QM Interpretations

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I think that in the MWI, the Born rule can be derived from the weaker assumption that measuring an observable of a system that is in an eigenstate will yield the corresponding eigenvalue with certainty.
 

Fredrik

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I think that in the MWI, the Born rule can be derived from the weaker assumption that measuring an observable of a system that is in an eigenstate will yield the corresponding eigenvalue with certainty.
That's what Wikipedia claims (here), and their reference for that is this 1968 article by James Hartle. I checked it out some time ago and he's clearly also assuming that the Hilbert space of a physical system is the tensor product of the Hilbert spaces of its subsystems. That's a very strong assumption. I don't have all the details figured out, but it seems to me that this assumption is essentially equivalent to assuming that the Born rule holds. The weak assumption that you mentioned is probably just the piece that needs to be added to make them completely equivalent.
 

Fredrik

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...an internal sentient observer would be perfectly able to describe his own history and experience using, say, Copenhagen and wavefunction collapse, and treat everything about splitting of himself as Occam razor violating nonsense.
I still don't think that QM is anything more than a set of rules that tells us how to calculate probabilities of possibilities, but I think that using Occam's razor as an argument against the MWI makes about as much sense as using it against special relativity because it includes more than one inertial frame. If (the Dirac-von Neumann version of) quantum mechanics actually describes reality (which is hard to dismiss based only on Occam, considering that no other theory does a better job), this reality clearly must include many worlds. Even if some other version of QM is an accurate description of the world, then why would we consider it "simpler"? I don't think e.g. Bohm or a realist intepretation of the path integrals formulation is any simpler. The worlds are only "points of view" in the linear and deterministic evolution of a single point in a vector space, and I think the people who try to use Occam against the MWI have completely failed to understand this point.

I wonder if it's possible to set up an experiment that creates a superposition of a human being in different states.
I think every human is always in a superposition in most decompositions into subsystems, but the only decomposition that mattters to that human is the one that describes the universe as consisting of his memory and everything else, and in that decomposition, his memory states keep developing correlations with eigenstates of whatever he observes. Conscious experience is the development of such correlations. An important detail here is that the correlations form so quickly that the human won't ever notice that he failed to experience the time of decoherence.

This all depends on what you mean by "world". There's a definite branching of sentient beings and of classical worlds. There isn't any branching of the whole state of the universe in the big Hilbert space - that one simply evolves according to Schrodinger's equation.
Good point. I agree.
 
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What's the advantage of the MWI?
(What's the aim of any interpretation in general?)

To me it seems MWI makes an even more abstract mess than what we had before with "conventional" thinking.
 

Fredrik

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What's the advantage of the MWI?

(What's the aim of any interpretation in general?)
It's a way to interpret QM as a description of what actually happens, instead of as nothing more than a set of rules that tells us how to calculate probabilities of possibilities. That's exactly what interpretations are about.

To me it seems MWI makes an even more abstract mess than what we had before with "conventional" thinking.
What do you consider conventional?
 
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Hmm, then I personally prefer the set of rules without the complicating extension like MWI around it.

Conventional I consider the Copenhagen interpretation I suppose.

Actually I do favour interpreting the set of rules of QM as to make the picture either more intuitive or so that one can grasp QM effects better in the mind or make the picture more well defined so that one has never doubt about what the result of a question might be. I cannot see MWI achieving either of these two. I also believe a good interpretation will give as a more correct and complete version of QM.

Admittedly I don't understand MWI fully, but that also shows that it doesn't simplify so much?

What is your opinion? Why do we need an interpretation and what should it achieve?
(I repeat that my opinion is that it should make QM either easier or extend it)
 
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Actually I do favour interpreting the set of rules of QM as to make the picture either more intuitive or so that one can grasp QM effects better in the mind or make the picture more well defined so that one has never doubt about what the result of a question might be. I cannot see MWI achieving either of these two. I also believe a good interpretation will give as a more correct and complete version of QM.
Hi Gerenuk,

Hmmm... you seem to be describing de Broglie-Bohm..

I found the following on-line lecture course helpful: http://www.tcm.phy.cam.ac.uk/~mdt26/pilot_waves.html
 
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Thanks a lot for the link. These kind of interpretation are actually really my favourite, but I also haven't studied them yet. So I've collected information and books and will study them soon.
My first impression was that the pilot wave is still a bit awkward.

Are there any less-known interpretations that are similar to that?
 
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Thanks a lot for the link. These kind of interpretation are actually really my favourite, but I also haven't studied them yet. So I've collected information and books and will study them soon.
My first impression was that the pilot wave is still a bit awkward.

Are there any less-known interpretations that are similar to that?
No there aren't - as far as I know.

Your impression that the de Broglie-Bohm pilot wave approach is awkward, I would respectfully suggest, is not true.

Tell you what. You don't have to go through the full Cambridge de Broglie-Bohm course. I note that the guy recently added a popular lecture to the bottom of his slides page. Why don't you read that (should take half an hour) then, if you still think it's awkward, we can talk.. (the many worlds guys have had their four pages).
 

Fredrik

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Why do we need an interpretation and what should it achieve?
We don't need one, but it would be nice to have one, if it really does describe what actually happens.

Hmm, then I personally prefer the set of rules without the complicating extension like MWI around it.
I do too.

I also believe a good interpretation will give as a more correct and complete version of QM.
A correct interpretation would certainly do that, but I doubt that there is such a thing.

Conventional I consider the Copenhagen interpretation I suppose.
I don't think there's a universally accepted definition of the Copenhagen interpretation, but most people would say that it asserts that the laws of QM do not apply to measuring devices(!) even though it applies to the components they're made of(!!), and that measurements "collapse" wave functions into eigenstates. The problem with this is that it's complete rubbish that no one has ever believed is true. The first assumption introduces an obvious inconsistency into the theory, and the second implies that we have not one, but two rules that specify how systems change with time. That makes another inconsistency possible.

I would say that the MWI must be defined by the assumption that QM actually describes something, and that all that stuff about "worlds" follows logically from that assumption. (Those logical arguments are quite complicated, as you can see in my posts above). We may not like it, but it's certainly better than the version of the Copenhagen interpretation that I described above.
 

Hurkyl

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Hmm, then I personally prefer the set of rules without the complicating extension like MWI around it.
The main epistemological point of MWI is that it is essentially the only interpretation that is not an extension.

The extra "complication" is mainly because it looks different. Also, it's partly because it does not use any extensions -- like collapse -- that could be used to simplify things.
 

Hurkyl

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For Fredrik:

There may be a higher-level difference in how we picture QM.

I prefer something more like the C*-algebra picture. The main thing is the algebra of observables. Quantum states are functions that map observables to complex numbers that satisfy certain properties. (we might call the value of such a function the "expected value" of the observable on the state)

For any particular state, we can apply the GNS construction to create a Hilbert space in which our state is represented by a ket. The Born rule simply comes from the definition of what it means for a ket to represent a quantum state -- i.e. that [itex]\rho(O) = \langle \rho | O | \rho \rangle[/itex].

While the ket picture is useful for some calculations, it obscures what's happening when we want to restrict to subsystems or whatever.
 
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That's what Wikipedia claims (here), and their reference for that is this 1968 article by James Hartle. I checked it out some time ago and he's clearly also assuming that the Hilbert space of a physical system is the tensor product of the Hilbert spaces of its subsystems. That's a very strong assumption. I don't have all the details figured out, but it seems to me that this assumption is essentially equivalent to assuming that the Born rule holds. The weak assumption that you mentioned is probably just the piece that needs to be added to make them completely equivalent.
Perhaps I'm missing your point, but how could the Hilbert space of a physical system NOT be the tensor product of its subsystems? That seems axiomatic to me.

using Occam's razor as an argument against the MWI makes about as much sense as using it against special relativity because it includes more than one inertial frame. If (the Dirac-von Neumann version of) quantum mechanics actually describes reality (which is hard to dismiss based only on Occam, considering that no other theory does a better job), this reality clearly must include many worlds.
The observer is never able to experience a splitting of himself, because he's always in a state of definite memory. In places where MWI says that the observer is split, the observer instead observes wavefunction collapse. So, from the point of view of the observer, those "other" states of him are unobservable and do not exist. From his point of view, the Occam-minimal, positivist interpretation is Copenhagen and not MWI. Even if MWI is the proper description of the totality of the universe.

It would be more interesting to design an experiment that "proves" to an observer that he did, in fact, split, but, short of quantum suicide, nothing good comes to mind.
 

Fredrik

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Perhaps I'm missing your point, but how could the Hilbert space of a physical system NOT be the tensor product of its subsystems? That seems axiomatic to me.
There's more than one way to use two Hilbert spaces to construct a third. We use the tensor product because we want to make sure that the probability of obtaining two specific results in two independent measurements on two non-interacting systems is the product of the two probabilities assigned by the Born rule. See this post for a few more details about this, and this one for more about the tensor product in general.
 

Fredrik

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I prefer something more like the C*-algebra picture.
I think I will too when I have learned it. I have bought the books already. 1, 2. I just need to get through them. It looks like it will take a long time. I'm going to finish another book (3) before I get deep into these two.

Nothing in your summary was new to me, but it covers most of what I know already, which is just the "big picture" and none of the details. One of the things I feel that I do understand is that there isn't a huge difference between the C*-algebra formulation and the Dirac-von Neumann (Hilbert space) formulation. It avoids superselection rules, but those aren't relevant here since we can consider a quantum theory that doesn't have any. It may be a better starting point for derivations of rigorous theorems, but that doesn't seem to be very important here either. It's prettier, but...you get the idea.

While the ket picture is useful for some calculations, it obscures what's happening when we want to restrict to subsystems or whatever.
OK, that's a statement I haven't heard before. How does the C*-algebra formulation deal with subsystems, and how is it relevant? Does it imply that something I said is wrong?

By the way, I'm quite fascinated by the fact that there are so many different approaches that lead to essentially the same thing. The Dirac-von Neumann approach defines a mathematical structure (a complex separable Hilbert space) to represent the states of a physical system. The C*-algebra approach defines a mathematical structure (a non-abelian C*-algebra) to represent the observables, and the quantum logic approach defines a mathematical structure (a something something orthomodular lattice that something something :smile:) to represent experimentally verifiable statements. OK, I know even less about quantum logic than about C*-algebras, but I've bought a book about that too. 4. If I can get through all of these by the end of 2010, I'll be quite pleased with myself.
 
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There's more than one way to use two Hilbert spaces to construct a third. We use the tensor product because we want to make sure that the probability of obtaining two specific results in two independent measurements on two non-interacting systems is the product of the two probabilities assigned by the Born rule. See this post for a few more details about this, and this one for more about the tensor product in general.

In order for the tensor product construction to work, all we need is for the two Hilbert spaces to be orthogonal, which is automatically true in all interpretations of QM as long as two systems are non-overlapping.
 

Fredrik

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In order for the tensor product construction to work, all we need is for the two Hilbert spaces to be orthogonal, which is automatically true in all interpretations of QM as long as two systems are non-overlapping.
The subsystems aren't represented by orthogonal subspaces. For example, if you take the tensor product of a 2-dimensional and a 3-dimensional Hilbert space, the result is 6-dimensional, not 5-dimensional. The choice to use the tensor product is definitely non-trivial.

(I'm going to bed now, so I won't be writing any more replies for at least 8 hours).
 
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Returning to the Born rule… I am keeping to have some kind of internal dialog with myself, and cant escape this trap. May be someone can help me. As a reminder, I like MWI, but the Born rule… personally, I think for MWI it must be interpreted differently. So:

“I am driving to work. But there is a branch where (because my brain malfunctioned) I killed/attacked people and ended in a jail/got killed”
“Yes, such branch exists, but the probability is very very low”
“But our sense of “being real” does not depend of “intensity” of a branch!”
“How is it possible?”
“Generate 1000 random decimal digits and read this number. Now you are on one of 10^1000 branches. Do you feel 10^1000 times less real after you did it?”
“Definitely not. Then intensity is not important. Even if we have Frequent event (90%) and Rare event (10% probability), and we make 100 tries, then all combinations are possible, like FFFFFFFFFFFF… (100 Fs), and RRRRRRRRR (100Rs which is also rare). All 2^100 branches must exist! There are 2^100 observers observing all these branches”
“Lets make that experiment. I bet we get about 85-95Fs and 5-15Rs. What is a prediction of MWI?”
“Hmmmm…. Everything is possible…”
I am blocked at this point.
 
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P.S.
If anyone claims that Born rule is proven in MWI first I need to know, how Born rule is defined, because there is NO probability MWI. It must be defined is other terms, like, total number of observers observing X divided by the total number of observers in some subbranch on a given basic...
 

Fra

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What is your opinion? Why do we need an interpretation and what should it achieve?
(I repeat that my opinion is that it should make QM either easier or extend it)
I think this is a motivated question. I posed the same in post 43, where I gave my view.

(I guess an addition would be to note that my interpretation is also somewhat related to the version of MWI called "many minds" instead of many worlds; which is basically the idea that the different worlds are simply the different views the actual observers have that populate our one universe. The problems with this appoach, are then solved by letting the popultion and thus worlds evolve - in this picture the different worlds do interact; which is why it's better seen as many minds rather as many worlds, and from where I see it, this view gives a very good stance for expansion and unification of current theory - in line with my "pet views")

Edit: To respond again in little more detail to the point with my preffered view - it apparently holds the potential of unification since INTERACTIONS can probably be inferred from the rational player analogy (where in economy the dynamics of the economical system is inferred from the assumption of each player acting rationally) where each subsystem of the universe acts originally at will, but an evolution where selection for rational actions takes place. This is my motivation for my view. It is a vision, not finished theory, but the rationality for my preference lies in that I see it as a very natural and promising stance for extending current models and solving some of the open problems.

What the point is with the regular standard MWI I don't know. I don't see it either :)

/Fredrik
 
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Fra

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How about this?

“Definitely not. Then intensity is not important. Even if we have Frequent event (90%) and Rare event (10% probability), and we make 100 tries, then all combinations are possible, like FFFFFFFFFFFF… (100 Fs), and RRRRRRRRR (100Rs which is also rare). All 2^100 branches must exist! There are 2^100 observers observing all these branches”
“Lets make that experiment. I bet we get about 85-95Fs and 5-15Rs. What is a prediction of MWI?”
“Hmmmm…. Everything is possible…”
I am blocked at this point.
Let's play with idea of the many observer view rather than many world view? (or just think of MWI, but where there is a physical basis for each world, which is an subjective view)

If we instead acknowledge that each observer, actually sees a different statistical basis, and thus has acquired different priors. No finite real inside observer have something we can call complete statistical basis, or a "fair sampling".

This explains (assuming the observer is rational) why the different observers in a given population act differently. Each observers rationally act upon his own history only.

I think it's central to ask what is the point of "making a prediction"? Clearly the rational action of one observer, depends on the expected future. "anything is possible" would be a useless constraint. But otoh, a particular observer would not infer that anything is possible, since the distinguishable state space of a given observe is truncated.

I'm not sure if this makes sense to you, but I don't see this as as problem. But then I don't have the degree of realist desire you have :)

/Fredrik
 

Fra

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Quite relevant to what I tried to convey relates to what user=Fredrik (not Fra) said about his impression of MWI

The same as the axioms for the statistical interpretation (Link), plus the additional assumptions that it makes sense to consider the Hilbert space of the universe (even though it includes yourself), and that a state vector in that Hilbert space is a representation of all the properties of a physical system (the omnium). (The statistical interpretation doesn't assume that, and it never includes the observer in the Hilbert space).
According to my way of reasoning, these two ideas does not even mix consistently.

IMHO a possible corrected version of something close to it, but still very different is to consider a kind of holographic picture where each observer encodes an image of it's own environment (the remainder of the universe). But obviously each observers has encoded a different version of the universe, and more so, only the OBSERVABLE part of the universe. This should even constrain the size of the observable universe, and relate it to the observers complexity in a kind of holographic spirit.

A kind of statistical interpretation can still be maintained, but it's of subjective nature - which is no problem per see.

And instead of a an very ambiougs and unclear infinite superposition of universes, we instead have a set of interacting VIEWS of worlds, represented by a population of observers in our one evolving universe (since the observes aren't static).

/Fredrik
 

Demystifier

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I think that in the MWI, the Born rule can be derived from the weaker assumption that measuring an observable of a system that is in an eigenstate will yield the corresponding eigenvalue with certainty.
The problem with such an additional assumption is that it destroys all the beauty of pure MWI without that assumption. This is because such an assumption raises questions that cannot be answered within MWI:
What does it mean "to measure"?
Are measurements and/or observers described by the Schrodinger equation?
Or are they something external?
In short, with this additional assumption, MWI is not much different from the Copenhagen interpretation.

On the other hand, without that assumption (or a similar one) MWI is really beautiful and elegant, but unfortunately - physically empty. It's physically empty because it cannot explain the emergence of the Born rule - the crucial part of standard QM without which QM is physically empty.

So there are only 2 possibilities:
1. Abandon MWI completely, or
2. Accept MWI but add something additional that will destroy a part of its beauty.

If you choose 2., then you can add something that will make it either vague (like the assumption above), or not vague. The price for choosing something not vague is that it will probably look somewhat ad hoc. The best known example of non-vague but ad hoc assumption that can be added to MWI in order to recover the Born rule is - the Bohmian particle trajectories.

Personally, I find the last choice most appealing because this "ad hoc" assumption does not look to me so much ad hoc at all. Unfortunately, there is no objective quantitative measure of "ad hocness", so I cannot present a proof that the Bohmian trajectories are not so much ad hoc as many think that they are. I can present arguments, but not the proof.

In short, I believe that pure MWI is correct, but not complete. A possible completion of MWI is provided by the Bohmian interpretation.
 
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Fredrik

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the Born rule… personally, I think for MWI it must be interpreted differently.
...like, total number of observers observing X divided by the total number of observers in some subbranch on a given basic...
We can and should interpret it differently, but I don't know if there's a way to interpret it the way you're suggesting. There are several different ways to derive the Born rule's assignment of probabilities from the assumption that the Hilbert space of the omnium can be decomposed into a tensor product of Hilbert spaces of subsystems (and some minor technical assumption such as the one Count Iblis mentioned). We can also prove the converse. This suggests that in the MWI, we should think of the Born rule as the assumption that these decompositions are allowed. The actual probability assignment should be thought of as a result derived from that axiom. Just don't forget that the this decomposition axiom is very non-trivial.

The decompositions are not only allowed, they're what turns the model into a theory. In the MWI, Hilbert space with the Schrödinger equation and without the Born rule, is a perfectly valid mathematical model of reality, but it's not a theory because it doesn't make any predictions about results of experiments. An experiment is an interaction between subsystems, so we can't even begin to think about predictions until we have decomposed the omnium into the appropriate subsystems. The most useful way seems to be to decompose the omnium into the tensor product of "the system" and "the environment". The observer is part of the environment.

What you said about probability is how I was thinking about the MWI before this thread. That's actually the main reason why I felt that the MWI was complete nonsense. I have never seen anyone even try to define the "branches", or to quantitatively define probability in terms similar to what you're talking about, and without that, I felt that the MWI was at best a few "loosely stated ideas about what sort of things are happening". (And that really was the most positive way I could describe it. Most of the time I wouldn't be so kind).

Edit: I don't fully understand any of the derivations of the Born rule's assignment of probabilities from the decomposition assumption + minor technicalities. It's possible that someone who does would think that this approach does explain how to think about probabilities in a way that's similar to what you describe.
 
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Demystifier

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There are several different ways to derive the Born rule's assignment of probabilities from the assumption that the Hilbert space of the omnium can be decomposed into a tensor product of Hilbert spaces of subsystems (and some minor technical assumption such as the one Count Iblis mentioned).
Such as?
(I ask for a reference where such a derivation can be seen.)

Edit: Now I have seen your edit in which you admit that you do not fully understand any of such derivations.
 

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