrix which is almost diagonal in the preferred basis, but this density matrix can be interpreted in many different ways. One of them is as an ensemble of systems (many worlds). Another is as a single system in a specific but unknown state. I actually prefer a third option: For a theory to be scientific, it's not necessary that it describes reality. It's sufficient that it tells us how to compute the probabilities of possible results of experiments. It's actually rather naive to expect that if science has something to say about a phenomenon, it's always in the form of a model with the property that every concept in the model has a counterpart in the real world. I just think QM is the first example of a theory that's non-trivial in the sense that it isn't a description of reality. (And no, this isn't quite the same as "shut up and calculate". It's a bit more sophisticated than that ).

Frederick,

Regarding the circularity issue, it does not mean that it is wrong. As a reminder, the very first (naive) explanations of HUP (using a though experiment with a microscope) were also circular: lets assume that the particles (say, photons) have the following dependency of momentum/position. Now we are trying to measure a position and momentum of another particle (a photon). We find the same HUP. Now we assume that ALL particles have the same property...

So I agree with you, it might not be a proof but it is a good indication that the theory is self-consistent. In order to finalize a proof, one need to assume that born rule is violated and show that it leads to incosistency.

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Frederick,

I dont think that you can ever make a theory where other worlds are NOT real. This is because the Quantum Decoherence is a GRADUAL process. 2 systems exchage, say, the photons and they gradulally decohere (in each branch we observe this process as they become somehow 'synchronized' in terms of macroscopic reality - they become 'diagonal')

Based on experiments, to completely decohere you need several photons, about 5. I hope you dont want to claim that say after an exchange of 3 photons ohter branch is real, but then, after some treshold is reached it 'suddenly' becomes non real? This looks very artificial, right?

Sorry, this is the good one:

http://vergil.chemistry.gatech.edu/notes/quantrev/node20.html" [Broken]

If we dont discuss for not Born's rule (Sept 2007 issue) then most of these axioms are valid, except the word "measurement", because MWI explains what the measurement is hence it is not an 'atomic' operation and it CAN NOT be included in the axiomatic.

To be more specific, check Postulate 2. There are no 'observables' except the states of the measurement systems. So in MWI you cant simply say 'This is momentum'. instead, you need to show, that a measurement device, built to measure a what we call a momentum, after interaction with a particle will show on it's indicator the value we call a momentum and it is the same defined by the operator. What is more difficult, you need to show that ALL different apparatus designed to measure the momentum will do the same.

It is very close to the 'Physics from scrach' approach by Max Tegmark, so MWI can make his (and mine) dream come true.

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Fredrik
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Regarding the circularity issue, it does not mean that it is wrong.
It only means that there's still no proof of the claim that in QM without the Born rule as an explicit axiom, interactions between the system and the environment make the density matrix of the combined system approximately diagonal in a basis selected dynamically by the interactions.

In order to finalize a proof, one need to assume that born rule is violated and show that it leads to incosistency.
What we would need to do (What Zurek claimed to have done) is to prove that the Born rule* follows from the other axioms.

*) To be more specific: A version of the Born rule that says that the collapse is only approximate.

I dont think that you can ever make a theory where other worlds are NOT real. This is because the Quantum Decoherence is a GRADUAL process. 2 systems exchage, say, the photons and they gradulally decohere (in each branch we observe this process as they become somehow 'synchronized' in terms of macroscopic reality - they become 'diagonal')
I don't follow your argument here. If decoherence had been absolute and instantaneous instead of gradual, we would still have at least two possible interpretations: Wavefunction collapse, and many worlds. What the gradual nature of decoherence suggests is just that the collapse isn't absolute, and the universe doesn't actually split into many copies. It's more subtle than that.

Based on experiments, to completely decohere you need several photons, about 5. I hope you dont want to claim that say after an exchange of 3 photons ohter branch is real, but then, after some treshold is reached it 'suddenly' becomes non real?
I assume that what you mean by "completely" is "approximately". Otherwise it would contradict what you said about dechoherence being gradual. Of course I don't want to claim anything like what you're suggesting here. The "branches" are either never real or always real, even before the measurement.

By the way, I took a look at the 'Against "Against many worlds interpretations" ' preprint that Tegmark referenced. It looks like a bunch of incoherent nonsense. I wonder if Tegmark ever tried to understand a single sentence in that paper except for the ones that claimed (incorrectly) that Kent doesn't even understand what the MWI is. Sakaguchi didn't seem to understand what he was talking about, and this paper was never published anywhere as far as I can tell, so it's really weird that Tegmark used it as a reference.

Fredrik
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Regarding the derivation of the Born rule...This article looks interesting. I just found it, so I haven't read it yet, but I intend to.

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Fredrik
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According to Tegmark, the only axiom of many-worlds QM is that all isolated systems evolve in time according to the Schrödinger equation. But seriously, that can't possibly define a theory. I'm not referring to the fact that he didn't mention the usual stuff about how states are represnted by unit rays and so on. (He's obviously just thinking that that's so standard that it's not even worth mentioning). I'm talking about the fact that there are no axioms that tell us how to interpret the mathematics as predictions about results of experiments.

Let's say that we know that the "outside view" of the evolution of the universe (decomposed into the subsystems qubit+observer+environment) is

|q>|>|X> → |0>|>|X'>+|1>|>|X''>

Now what exactly is the justification for the claim that the first term describes an observer who's happy because she found the qubit to be in state |0>? I really don't see any. I don't even see any justification for the decomposition of the universe into subsystems. Maybe there's some theorem about Hilbert spaces that gurantees that it can be expressed as a tensor product of a bunch of subsystems, but even if that's the case, it only solves one of the smaller problems.

Kent pointed out that there's no mechanism that can select the basis that defines the "worlds". Tegmark dismisses Kent's arguments as if they were the incoherent ramblings of a madman, and references an unpublished article to support this. But that article is really bad and doesn't make much sense. That's probably why it wasn't published.

Tegmark claims that decoherence is the mechanism that dynamically selects the basis, but it seems to me (and to this guy) that decoherence relies on additional axioms. In particular, we need to be able to express the Hilbert space as a tensor product of the Hilbert spaces of subsystems, and to compute the reduced density matrix of the qubit+observer system by "tracing out" the environment's degrees of freedom. The justification for these things is the Born rule, i.e. the axiom that says that if we measure an observable B when the system is in state |u>, the probability that we'll get the result b is |<b|u>|2.

So I think I have to metaphorically join the club of people who think that there is no many-worlds theory. It seems that no one has been able to properly define Everett's MWI, and the foreword to the 1997 version of Kent's article suggests that other many-worlds theories like consistent histories have similar problems.

One more thing...The many-worlds axiom is somewhat ill-defined, since the term "isolated system" hasn't been defined in advance. It seems more natural to take corollary 1 as the axiom (the universe evolves according to the Schrödinger equation) and then define a subsystem to be "isolated" if it evolves according to the Schrödinger equation.

Tegmark claims that decoherence is the mechanism that dynamically selects the basis, but it seems to me (and to this guy) that decoherence relies on additional axioms. In particular, we need to be able to express the Hilbert space as a tensor product of the Hilbert spaces of subsystems, and to compute the reduced density matrix of the qubit+observer system by "tracing out" the environment's degrees of freedom. The justification for these things is the Born rule, i.e. the axiom that says that if we measure an observable B when the system is in state |u>, the probability that we'll get the result b is |<b|u>|2.

So I think I have to metaphorically join the club of people who think that there is no many-worlds theory.

I'm in this club too. By the way, different choices of tensor product structures give even different physics. http://arxiv.org/abs/0901.3262" [Broken]

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1 According to Tegmark, the only axiom of many-worlds QM is that all isolated systems evolve in time according to the Schrödinger equation.

2 Now what exactly is the justification for the claim that the first term describes an observer who's happy because she found the qubit to be in state |0>? I really don't see any.

3 I don't even see any justification for the decomposition of the universe into subsystems. Maybe there's some theorem about Hilbert spaces that gurantees that it can be expressed as a tensor product of a bunch of subsystems, but even if that's the case, it only solves one of the smaller problems.

1 Check better the Wiki definition. MWI does not have any new axioms, instead, it denies the existence of some. What Tegmark was saying (as I understand) is just 'You dont need any sort of collapses' - so he just expressed the denial

Again,

http://en.wikipedia.org/wiki/Many_worlds_interpretation#Axiomatics
The existence of many worlds in superposition is not accomplished by introducing some new axiom to quantum mechanics, but on the contrary by removing the axiom of the probabilistic collapse of the wave packet: All the possible consistent

2 You are happy because cat is not dead. This works that way because your brain works. For a sadist, for example, it can be vice verse, and MWI is not supposed to explain the psycology.

3 Again, I have to repeat (looks like we are going in circles)
Decomposition is a parameter for a quantum decoherence.
If you look at the Universe from birds view you dont need one at all.
If you ask "but wait, why do I see a dead cat..."
And I interrupt you immediately saying
"So admit that YOU, not MWI has just made a decomposition of a Universe into YOU as an observer, a cat and the rest of the Universe, and YOU, not MWI has just defined a preferred basis for a decoherence asking to calculate what YOU, not any other observer, would see.

Hmmm, didn't Kent himself do a lot of work on Consistent Histories?

Fredrik
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Gold Member
1 Check better the Wiki definition. MWI does not have any new axioms, instead, it denies the existence of some. What Tegmark was saying (as I understand) is just 'You dont need any sort of collapses' - so he just expressed the denial
I don't know what gave you the idea that I don't know that. I didn't say that Tegmark or anyone else has suggested additional axioms. What I'm saying is that if you remove the "collapse according to the Born rule" axiom, what you have left isn't even a theory. If you disagree, then please tell me how to make a prediction in the MWI framework. Any prediction.

The Wikipedia section you linked to has a reference to an article by Hartle that claims to derive the Born rule. I haven't read it yet, so I can't really comment, except that to say that if he did in fact prove this back in 1968, then it's pretty weird that these newer articles I've come across haven't mentioned it (at least not in a way that caught my attention).

2 You are happy because cat is not dead. This works that way because your brain works. For a sadist, for example, it can be vice verse, and MWI is not supposed to explain the psycology.
I'm definitely not talking about psychology, and I don't know why you are. I think you're missing my point. What makes you think that the Hilbert space of the universe can even be decomposed into a tensor product of Hilbert spaces that represent subsystems? There isn't even any justification for that, so we don't even know that the left-hand side of what I wrote makes sense. And even if it does, we still don't have any justification for the interpretation of the individual terms on the right as representing how you would describe the cat.

3 Again, I have to repeat (looks like we are going in circles)
Decomposition is a parameter for a quantum decoherence.
If you look at the Universe from birds view you dont need one at all.
If you ask "but wait, why do I see a dead cat..."
And I interrupt you immediately saying
"So admit that YOU, not MWI has just made a decomposition of a Universe into YOU as an observer, a cat and the rest of the Universe, and YOU, not MWI has just defined a preferred basis for a decoherence asking to calculate what YOU, not any other observer, would see.
I don't think that makes sense. If you're going to derive the Born rule from the time evolution axiom, that axiom (or a general theorem of Hilbert spaces, but definitely not you or I) must imply that a tensor product decomposition of the Hilbert space exists. If you choose to assume that such decompositions exist, you have essentially reintroduced the axiom we dropped, because the Born rule is the reason why a tensor product is the appropriate way to represent a composite system.

Fredrik
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Hmmm, didn't Kent himself do a lot of work on Consistent Histories?
Yes, I read parts of this paper some time ago. It was pretty interesting. I think I'm going to have to read the rest soon. As I recall, they did come to some disturbing conclusions about the consistent histories formalism as well, but I may have misunderstood that since I didn't read it carefully enough. I remember something about how dinosaur fossils in the ground don't imply that dinosaurs once lived on Earth. (Just wait until the creationists get hold of that one ). But maybe that was just Smolin's interpretation of this whole thing. He mentioned that he attended a talk about this article in his book.

This quote suggests that if they think consistent histories have problems, they are even more critical of Everett's MWI. I'm emphasizing the if, because I really didn't read enough to know what they think.

Briefly, our view is that although the ideas of Everett et al. have motivated interesting work, including some of the papers we shall discuss, no welldefined scientific theory has yet been described in the many-worlds literature except for Bell’s intentionally pathological Everett-de Broglie-Bohm hybrid,[15] and that most of the ideas hinted at in earlier many-worlds papers can more naturally be understood in the language of consistent histories.

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Fra, I am trying to understand your question.

So, you are not sure that a decomposition is possible. Definitely, we know how separate systems spacially, like saying, here is a table, and here is a chair, table ends here, and a chair starts there.

But we are not sure that we can separate systems in an information space. Entangled particles share the same qbits no matter how far the particles in space are, and as all particles were born sometimes... there are probably entanglement links connecting everything in our Universe like spahetti in a way we cant imagine, so decomposition based on just some location is space is not valid.

Is my understanding correct?

Fredrik
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Gold Member
Fra, I am trying to understand your question.
I knew that guy would eventually get someone to confuse the two of us by ending all his posts with "/Fredrik".

So, you are not sure that a decomposition is possible. Definitely, we know how separate systems spacially, like saying, here is a table, and here is a chair, table ends here, and a chair starts there.
Yes, we can decompose the real world into systems in an intuitive way, and that suggests that we should look for a theory in which the mathematical model of the universe can be decomposed into mathematical models of the subsystems.

But we are not sure that we can separate systems in an information space. Entangled particles share the same qbits no matter how far the particles in space are, and as all particles were born sometimes... there are probably entanglement links connecting everything in our Universe like spahetti in a way we cant imagine, so decomposition based on just some location is space is not valid.

Is my understanding correct?
Those things are not what concern me. Consider the standard ("Copenhagenish") formulation of QM, which contains the axiom that if a system is in state |u> (in Hilbert space H1) and we measure observable B, the probability of result b is

Pu(b)=|<b|u>|2.​

Now consider a second system that's isolated from the first, and in state |v> (in Hilbert space H2), as we measure observable C. The probability of result c is

Pv(c)=|<c|v>|2.​

The probability that simultaneous measurements of B and C will yield results b and c is of course

Pu(b)Pv(c)=|<b|u>|2|<c|v>|2 =|<b|u><c|v>|2 =| (<b| ¤ <c|) (|u> ¤ |v>) |2,​

where I'm using the symbol ¤ as "tensor product" (LaTeX code "\otimes").

This identity is the reason why we use the tensor product to represent the combined system. In this case, the Hilbert space of the combined system is H=H1¤H2. It's very important to realize that this is a consequence of the Born rule.

Now, if we instead start with the axiom that the Hilbert space of the combined system is H, and that its states evolve in time according to the Schrödinger equation, then how can we possibly justify writing H=H1¤H2? My answer to that question is that the assumption that we can write H=H1¤H2 is a new axiom in the theory, and it's essentially equivalent to the Born rule.

So it isn't really surprising or at all remarkable that people have been able to derive the Born rule from these two axioms. What I'm objecting to is that people (not just you, but also e.g. Tegmark and Wikipedia), are claiming that the first axiom is all we need, when they have in fact used an alternative axiom which seems to be equivalent to the one they dropped.

I think what is less controversial is that in the MWI you can derive the Born rule from the special case that measuring an observable if the system is in an eigenstate of that observable, will yield the corresponding eigenvalue with certainty.

This is weaker form of the Born rule actually is a more "realistic" rule for the MWI, because one should not invoke probablities to define a deterministic theory, so all statements about probabilities must be derived from rules that do not invoke probabilities.

Fredrik
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Gold Member
Yes, that makes a lot more sense. I'd like to read Hartle's article about this, but I'm not going to pay 21 USD for it. Who pays for this kind of stuff anyway?

Again, I have to repeat (looks like we are going in circles)
Decomposition is a parameter for a quantum decoherence.
If you look at the Universe from birds view you dont need one at all.
If you ask "but wait, why do I see a dead cat..."
And I interrupt you immediately saying
"So admit that YOU, not MWI has just made a decomposition of a Universe into YOU as an observer, a cat and the rest of the Universe, and YOU, not MWI has just defined a preferred basis for a decoherence asking to calculate what YOU, not any other observer, would see.

It's not me who makes the decomposition. The decomposition has to be real for MWI to make sense as a realistic interpretation.

In http://arxiv.org/abs/0903.4657" [Broken] i show not only that different decompositions exist, but also that they are physically different, and that it is extremely improbable that we are one random choice of these.

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